Quasiconformal mappings and their applications. IWQCMA05

International Workshop on QUASICONFORMAL MAPPINGS AND THEIR APPLICATIONS December 27, 2005 - January 01, 2006 Indian ...

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International Workshop on QUASICONFORMAL MAPPINGS AND THEIR APPLICATIONS

December 27, 2005 - January 01, 2006

Indian Institute of Technology Madras Department of Mathematics

Proceedings of the International Workshop on Quasiconformal Mappings And Their Applications (IWQCMA05) December 27, 2005 - January 01, 2006

Edited by S. Ponnusamy T. Sugawa M. Vuorinen

Co-organized by • Chennai Mathematical Institute, Chennai, India • Institute of Mathematical Sciences, Chennai, India

Sponsored mainly by • National Board for Higher Mathematics (DAE), India • Forum d’Analystes, Chennai, India • National Science Foundation, USA • The Abdus Salam International Center for Theoretical Physics, Italy • Commission on Development and Exchanges of the International Mathematical Union • Indian National Science Academy, India • Council of Scientific and Industrial Research, India • Department of Science and Technology, India

Preface The Department of Mathematics, IIT Madras, Chennai hosted International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05) December 27, 2005- January 1, 2006. This event was the first one in India on this active research area which has its roots in geometric function theory and which is closely connected with several topics of mathematical analysis. The organizers gratefully acknowledge the financial support of 1. 2. 3. 4. 5. 6. 7. 8.

National Board for Higher Mathematics (DAE), India National Science Foundation, USA The Abdus Salam International Center for Theoretical Physics, Italy Commission on Development and Exchanges of the International Mathematical Union, Italy Indian National Science Academy, India Council of Scientific and Industrial Research, India Department of Science and Technology, India Forum d’Analystes, Chennai, India.

We also thank 1. Theivanai Ammal College for Women, Viluppuram, India 2. Canara Bank, IIT Madras Campus 3. State Bank of India, IIT Madras Campus for their support. Many people have given us help, in particular the students of Prof. S.Ponnusamy. Prior to the start of the workshop, preworkshop lectures were given by Dr. Antti Rasila and Prof. Raimo Näkki from Finland. We thank both of them. Also, we take this opportunity to thank Prof. R. Balasubramanian, Prof. R. Parvatham, and Prof. C. S. Seshadri for their continued encouragement and helpful advice. The participants, who represented many different countries, received in most cases financial support from their national funding organizations to cover their expenses. ICTP’s generous support was useful in supporting mathematicians from developing countries. Without the invaluable support from the aforementioned organizations, this conference would not have been possible in its present form. The main goal of the conference was to bring together internationally wellknown experts representing geometric function theory and some related topics. They were requested to deliver a series of lectures for postgraduate students on their respective areas. The audience consisted of mathematicians ranging from graduate students to well-known experts from all the participating countries. Conformal invariance and conformally invariant metrics have been important research topics in geometric function theory during the past century. These topics also were discussed or mentioned in several of the lectures. The organizing committee was pleased to observe that the lectures were very well received and

lead to many lively discussions afterwards. We were also pleased to receive positive response from the speakers to our request to contribute their lectures for the proceedings. It is our hope that the publication of these proceedings will make the results presented in this Workshop and also this research area and its challenging open problems more widely known for a wide readership than what is the case presently. The editorial work was carried out at IIT Madras, and the www-pages http://mat.iitm.ac.in/ samy/ http://www.cajpn.org/madras/ http://www.math.utu.fi/proceedings/madras contain a copy of these proceedings. Special thanks to Mr. Swadesh Kumar Sahoo, and Mrs. P. Vasundhra for their help in the organization of the meeting. We take this opportunity to thank Prof. Roger W. Barnard for his support in getting NSF grant for supporting US participants, and for its partial support in bringing out this proceedings. On behalf of the Organizing committee S. Ponnusamy IIT Madras T. Sugawa Hiroshima University M. Vuorinen University of Turku

Preface

Contents

Roger W. Barnard, Clint Richardson, Alex Yu. Solynin A note on a minimum area problem for non-vanishing functions

1

Alan F. Beardon and David Minda The hyperbolic metric and geometric function theory

9

Peter H¨ ast¨ o Isometries of relative metrics

57

David A Herron Uniform spaces and Gromov hyperbolicity

79

Ilkka Holopainen, and Pekka Pankka p-Laplace operator, quasiregular mappings, and Picard-type theorems

117

Henri Lind´ en Hyperbolic-type metrics

151

Williams Ma and David Minda Geometric properties of hyperbolic geodesics

165

Olli Martio Quasiminimizers and potential theory

189

R. Michael Porter History and recent developments in techniques for numerical conformal mapping

207

Antti Rasila Introduction to quasiconformal mappings in n-space

239

Toshiyuki Sugawa The universal Teichm¨ uller space and related topics

261

Matti Vuorinen Metrics and quasiregular mappings

291

G. Brock Williams Circle packing, quasiconformal mappings, and applications

327

List of Participants

349

Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05)

A Note on a Minimum Area Problem for Non-Vanishing Functions Roger W. Barnard, Clint Richardson, Alex Yu. Solynin Abstract. We find the minimal area covered by the image of the unit disk for nonvanishing univalent functions normalized by the conditions f (0) = 1, f ′ (0) = α. We discuss two different approaches, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where we can employ the well known integral representation to obtain the solution upon prior knowledge about the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an explicit solution. Keywords. Symmetrization, Minimal Area Problem. 2000 MSC. 30C70.

Contents 1. Introduction

1

2. Outline of Our Method

4

3. The Iceberg Problem

6

References

8

1. Introduction Z p p Let D = {z : |z| < 1} and A = f analytic in D : |f (z)| dA = ||f ||Ap < ∞ , p

the Bergman space of analytic functions in D.

D

Recently, Aharanov, Beneteau, Khavinson, and Shapiro [2] considered a general minimization problem on Ap inf{||f ||Ap : f ∈ Ap , ℓi (f ) = ci , i = 1, . . . , n}

where ℓi are bounded linear functionals on Ap , p > 1. They proved several general results about this problem. Version October 19, 2006. Supported by NSF grant DMS-0412908.

2

Roger W. Barnard, Clint Richardson, Alex Yu. Solynin

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As we know, in recent years tremendous progress has been achieved in the study of Bergman spaces. For a detailed account of this progress, we refer to the recent monograph by Peter Duren and Alex Schuster, Bergman Spaces, [7]. Aharanov, Beneteau, Khavinson, and Shapiro [2] also mentioned that to obtain a complete solution of a particular problem, one often needs additional information which does not follow from their methods. A particular example is the following open problem: inf

Z

D

′

2

|f | dA : f 6= 0 in D, f (0) = 1, f (0) = α

This is a “typical” extremal problem on the class of non-vanishing analytic functions. The nonlinearity of the class is the obvious obstacle here. But, we have a method which allows us to solve some problems similar to this one. Let Nα = {f : f is univalent, and non-vanishing on D, f (z) = 1 + a1 (f )z + . . . ,

normalized by a1 (f ) = α } The area of the image f (D) is given by Z ∞ X ′ 2 n|an (f )|2 . D(f ) = |f | dA = π D

n=1

Thus

D(f ) ≥ πα2 ,

with equality iff f (z) = 1 + αz. Since this map f is in Nα , 0 < α ≤ 1, Koebe’s 1/4 Theorem implies Nα = ∅ for α > 4. So the nontrivial range is 1 < α < 4 . For the non-trivial range, the minimal area problem for Nα is solved by Theorem 1.1. For 1 < α < 4, let f ∈ Nα . Then 2 √ √ √ (1.1) αa2 − 2 a2 − 1 a + a2 − 1 D(f ) ≥παa2 a + a2 − 1

where a = a(α) is the solution to " # √ √ √ 1 = a2 1 − a2 − a(a + a2 − a)3 log (a + a2 − 1)4 /16a2 (a2 − 1) , (1.2) α which is unique in the interval 1 < α < ∞. Equality in (1.1) holds iff f = fα defined by p Z z −β ξ 2 − a2 dz fα (z) = 2 p p √ z −1 ξ + ξ2 + 1 a ξ 2 − 1 + ξ a2 − 1

A Note on a Minimum Area Problem for Non-Vanishing Functions

3

100

πα

80

2

60

AHΑL 40

Α(α)

20

1

2

3

4

Α Figure 1. The graph of A(α).

Α=3

4

4

-4

8

-4

Figure 2. The extremal domain Dα = fα (D) for α = 3. with ξ =

ia 1−z √ 2 z

and β = αa2 a +

For 0 < α < 4, let

√

a2 + 1 .

A(α) = min D(f ) f ∈Nα

denote the minimal area covered by the images of functions in the class Nα . Note A(α) is convex and increasing. This can be proven from the formulas, geometry, and variational arguments. See Figure 1.

4


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2. Outline of Our Method First consider the minimal area problem on Tα , the typically real nonvanishing functions (not necessarily univalent). Use the linear structure of Tα and reformulate to show uniqueness and get simple “sufficient conditions” for extremality corresponding to linearized functions. This gives Theorem 2.1. For 1 < α < 4, let f ∈ Tα . Then (1.1) holds with the same cases of equality. The technique of this proof was developed earlier in [1]. What is missing is how to construct the extremal function! Next. Assuming sufficient smoothness, we can apply a variant of Julia’s Variational Formula in [5]. This leads to boundary conditions for an extremal analytic function. To obtain this “conditional” solution requires a priori smoothness. Next, to achieve the “required smoothness,” we exploit geometric control of the mapping radius and apply standard symmetrization techniques to obtain the sufficient initial Jordan rectifiability as in [4]. Then we can apply earlier smoothing variations developed by Barnard and Solynin in [5] giving “required smoothness.” Thus the “conditional” proof becomes a true proof. We then verify that the function recovered from the first step satisfies the sufficient conditions of extremality which also leads then to a complete solution of the problem. For a first step on Tα , we renormalize so that f (0) = 1, f ′ (0) = α. Subordination implies 0 < α ≤ 4. Since Tα is compact and convex, the minimizer exists and is unique. The uniqueness follows by letting f1 and f2 be two minimizers. Then Z 1 (2.1) |f1′ + f2′ |2 dσ D((f1 + f2 )/2) = 4 D Z Z 1 ′ 2 ′ 2 |f1 | dσ + |f2 | dσ ≤ 2 D D 1 = (D(f1 ) + D(f2 )) , 2 with equality iff f1′ ≡ f2′ . We note here that the uniqueness obtained here is fortunate, since uniqueness is in general not obtained when variational and approximation methods are used. Reformulating the problem using the linearity of Tα , we use the following lemma from [1, 3] Lemma 2.2. For fα′ continuous on D, fα minimizes D(f ) on Tα iff fα minimizes Z L(f ) = ℜ fα′ (z)f ′ (z) dσ D

on Tα .


5

Proof. See Lemma 1 of [3]. Lemma 2.3. If fα′ is continuous on D, then Z π L(f ) = Kα (t)dµf (t), 0

where

Kα (t) = Proof. See [3].

2πα it ′ it ℑ e f (e ) . sin t

Proof of Theorem 2.1 for Tα . For Dα = fα (D), first show (0, 1] ⊂ Dα by considering 1 fα (τ z) fe(z) = 1 − + τ τ e for τ < 1 and compare D(f ) with D(fα ). Then fα (−1) = 0 since fα is not identically 0 and fα ∈ Tα . Thus with Lemmas 2.2 and 2.3, fα minimizes D(f ) on Tα iff fα minimizes L(f ) under the constraints Z π 2 dµf = 1 0 Z π 2 t 2 dµf = sec 2 α 0

Now we can use well known results to show fα is extremal iff Kα satisfies t 2 ∀ t ∈ Supp (µfα ) Kα (t) = λ0 + λ1 sec 2 t 2 Kα (t) ≥ λ0 + λ1 sec ∀t ∈ / Supp (µfα ), 2 where λ0 , λ1 are real constants.

Long computations, see [3], show our fα gives Kα that satisfies these conditions! Next we characterize the geometry of extremal domains for Nα . Lemma 2.4. 1. ∀ α, 1 < α < 4, an extremal fα minimizing D(f ) exists in Nα . 2. If fα is extremal, then fα (D) is bounded, starlike with respect to 1, and circularly symmetric with respect to rays ℓτ = {z = x + iy : y = 0, x ≥ τ, ∀ τ, 0 ≤ τ ≤ 1}.

3. The min area A(α) := D(fα ) for 1 < α < 4.

Proof. Apply circular and radial symmetrizations, then polarizations similar to arguments in [5].

6


f

IWQCMA05

Α=3

α

L fr 4

1

4

-4

L nf

8

-4

Figure 3. The free (Lf r ) and non-free (Lnf ) portions of the boundary. Now combine Theorem 2.1 and Lemma 2.2 to see that if fα is extremal in Nα , then since fα ∈ Tα , Theorem 2.1 implies Theorem 1.1.

Next we show how the extremal fα in Nα can be recovered from its boundary values. Lemma 2.5. Let fα be extremal for Nα . Then f ′ is continuous on D and |f ′ | ≡ β ≥ α ∀ z ∈ ℓf r . See Figure 2. Proof. Apply the deep “two point variation techniques” from [5] twice giving f ′ these properties on ℓf r . Then use the Julia-Wolff Theorem and boundary behavior properties from Pommerenke [6], giving f ′ these properties everywhere.

Lemma 2.6. If fα is extremal, ϕ(z) = log(zfα′ (z)) maps as described in Figure 2, with i(1 − z) q1 (z) = ϕ √ 2 sin( 20 ) z Z ξ t2 − b2 √ ϕ2 (ξ) = ci dt + s. 2 2 2 0 (t − a ) 1 − t Long computations are used to show monotonicity, then we use line integral formulae to compute the area as in [4].

3. The Iceberg Problem A related problem is known as the Iceberg Problem: Given a fixed volume above the water, how deep can the iceberg go? See Figure 3. This problem can be modeled by supposing a slice I is a continuum in C and E = {II : cap I = 1, area [II ∩ UHP] = α}.


ϕ

−1

τ

0

q

ϕ

1

α = log |f ’(1)|

2

UHP

Figure 4. The mapping ϕ(z) = log(zfα′ (z)).

I

h?

Figure 5. The Iceberg Problem. We anticipate using similar arguments to those in this paper to find h = min min{ℑz : z ∈ I }. I ∈E

7

8


IWQCMA05

References [1] D. Aharonov, H. S. Shapiro, and A. Yu. Solynin, Minimal area problems for functions with integral representation, J. Analyse Math., to appear. [2] Dov Aharonov, Catherine Bénéteau, Dmitry Khavinson, and Harold Shapiro, Extremal problems for nonvanishing functions in Bergman spaces, Selected topics in complex analysis, Oper. Theory Adv. Appl., vol. 158, Birkhäuser, Basel, 2005, pp. 59–86. [3] R. W. Barnard, C. Richardson, and A.Yu Solynin, A minimal area problem for nonvanishing functions, Analysis and Algebra, accepted. [4] R.W. Barnard, C. Richardson, and A.Yu Solynin, Concentration of area in half planes, Proc. Amer. Math. Soc. 133 (2005), no. 7, 2091–99. [5] R.W. Barnard and A.Yu Solynin, Local variations and minimal area problems, Indiana Univ. Math. J. 53 (2004), no. 1, 135–167. [6] Ch. Pommerenke, Boundary behavior of conformal maps, Springer-Verlag, 1992. [7] Alex Schuster and Peter Duren, Bergman spaces of analytic function, Mathematical Surveys and Monographs, no. 100, American Mathematical Society, Providence, RI, 2004. Roger W. Barnard, Clint Richardson, Alex Yu. Solynin Address: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 Address: Department of Mathematics and Statistics, Stephen F. Austin University, Nacogdoches, Texas 75962 Address: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]


The hyperbolic metric and geometric function theory A.F. Beardon and D. Minda Abstract. The goal is to present an introduction to the hyperbolic metric and various forms of the Schwarz-Pick Lemma. As a consequence we obtain a number of results in geometric function theory. Keywords. hyperbolic metric, Schwarz-Pick Lemma, curvature, Ahlfors Lemma. 2000 MSC. Primary 30C99; Secondary 30F45, 47H09.


10

2. The unit disk as the hyperbolic plane

11

3. The Schwarz-Pick Lemma

16

4. An extension of the Schwarz-Pick Lemma

19

5. Hyperbolic derivatives

21

6. The hyperbolic metric on simply connected regions

24

7. Examples of the hyperbolic metric

28

8. The Comparison Principle

33

9. Curvature and the Ahlfors Lemma

36

10. The hyperbolic metric on a hyperbolic region

41

11. Hyperbolic distortion

45

12. The hyperbolic metric on a doubly connected region

47

12.1. Hyperbolic metric on the punctured unit disk

47

12.2. Hyperbolic metric on an annulus

49

13. Rigidity theorems

51

14. Further reading

54

References

55

Version October 19, 2006. The second author was supported by a Taft Faculty Fellowship and wishes to thank the University of Cambridge for its hospitality during his visit November 2004 - April, 2005.

10

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1. Introduction The authors are writing a book, The hyperbolic metric in complex analysis, that will include all of the material in this article and much more. The material presented here is a selection of topics from the book that relate to the Schwarz-Pick Lemma. Our goal is to develop the main parts of geometric function theory by using the hyperbolic metric and other conformal metrics. This paper is intended to be both an introduction to the hyperbolic metric and a concise treatment of a few recent applications of the hyperbolic metric to geometric function theory. There is no attempt to present a comprehensive presentation of the material here; rather we present a selection of several topics and then offer suggestions for further reading. The first part of the paper (Sections 2-5) studies holomorphic self-maps of the unit disk D by using the hyperbolic metric. The unit disk with the hyperbolic metric and hyperbolic distance is presented as a model of the hyperbolic plane. Then Pick’s fundamental invariant formulation of the Schwarz Lemma is presented. This is followed by various extensions of the Schwarz-Pick Lemma for holomorphic self-maps of D, including a Schwarz-Pick Lemma for hyperbolic derivatives. The second part of the paper (Sections 6-9) is concerned with the investigation of holomorphic maps between simply connected proper subregions of the complex plane C using the hyperbolic metric, as well as a study of negatively curved metrics on simply connected regions. Here ‘negatively curved’ means metrics with curvature at most −1. The Riemann Mapping Theorem is used to transfer the hyperbolic metric to any simply connected region that is conformally equivalent to the unit disk. A version of the Schwarz-Pick Lemma is valid for holomorphic maps between simply connected proper subregions of the complex plane C. The hyperbolic metric is explicitly determined for a number of special simply connected regions and estimates are provided for general simply connected regions. Then the important Ahlfors Lemma, which asserts the maximality of the hyperbolic metric among the family of metrics with curvature at most −1, is established; it provides a vast generalization of the Schwarz-Pick Lemma. The representation of metrics with constant curvature −1 by bounded holomorphic functions is briefly mentioned. The third part (Sections 10-13) deals with holomorphic maps between hyperbolic regions; that is, regions whose complement in the extended complex plane C∞ contains at least three points, and negatively curved metrics on such regions. The Planar Uniformization Theorem is utilized to transfer the hyperbolic metric from the unit disk to hyperbolic regions. The Schwarz-Pick and Ahlfors Lemmas extend to this context. The hyperbolic metric for punctured disks and annuli are explicitly calculated. A new phenomenon, rigidity theorems, occurs for multiply connected regions; several examples of rigidity theorems are presented. The final section offers some suggestions for further reading on topics not included in this article.

The hyperbolic metric and geometric function theory

11

2. The unit disk as the hyperbolic plane We assume that the reader knows that the most general conformal automorphism of the unit disk D onto itself is a Möbius map of the form az + c¯ , a, c ∈ C, |a|2 − |c|2 = 1, (2.1) z 7→ cz + a ¯ or of the equivalent form z−a , θ ∈ R, a ∈ D. (2.2) z 7→ eiθ 1−a ¯z It is well known that these maps form a group A(D) under composition, and that A(D) acts transitively on D (that is, for all z and w in D there is some g in A(D) such that g(z) = w). Also, A(D, 0), the subgroup of conformal automorphisms that fix the origin, is the set of rotations of the complex plane about the origin. The hyperbolic plane is the unit disk D with the hyperbolic metric 2 |dz| λD (z)|dz| = . 1 − |z|2 This metric induces a hyperbolic distance dD (z, w) between two points z and w in D in the following way. We join z to w by a smooth curve γ in D, and define the hyperbolic length ℓD (γ) of γ by Z ℓD (γ) = λD (z) |dz|. γ

Finally, we set

dD (z, w) = inf ℓD (γ), γ

where the infimum is taken over all smooth curves γ joining z to w in D. It is immediate from the construction of dD that it satisfies the requirements for a distance on D, namely (a) dD (z, w) ≥ 0 with equality if and only if z = w; (b) dD (z, w) = dD (w, z); (c) for all u, v, w in D, dD (u, w) ≤ dD (u, v) + dD (v, w). The hyperbolic area of a Borel measurable subset of D is Z Z aD (E) = λ2D (z)dxdy. E

We need to identify the isometries of both the hyperbolic metric and the hyperbolic distance. A holomorphic function f : D → D is an isometry of the metric λD (z) |dz| if for all z in D, (2.3) λD f (z) |f ′ (z)| = λD (z), and it is an isometry of the distance dD if, for all z and w in D, (2.4) dD f (z), f (w) = dD (z, w).

In fact, the two classes of isometries coincide, and each isometry is a Möbius transformation of D onto itself.

12

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Theorem 2.1. For any holomorphic map f : D → D the following are equivalent: (a) f is a conformal automorphism of D; (b) f is an isometry of the metric λD ; (c) f is an isometry of the distance dD . Proof. First, (a) implies (b). Indeed, if (a) holds, then f is of the form (2.1), and a calculation shows that |f ′ (z)| 1 = , 2 1 − |f (z)| 1 − |z|2 so (b) holds. Next, (b) implies (a). Suppose that (b) holds; that is, f is an isometry of the hyperbolic metric. Then for any conformal automorphism g of D, h = g ◦ f is again an isometry of the hyperbolic metric. If we choose g so that h(0) = g(f (0)) = 0, then 2|h′ (0)| = λD (h(0)|h′ (0)| = λD (0) = 2. Thus, h is a holomorphic self-map of D that fixes the origin and |h′ (0)| = 1, so Schwarz’s Lemma implies h ∈ A(D, 0). Then f = g −1 ◦ h is in A(D). We have now shown that (a) and (b) are equivalent. Second, we prove (a) and (c) are equivalent. If f ∈ A(D), then f is an isometry of the metric λD . Hence, for any smooth curve γ in D, Z Z ℓD f ◦ γ = λD (w) |dw| = λD f (z) |f ′ (z)| |dz| = ℓD (γ). f ◦γ

γ

This implies that for all z, w ∈ D, dD (f (z), f (w)) ≤ dD (z, w). Because f ∈ A(D), the same argument applies to f −1 , and hence we may conclude that f is a dD – isometry. Finally, we show that (c) implies (a). Take any f : D → D that is holomorphic and a dD –isometry. Choose any g of the form (2.1) that maps f (0) to 0 and put h = g ◦ f . Then his holomorphic, a dD –isometry, and h(0) = 0. Thus dD 0, h(z) = dD h(0), h(z) = dD (0, z). This implies that |h(z)| = |z| and hence, that h(z) = eiθ z for some θ ∈ R. Thus h ∈ A(D, 0) and, as f = g −1 ◦ h, f is also in A(D). In summary, relative to the hyperbolic metric and the hyperbolic distance, the group A(D) of conformal automorphism of the unit disk becomes a group of isometries. Theorem 2.2. The hyperbolic distance dD (z, w) in D is given by (2.5)

dD (z, w) = log

1 + pD (z, w) = 2 tanh−1 pD (z, w), 1 − pD (z, w)

where the pseudo-hyperbolic distance pD (z, w) is given by z−w . (2.6) pD (z, w) = 1 − z w¯


13

Proof. First, we prove that if −1 < x < y < 1 then ! y−x 1 + 1−xy . (2.7) dD (x, y) = log y−x 1 − 1−xy Consider a smooth curve γ joining x to y in D, and write γ(t) = u(t) + iv(t), where 0 ≤ t ≤ 1. Then Z 1 Z 1 2 |γ ′ (t)| dt 2 u′ (t) dt ℓD (γ) = ≥ 2 2 0 1 − |γ(t)| 0 1 − u(t) because |γ(t)|2 ≥ |u(t)|2 = u(t)2 and |γ ′ (t)| ≥ |u′ (t)| ≥ u′ (t). The second integral can be evaluated directly and gives ! y−x 1 + 1+y 1−x 1−xy = log . ℓD (γ) ≥ log y−x 1−y 1+x 1 − 1−xy Because equality holds here when γ(t) = x + t(y − x), 0 ≤ t ≤ 1, we see that (2.7) holds, so (2.5) is valid for −1 < x < y < 1.

Now we have to extend (2.5) to any pair of points z and w in D. Theorem 2.1 shows that each Euclidean rotation about the origin is a hyperbolic isometry and this implies that, for all z, dD (0, z) = dD (0, |z|). Now take any z and w in D, and let f (z) = (z − w)/(1 − z w). ¯ Then f is a conformal automorphism of D, and so is a hyperbolic isometry. Thus dD (z, w) = dD (w, z)

= dD f (w), f (z) = dD 0, f (z) = dD 0, |f (z)| = dD 0, pD (z, w) ,

which, from (2.7) with x = 0 and y = pD (z, w), gives (2.5). Note that (2.5) produces dD (0, z) = log Also,

1 + |z| , 1 − |z|

dD (0, z) = 2 tanh−1 |z|.

dD (z, w) pD (z, w) = λD (w) = 2 lim . z→w |z − w| z→w |z − w| lim

A careful examination of the proof of (2.5) shows that if γ is a smooth curve that joins x to y, where −1 < x < y < 1, then ℓD (γ) = dD (0, x) if and only if γ is the simple arc from x to y along the real axis. As hyperbolic isometries map circles into circles, map the unit circle onto itself, and preserve orthogonality, we can now make the following definition.

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Definition 2.3. Suppose that z and w are in D. Then the (hyperbolic) geodesic through z and w is C ∩ D, where C is the unique Euclidean circle (or straight line) that passes through z and w and is orthogonal to the unit circle ∂D. If γ is any smooth curve joining z to w in D, then the hyperbolic length of γ is dD (z, w) if and only if γ is the simple arc of C in D that joins z and w. The unit disk D together with the hyperbolic metric is called the Poincaré model of the hyperbolic plane. The “lines” in the hyperbolic plane are the hyperbolic geodesics and the angle between two intersecting lines is the Euclidean angle between the Euclidean tangent lines at the point of intersection. The hyperbolic plane satisfies all of the axioms for Euclidean geometry with the exception of the Parallel Postulate. It is easy to see that if γ is a hyperbolic geodesic in D and a ∈ D is a point not on γ, then there are infinitely many geodesics through a that do not intersect γ and so are parallel to γ. We shall now show that the hyperbolic distance dD is additive along geodesics. By contrast, the pseudo-hyperbolic distance pD is never additive along geodesics. Theorem 2.4. If u, v and w are three distinct points in D that lie, in this order, along a geodesic, then dD (u, w) = dD (u, v) + dD (v, w). For any three distinct points u, v and w in D, pD (u, w) < pD (u, v) + pD (v, w). Proof. Suppose that u, v and w lie in this order, along a geodesic. Then there is an isometry f that maps this geodesic to the real diameter (−1, 1) of D, with f (v) = 0. Let x = f (u) and y = f (w), so that −1 < x < 0 < y < 1. It is sufficient to show that dD (x, 0) + dD (0, y) = dD (x, y); this is a direct consequence of (2.7). It is easy to verify that pD a distance function on D, except possibly for the verification of the triangle inequality. This holds because, for any distinct u, v and w, 1 pD (u, w) = tanh dD (u, w) 2 1 ≤ tanh [dD (u, v) + dD (v, w)] 2 tanh 21 dD (u, v) + tanh 12 dD (v, w) = 1 + tanh 12 dD (u, v) tanh 21 dD (v, w) 1 1 < tanh dD (u, v) + tanh dD (v, w) 2 2 = pD (u, v) + pD (v, w). This also shows that there is always a strict inequality in the triangle inequality for pD for any three distinct points. The following example illustrates how the hyperbolic distance compares with the Euclidean distance in D.


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Example 2.5. The Poincaré model of the hyperbolic plane does not accurately reflect all of the properties of the hyperbolic plane. For example, the hyperbolic plane is homogeneous; this means that for any pair of points a and b in D there is an isometry f with f (a) = b. Intuitively this means that the hyperbolic plane looks the same at each point just as the Euclidean plane does. However, with our Euclidean eyes, the origin seems to occupy a special place in the hyperbolic plane. In fact, in the hyperbolic plane the origin is no more special than any point a 6= 0. Here is another way in which the Poincaré model deceives our Euclidean eyes. Let x0 , x1 , x2 , . . . be the sequence 0, 21 , 34 , 87 , . . ., so that xn = (2n − 1)/2n , and xn+1 is halfway between xn and 1 in the Euclidean sense. A computation using (2.5) shows that dD (0, xn ) = log(2n+1 − 1). We conclude that dD (xn , xn+1 ) → log 2 as n → ∞; thus the points xn are, for large n, essentially equally spaced in the hyperbolic sense along the real diameter of D. Moreover, in any figure representing the Poincaré model the points xn , for n ≥ 30, are indistinguishable from the point 1 which does not lie in the hyperbolic plane. In brief, although the hyperbolic plane contains arbitrarily large hyperbolic disks about the origin, our Euclidean eyes can only see hyperbolic disks about the origin with a moderate sized hyperbolic radius. Let us comment now on the various formulae that are available for dD (z, w). It is often tempting to use the pseudohyperbolic distance pD rather than the hyperbolic distance dD (and many authors do) because the expression for pD is algebraic whereas the expression for dD is not. However, this temptation should be resisted. The distance pD is not additive along geodesics, and it does not arise from a Riemannian metric. Usually, the solution is to use the following functions of dD , for it is these that tend to arise naturally and more frequently in hyperbolic trigonometry: |z − w|2 1 2 1 = |z − w|2 λD (z)λD (w), (2.8) sinh dD (z, w) = 2 2 2 (1 − |z|) (1 − |w| ) 4 and |1 − z w| ¯2 1 1 = |1 − z w| ¯ 2 λD (z)λD (w). cosh2 dD (z, w) = 2 2 2 (1 − |z| )(1 − |w| ) 4 These can be proved directly from (2.5), and together they give the familiar formula z−w 1 = pD (z, w). tanh dD (z, w) = 2 1 − z w¯ We investigate the topology defined on the unit disk by the hyperbolic distance. For this we study hyperbolic disks since they determine the topology. The hyperbolic circle Cr given by {z ∈ D : dD (0, z) = r} is a Euclidean circle with Euclidean center 0 and Euclidean radius tanh 12 r. Now let C be any hyperbolic circle, say of hyperbolic radius r and hyperbolic center w. Then there is a hyperbolic isometry f with f (w) = 0, so that f (C) = Cr . As Cr is a Euclidean circle, so is f −1 (Cr ), which is C. Conversely, suppose that C is a Euclidean circle in D.

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Then there is a hyperbolic isometry f such that f (C) is a Euclidean circle with center 0, so that f (C) = Cr for some r. Thus, as f is a hyperbolic isometry, f −1 (Cr ) = C, is also a hyperbolic circle. This shows that the set of hyperbolic circles coincides with the set of Euclidean circles in D. As the same is obviously true for open disks (providing that the closed disks lie in D), we see that the topology induced by the hyperbolic distance on D coincides with the Euclidean topology on the unit disk. Theorem 2.6. The topology induced by dD on D coincides with the Euclidean topology. The space D with the distance dD is a complete metric space. Proof. We have already proved the first statement. Suppose, then, that (zn ) is a Cauchy sequence with respect to the distance dD . Then (zn ) is a bounded sequence with respect to dD and, as we have seen above, this means that the (zn ) lie in a compact disk K that is contained in D. As λD ≥ 2 on D, we see immediately from (2.8) that (zn ) is a Cauchy sequence with respect to the Euclidean metric, so that zn → z ∗ , say, where z ∗ ∈ K ⊂ D. It is now clear that dD (zn , z ∗ ) → 0 so that D with the distance dD is complete. The Euclidean metric on D arises from the fact that D is embedded in the larger space C and is not complete on D. By contrast, an important property of the distance dD is that dD (0, |z|) → +∞ as |z| → 1; informally, the boundary ∂D of D is ‘infinitely far away’ from each point in D. This is a consequence of the fact that D equipped with the hyperbolic distance dD is a complete metric space and is another reason why dD should be preferred to the Euclidean metric on D. Exercises. 1. Verify that (2.1) and (2.2) determine the same subgroup of Möbius transformations. 2. Suppose equality holds in the triangle inequality for the hyperbolic distance; that is, suppose u, v, w in D and dD (u, w) = dD (u, v) + dD (v, w). Prove that u, v and w lie on a hyperbolic geodesic in this order. 3. Verify that the hyperbolic disk DD (a, r) is the Euclidean disk with center c and radius R, where a 1 − tanh2 (r/2) (1 − |a|2 ) tanh(r/2) c= and R = . 1 − |a|2 tanh2 (r/2) 1 − |a|2 tanh2 (r/2) 4. (a) Prove that the hyperbolic area of a hyperbolic disk of radius r is 4π sinh2 (r/2). (b) Show that the hyperbolic length of a hyperbolic circle with radius r is 2π sinh r.

3. The Schwarz-Pick Lemma We begin with a statement of the classical Schwarz Lemma.


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Theorem 3.1 (Schwarz’s Lemma). Suppose that f : D → D is holomorphic and that f (0) = 0. Then either (a) |f (z)| < |z| for every non-zero z in D, and |f ′ (0)| < 1, or (b) for some real constant θ, f (z) = eiθ z and |f ′ (0)| = 1. The Schwarz Lemma is proved by applying the Maximum Modulus Theorem to the holomorphic function f (z)/z on the unit disk D. It says that if a holomorphic function f : D → D fixes 0 then either (a) f (z) is closer to 0 than z is, or (b) f is a rotation of the plane about 0. Although both of these assertions are true in the context of Euclidean geometry, they are only invariant under conformal maps when they are interpreted in terms of hyperbolic geometry. Moreover, as Pick observed in 1915, in this case the requirement that f has a fixed point in D is redundant. We can now state Pick’s invariant formulation of Schwarz’s Lemma [33]. Theorem 3.2 (The Schwarz-Pick Lemma). Suppose that f : D → D is holomorphic. Then either (a) f is a hyperbolic contraction; that is, for all z and w in D, (3.1) dD f (z), f (w) < dD (z, w), λD f (z) |f ′ (z)| < λD (z), or (b) f is a hyperbolic isometry; that is, f ∈ A(D) and for all z and w in D, (3.2) dD f (z), f (w) = dD (z, w), λD f (z) |f ′ (z)| = λD (z)

Proof. By Theorem 2.1, f is an isometry if and only if one, and hence both, of the conditions in (3.2) hold. Suppose now that f : D → D is holomorphic but not an isometry. Select any two points z1 and z2 in D. Here is the intuitive idea behind the proof. Because the hyperbolic plane is homogeneous, we may assume without loss of generality that both z1 and f (z1 ) are at the origin. In this special situation (3.1) follows directly from part (b) of Theorem 3.1. Now we write out a formal argument. Let g and h be conformal automorphisms (and hence isometries) of D such that g(z1 ) = 0 and h f (z1 ) = 0. Let F = hf g −1 ; then F is a holomorphic self-map of D that fixes 0. As g and h are isometries, F is not an isometry too. Therefore, by Schwarz’s Lemma, or else f would be ′ for all z, dD 0, F (z) < dD (0, z) and |F (0)| < 1. Thus, as F g = hf and g, h are hyperbolic isometries, dD f (z1 ), f (z2 ) = dD hf (z1 ), hf (z2 ) = dD F g(z1 ), F g(z2 ) = dD 0, F g(z2 ) < dD 0, g(z2 ) = dD g(z1 ), g(z2 ) = dD (z1 , z2 ).

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This is the first inequality in (3.1). To obtain the second inequality, we apply the Chain Rule to each side of F g = hf and obtain |F ′ (0)| =

|f ′ (z1 )|(1 − |z1 |2 ) < 1. 1 − |f (z1 )|2

This gives the second inequality in (3.1) at an arbitrary point z1 . Often the Schwarz-Pick Lemma is stated in the following form: Every holomorphic self-map of the unit disk is a contraction relative to the hyperbolic metric. That is, if f is a holomorphic self-map of D, then (3.3) dD f (z), f (w) ≤ dD (z, w), λD f (z) |f ′ (z)| ≤ λD (z). If equality holds in either inequality, then f is a conformal automorphism of D. One should note that the two inequalities in (3.3) are equivalent. If the first inequality holds, then dD (z, w) dD (f (z), f (w)) |f (z) − f (w)| ≤ lim = λD (z). λD f (z) |f ′ (z)| = lim w→z |z − w| w→z |f (z) − f (w)| |z − w|

On the other hand, if the second inequality holds, then integration over any path γ in D gives ℓD (f ◦ γ) ≤ ℓD (γ). This implies the first inequality in (3.3). Hyperbolic geometry had been used in complex analysis by Poincaré in his proof of the Uniformization Theorem for Riemann surfaces. The work of Pick is a milestone in geometric function theory, it shows that the hyperbolic metric, not the Euclidean metric, is the natural metric for much of the subject. The definition of the hyperbolic metric might seem arbitrary. In fact, up to multiplication by a positive scalar it is the only metric on the unit disk that makes every holomorphic self-map a contraction, or every conformal automorphism an isometry.

Theorem 3.3. For a metric ρ(z)|dz| on the unit disk the following are equivalent: (a) For any holomorphic self-map of D and all z ∈ D, ρ(f (z))|f ′ (z)| ≤ ρ(z); (b) For any f ∈ A(D) and all z ∈ D, ρ(f (z))|f ′ (z)| = ρ(z); (c) ρ(z) = cλD for some c > 0. Proof. (a)⇒(b) Suppose f ∈ A(D). Then the inequality in (a) holds for f . The inequality in (a) also holds for f −1 ; this gives ρ(z) ≤ ρ(f (z))|f ′ (z)|. Hence, every conformal automorphism of D is an isometry relative to ρ(z)|dz|. (b)⇒(c) Define c > 0 by ρ(0) = cλD (0). Now, consider any a ∈ D. Let f be a conformal automorphism of D with f (0) = a. Then because f is an isometry relative to both ρ(z)|dz| and the hyperbolic metric, ρ(a)|f ′ (0)| = ρ(0) = cλD (0) = cλD (a)|f ′ (0)|. Hence, ρ(a) = cλD (a) for all a ∈ D.

(c)⇒(a) This is an immediate consequence of the Schwarz-Pick Lemma.


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Exercises. 1. Suppose f is a holomorphic self-map of the unit disk. Prove |f ′ (0)| ≤ 1. Determine a necessary and sufficient condition for equality. 2. If a holomorphic self-map of the unit disk fixes two points, prove it is the identity. 3. Let a and b be distinct points in D. (a) Show that there exists a conformal automorphism f of D that interchanges a and b; that is, f (a) = b and f (b) = a. (b) Suppose a holomorphic self-map f of D interchanges a and b; that is, f (a) = b and f (b) = a. Prove f is a conformal automorphism with order 2, or f ◦ f is the identity.

4. An extension of the Schwarz-Pick Lemma Recently, the authors [8] established a multi-point version of the Schwarz-Pick Lemma that unified a number of known variations of the Schwarz and SchwarzPick Lemmas and also has many new consequences. A selection of results from [8] are presented in this and the next section; for more results of this type, consult the original paper. We begin with a brief discussion of Blaschke products. A function F : D → D is a (finite) Blaschke product if it is holomorphic in D, continuous in D (the closed unit disk), and |F (z)| = 1 when |z| = 1. If F is a Blaschke product then so are the compositions g(F (z)) and F (g(z)) for any conformal automorphism g of D. In addition, it is clear that any finite product of conformal automorphisms of D is a Blaschke product. We shall now show that the converse is true. Suppose that F is a Blaschke product. If F has no zeros in D then, by the Minimum Modulus Theorem, F is a constant, which must be of modulus one. Now suppose that F does have a zero in D. Then it can only have a finite number of zeros in D, say a1 , . . . , ak (which need not be distinct), and Y k z − am F (z) 1−a ¯m z m=1

is a Blaschke product with no zeros in D. This shows that F is a Blaschke product if and only if it is a finite product of automorphisms of D. We say that F is of degree k if this product has exactly k non-trivial factors.

We now discuss the complex pseudo-hyperbolic distance in D, and the hyperbolic equivalent of the usual Euclidean difference quotient of a function. Definition 4.1. The complex pseudo-hyperbolic distance [z, w] between z and w in D is given by z−w . [z, w] = 1 − wz We recall that the pseudo-hyperbolic distance is |[z, w]|; see (2.6).

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The complex pseudo-hyperbolic distance is an analog for the hyperbolic plane D of the real directed distance x − y from y to x, for points on the real line R. Definition 4.2. Suppose that f : D → D is holomorphic, and that z, w ∈ D with z 6= w. The hyperbolic difference quotient f ∗ (z, w) is given by f ∗ (z, w) =

[f (z), f (w)] . [z, w]

If we combine (2.6) with the Schwarz-Pick Lemma we see that pD f (z1 ), f (z2 ) ≤ pD (z1 , z2 ),

and that equality holds for one pair z1 and z2 of distinct points if and only if f is a conformal automorphism of D (in which case, equality holds for all z1 and z2 ). It follows that if f : D → D is holomorphic, then either f is a hyperbolic isometry and |f ∗ (z, w)| = 1 for all z and w, or f is not an isometry and |f ∗ (z, w)| < 1 for all z and w. We shall now discuss the hyperbolic difference quotient f ∗ (z, w). This is a function of two variables but, unless we state explicitly to the contrary, we shall regard it as a holomorphic function of the single variable z. Note that f ∗ (z, w) is not holomorphic as a function of the second variable w. The basic properties of f ∗ (z, w) are given in our next result. Theorem 4.3. Suppose that f : D → D is holomorphic, and that w ∈ D. (a) The function z 7→ f ∗ (z, w) is holomorphic in D. (b) If f is not a conformal automorphism of D, then z 7→ f ∗ (z, w) is a holomorphic self-map D. (c) The map z 7→ f ∗ (z, w) is a conformal automorphism of D if and only if f is a Blaschke product of degree two. Proof. Part (a) is obvious as w is a removable singularity of the function ! −1 z−w f (z) − f (w) ∗ . f (z, w) = 1 − wz 1 − f (w)f (z)

Now suppose that f is not a conformal automorphism of D. Then, as we have seen above, |f ∗ (z, w)| < 1 and this proves (b).

To prove (c) we note first that there are conformal automorphisms g and h (that depend on w) of D such that f ∗ (z, w) = g(f (z))/h(z) or, equivalently, f (z) = g −1 f ∗ (z, w)h(z) . Clearly, if f ∗ (z, w) is an automorphism then f is a Blaschke product of degree two. Conversely, suppose that f is a Blaschke product of degree two. Then g f (z) is also a Blaschke product, say B, of degree two and f ∗ (z, w) = B(z)/h(z). As f ∗ (z, w) is holomorphic in z, we see that B(z) = h(z)h1 (z) for some automorphism h1 . Thus f ∗ (z, w) = h1 (z) as required. We shall now derive a three-point version of the Schwarz-Pick Lemma. Because it involves three points rather than two points as in the Schwarz-Pick Lemma, the


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following theorem has extra flexibility and it includes all variations and extensions of the Schwarz-Pick Lemma that are known to the authors. We stress, though, that this theorem contains much more than simply the union of all such known results. Although several Euclidean variations of the Schwarz-Pick Lemma are known, in our view much greater clarity is obtained by a strict adherence to hyperbolic geometry. This and other stronger versions of the Schwarz-Pick Lemma appear in [8]. Theorem 4.4 (Three-point Schwarz-Pick Lemma). Suppose that f is holomorphic self-map of D, but not an automorphism of D. Then, for any z, w and v in D, (4.1) dD f ∗ (z, v), f ∗ (w, v) ≤ dD (z, w).

Further, equality holds in (4.1) for some choice of z, w and v if and only if f is a Blaschke product of degree two. Proof. As f is holomorphic in D, but not an automorphism, Theorem 4.3(b) shows that the left-hand side of (4.1) is defined. The inequality (4.1) now follows by applying the Schwarz-Pick Lemma to the holomorphic self-map z 7→ f ∗ (z, v) of D. The Schwarz-Pick Lemma also implies that equality holds in (4.1) if and only if f ∗ (z, w) is a conformal automorphism of D and, by Theorem 4.3(c), this is so if and only if f is a Blaschke product of degree two. Theorem 4.4 is a genuine improvement of the Schwarz-Pick Lemma. Suppose, for example that f : D → D is holomorphic, but not an automorphism, and that f (0) = 0. Then the Schwarz-Pick Lemma tells us only that f (z)/z lies in the hyperbolic plane D, and that |f ′ (0)| < 1. However, it we put w = 0 in (4.1), and then let v → 0, we obtain the stronger conclusion that f (z)/z lies in the hyperbolic disk with center f ′ (0) and hyperbolic radius dD (0, z). Exercises. 1. If f (z) is a Blaschke product of degree k, prove that f ∗ (z, w) is a Blaschke product of degree k − 1. 2. Verify the following Chain Rule for the ∗-operator: For all z and w in D, and all holomorphic maps f and g of D into itself, (f ◦ g)∗ (z, w) = f ∗ g(z), g(w) g ∗ (z, w).

5. Hyperbolic derivatives Since the hyperbolic metric is the natural metric to study holomorphic selfmaps of the unit disk, one should also use derivatives that are compatible with this metric. We begin with the definition of a hyperbolic derivative; just as the Euclidean difference quotient leads to the usual Euclidean derivative, the hyperbolic difference quotient results in the hyperbolic derivative.

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Definition 5.1. Suppose that f : D → D is holomorphic, but not an isometry of D. The hyperbolic derivative f h (w) of f at w in D is [f (z), f (w)] (1 − |w|2 )f ′ (w) = . z→w [z, w] 1 − |f (w)|2 The hyperbolic distortion of f at w is dD (f (z), f (w)) . |f h (w)| = lim z→w dD (z, w) f h (w) = lim

By Theorem 4.3, |f h (z)| ≤ 1, and equality holds for some z if and only if equality holds for all z, and then f is a conformal automorphism of D. Theorem 4.4 leads to the following upper bound on the magnitude of the hyperbolic difference quotient in terms of dD (z, w) and the derivative at any point v between z and w. Theorem 5.2. Suppose that f : D → D is holomorphic. Then, for all z and w in D, and for all v on the closed geodesic arc joining z and w, (5.1) dD 0, f ∗ (z, w) ≤ dD 0, f h (v) + dD (z, w). Proof. First, it is clear that for any z and w, |f ∗ (z, w)| = |f ∗ (w, z)|. Thus dD 0, f ∗ (z, w) = dD 0, f ∗ (w, z) . Next, Theorem 4.4 (applied twice) gives dD 0, f ∗ (z, w) ≤ dD 0, f ∗ (v, w) + dD f ∗ (v, w), f ∗ (z, w) ≤ dD 0, f ∗ (v, w) + dD (z, v) = dD 0, f ∗ (w, v) + dD (z, v) ≤ dD 0, f ∗ (u, v) + dD (w, u) + dD (z, v).

We now let u → v, where v lies on the geodesic between z and w, and as dD (z, v) + dD (v, w) = dD (z, w), we obtain (5.1). Our next task is to transform (5.1) into a more transparent inequality about f . This is the next result which we may interpret as a Hyperbolic Mean Value Inequality, a result from [7]. Theorem 5.3 (Hyperbolic Mean Value Inequality). Suppose that f : D → D is holomorphic. Then, for all z and w in D, and for all v on the closed geodesic arc joining z and w, (5.2) dD f (z), f (w) ≤ log cosh dD (z, w) + |f h (v)| sinh dD (z, w) .

This inequality is sharper than the Schwarz-Pick inequality for if we use |f h (v)| ≤ 1 and the identity cosh t + sinh t = et , we recapture the SchwarzPick inequality. It is known that equality holds in (5.2) if and only if f is a Blaschke product of degree two and has a unique critical point c, such that either c, z = v, w, or c, w = v, z, lie in this order along a geodesic. We refer


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the reader to [8] for a proof of this, and for the fact that a Blaschke product of degree two has exactly one critical point in D. Proof. First, we note that, for all u and v, 1 tanh dD (0, u) = |u|, 2 1 1 tanh dD (u, v) = pD (u, v) = tanh dD (0, [u, v]). 2 2 ∗ Next, using the definition of f (z, w), the inequality in Theorem 5.2, and the addition formula for tanh(s + t), we have 1 tanh dD (f (z), f (w)) = pD (f (z), f (w)) 2 = pD (z, w)|f ∗ (z, w)| 1 = pD (z, w) tanh dD 0, f ∗ (z, w) 2 i h1 1 h ≤ pD (z, w) tanh dD 0, f (v) + dD (z, w) 2 2 pD (z, w) + |f h (v)| . = pD (z, w) 1 + pD (z, w)|f h (v)| Now the increasing function x 7→ tanh( 12 x) has inverse x 7→ log(1 + x)/(1 − x), so we conclude that, with p = pD (z, w) and d = |f h (v)|, 1 + pd + p(d + p) 1 + p2 2p dD (f (z), f (w)) ≤ log = log +d , 1 + pd − p(d + p) 1 − p2 1 − p2 which is (5.2).

Next, we provide a Schwarz-Pick type of inequality for hyperbolic derivatives; recall that the hyperbolic derivative is not holomorphic. This result is based on the observation that if f : D → D is holomorphic, but not a conformal automorphism of D, then f h (z) and f h (w) lie in D so that we can measure the hyperbolic distance between these two hyperbolic derivatives. Theorem 5.4. Suppose that f : D → D is holomorphic but not a conformal automorphism of D. Then, for all z and w in D, (5.3) dD f h (z), f h (w) ≤ 2dD (z, w) + dD f ∗ (z, w), f ∗ (w, z) . Proof. Theorem 4.4 implies that for all z, w and v, dD f ∗ (z, w), f ∗ (v, w) ≤ dD (z, v). We let v → w and obtain

dD f ∗ (z, w), f h (w) ≤ dD (z, w)

and (by interchanging z and w),

dD f ∗ (w, z), f h (z) ≤ dD (z, w).

These last two inequalities and the triangle inequality yields (5.3).

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It is easy to see that if f (0) = 0 then f ∗ (z, 0) = f ∗ (0, z) = f (z)/z. Thus we have the following corollary originally established in [6]. Corollary 5.5. Suppose that f : D → D is holomorphic but not a conformal automorphism of D, and that f (0) = 0. Then, for all z, (5.4) dD f h (0), f h (z) ≤ 2dD (0, z), and the constant 2 is best possible.

Example 5.6. The preceding corollary is sharp for f (z) = z 2 . Note that f h (z) = 2z/(1 + |z|2 ) and dD (f h (z), f h (w)) = 2dD (z, w) whenever z, w lie on the same hyperbolic geodesic through the origin. Thus, z 7→ f h (z) doubles all hyperbolic distances along geodesics through the origin; this doubling is not valid in general because in hyperbolic geometry there are no similarities except isometries. Moreover, it is possible to verify that there is no finite K such that dD (f h (z), f h (w)) ≤ KdD (z, w) for all z, w ∈ D, so z 7→ f h (z) does not even satisfy a hyperbolic Lipschitz condition, so (5.4) is no longer valid when the origin is replaced by an arbitrary point of the unit disk. In spite of Example 5.6 a full-fledged result of Schwarz-Pick type is valid for the hyperbolic distortion. Corollary 5.7 (Schwarz-Pick Lemma for Hyperbolic Distortion). Suppose that f : D → D is holomorphic but not a conformal automorphism of D. Then for all z, w ∈ D, dD |f h (z)|, |f h (w)| ≤ 2dD (z, w).

Proof. Note that, from the proof of Theorem 5.4, dD |f ∗ (z, w)|, |f h (w)| ≤ dD f ∗ (z, w), f h (w) ≤ dD (z, w), and, similarly, dD |f ∗ (w, z)|, |f h (z)| ≤ dD (z, w). As |f ∗ (w, z)| = |f ∗ (z, w)|, the desired inequality follows.

Exercises. 1. Verify the claims in Example 5.6. 2. Suppose that f : D → D is holomorphic but not a conformal automorphism of D. Prove that for all conformal automorphisms S and T of D, and all z and w in D, |(S ◦ f ◦ T )∗ (z, w)| = |f ∗ (T (z), T (w))|. In particular, deduce that the hyperbolic derivative is invariant in the sense that |(S ◦ f ◦ T )h (z)| = |f h (T (z))|.

6. The hyperbolic metric on simply connected regions There are several equivalent definitions of what it means for a region in the complex plane to be simply connected. A region Ω in C is simply connected if and only if any one of the following (equivalent) conditions hold:


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(a) the set C∞ \Ω is connected; (b) if f is holomorphic and never zero in Ω, then there is a single-valued holomorphic choice of log f in Ω; (c) each closed curve in Ω can be continuously deformed within Ω to a point of Ω. A region in C∞ is simply connected if (a) or (c) holds. The regions D, C and C∞ are all simply connected; an annulus is not.

Two subregions regions of C are conformally equivalent if there is a holomorphic bijection of one onto the other. This is an equivalence relation on the class of subregions of C, and the fundamental result about simply connected regions is the Riemann Mapping Theorem.

Theorem 6.1 (The Riemann Mapping Theorem). A subregion of C is conformally equivalent to D if and only if Ω is a simply connected proper subregion of C. Moreover, given a ∈ Ω there is a unique conformal mapping f : Ω → D such that f (a) = 0 and f ′ (a) > 0. The Riemann Mapping Theorem enables us to transfer the hyperbolic metric from D to any simply connected proper subregion Ω of C. Definition 6.2. Suppose that f is a conformal map of a simply connected plane region Ω onto D. Then the hyperbolic metric λΩ (z)|dz| of Ω is defined by (6.1) λΩ (z) = λD f (z) |f ′ (z)|. The hyperbolic distance dΩ is the distance function on Ω derived from the hyperbolic metric.

We need to show λΩ is independent of the choice of the conformal map f that is used in (6.1), for this will imply that λΩ is determined by Ω alone. Suppose, then, that f is a conformal map of Ω onto D. Then the set of all conformal maps of Ω onto D is given by h ◦ f , where h ranges over A(D). Any conformal automorphism h of D is a hyperbolic isometry, so that for all w in D, λD (w) = λD h(w) |h′ (w)|. If we now let g = h ◦ f , w = f (z) and use the Chain Rule we find that λD g(z) |g ′ (z)| = λD h(f (z) |h′ (f (z))||f ′ (z)| = λD (f (z))|f ′ (z)|

so that λΩ as defined in (6.1) is independent of the choice of the conformal map f. Thus, Definition 6.2 converts every conformal map of a simply connected proper subregion of C onto the unit disk into an isometry of the hyperbolic metric. The hyperbolic distance dΩ on a simply connected proper subregion Ω of C can be defined in two equivalent ways. First, one can pull-back the hyperbolic distance on D to Ω by setting dΩ (z, w) = dD (f (z), f (w)) for any conformal map f : Ω → D and verifying that this is independent of the choice of the conformal

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mapping onto the unit disk. Alternatively, the hyperbolic length of a path γ in Ω is Z ℓΩ (γ) = λΩ (z)|dz|, γ

and one can define

dΩ (z, w) = inf ℓΩ (γ), where the infimum is taken over all piecewise smooth curves γ in Ω that join z and w. These two definitions of the hyperbolic distance are equivalent. The hyperbolic distance dΩ on Ω is complete. Moreover, a path γ in Ω connecting z and w is a hyperbolic geodesic in Ω if and only if f ◦ γ is a hyperbolic geodesic in D. Also, for any a ∈ Ω and r > 0, f (DΩ (a, r)) = DD (f (a), r).

In fact, the essence of Definition 6.2 is that the entire body of geometric facts about the Poincaré model D of the hyperbolic plane transfers, without any essential change, to an arbitrary simply connected proper subregion of C with its own hyperbolic metric. If f : Ω → D is any conformal mapping, then f is an isometry relative to the hyperbolic metrics and hyperbolic distances on Ω and D. The next result is an immediate consequence of Definition 6.2 and we omit its proof; it asserts that all conformal maps of simply connected proper regions are isometries relative to the hyperbolic metrics and hyperbolic distances of the regions. Theorem 6.3 (Conformal Invariance). Suppose that Ω1 and Ω2 are simply connected proper subregions of C, and that f is a conformal map of Ω1 onto Ω2 . Then f is a hyperbolic isometry, so that for any z in Ω1 , (6.2) λΩ2 f (z) |f ′ (z)| = λΩ1 (z), and for all z, w ∈ Ω1

dΩ2 (f (z), f (w)) = dΩ1 (z, w).

Note that if γ is a smooth curve in Ω1 , then (6.2) implies ℓΩ2 (f ◦ γ) = ℓΩ1 (γ).

Theorem 6.3 implies that each element of A(Ω), the group of conformal automorphisms of Ω, is a hyperbolic isometry. Theorem 6.4 (Schwarz-Pick Lemma for Simply Connected Regions). Suppose that Ω1 and Ω2 are simply connected proper subregions of C, and that f is a holomorphic map of Ω1 into Ω2 . Then either (a) f is a hyperbolic contraction; that is, for all z and w in Ω1 , dΩ2 f (z), f (w) < dΩ1 (z, w), λΩ2 f (z) |f ′ (z)| < λΩ1 (z),

or (b) f is a hyperbolic isometry; that is, f is a conformal map of Ω1 onto Ω2 and for all z and w in Ω1 , dΩ2 f (z), f (w) = dΩ1 (z, w), λΩ2 f (z) |f ′ (z)| = λΩ1 (z).


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Proof. Because of Theorem 6.3 we only need verify (a) when the holomorphic map f : Ω1 → Ω2 is not a holomorphic bijection. Choose any point z0 in Ω1 , and let w0 = f (z0 ). Next, construct a holomorphic bijection h of D, and a holomorphic bijection g of Ω1 onto Ω2 ; these can be constructed so that h(0) = z0 and g(z0 ) = w0 = f (z0 ). Now let k = (gh)−1 f h. Then k is a holomorphic map of D into itself and k(0) = 0. Moreover, k is not a conformal automorphism of D or else f would be a holomorphic bijection. Thus |k ′ (0)| < 1 and, using the Chain Rule, this gives |f ′ (z0 )| < |g ′ (z0 )|. With this, λΩ2 f (z0 ) |f ′ (z0 )| < λΩ1 (z0 ) follows as (6.2) holds (with f replaced by g).

This establishes the second strict inequality in (a); the first strict inequality for hyperbolic distances follows by integrating the strict inequality for hyperbolic metrics. This version of the Schwarz-Pick Lemma can be stated in the following equivalent form. If f : Ω1 → Ω2 is holomorphic, then for all z and w in Ω1 , (6.3)

and (6.4)

dΩ2 (f (z), f (w)) ≤ dΩ1 (z, w), λΩ2 f (z) |f ′ (z)| ≤ λΩ1 (z).

Further, if either equality holds in (6.3) for a pair of distinct points or at one point z in (6.4) , then f is a conformal bijection of Ω1 onto Ω2 . Corollary 6.5 (Schwarz’s Lemma for Simply Connected Regions). Suppose Ω is a simply connected proper subregion of Ω and a ∈ Ω. If f is a holomorphic self-map of Ω that fixes a, then |f ′ (a)| ≤ 1 and equality holds if and only if f ∈ A(Ω, a), the group of conformal automorphisms of Ω that fix a. Moreover, f ′ (a) = 1 if and only if f is the identity. Theorem 6.4 is the fundamental reason for the existence of many distortion theorems in complex analysis. Consider the class of holomorphic maps of Ω1 into Ω2 . Then any such map f will have to satisfy the universal constraints (6.3) and (6.4) where the metrics λΩ1 and λΩ2 are uniquely determined (albeit implicitly) by the regions Ω1 and Ω2 . Thus (6.3) and (6.4) are, in some sense, the generic distortion theorems for holomorphic maps. This is the appropriate place to point out that neither the complex plane C nor the extended complex plane C∞ has a metric analogous to the hyperbolic metric in the sense that the metric is invariant under the group of conformal automorphisms. Recall that A(C) is the set of all maps z 7→ az + b, a, b ∈ C and a 6= 0, and A(C∞ ) is the group M of Möbius transformations. The group A(C) acts doubly transitively on C; that is, given two pairs z1 , z2 and w1 , w2 of distinct points in C there is a conformal automorphism f of C with f (zj ) = wj , j = 1, 2. Similarly, M acts triply transitively on C∞ . If there were a conformal metric on either C or C∞ invariant under the full conformal automorphism group, then

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the distance function induced from this metric would also be invariant under the action of the full group of conformal automorphisms. The following result shows that only trivial distance functions are invariant under A(C) or A(C∞ ). Theorem 6.6. If d is a distance function on C or C∞ that is invariant under the full group of conformal automorphisms, then there exists t > 0 such that d(z, w) = 0 if z = w and d(z, w) = t otherwise. Proof. Let d be a distance function on C that is invariant under A(C). Set t = d(0, 1). Consider any distinct z, w ∈ C. Because A(C) acts doubly transitively on C, there exists f ∈ A(C) with f (0) = z and f (1) = w. The invariance of d under A(C) implies d(z, w) = d(f (0), f (1)) = d(0, 1) = t. The same argument applies to C∞ . The Euclidean metric |dz| on C is invariant under the proper subgroup of A(C) given by z 7→ az + b, where |a| = 1 and b ∈ C. The spherical metric 2|dz|/(1 + |z|2 ) on C∞ is invariant under the group of rotations of C∞ , that is, Möbius maps of the form az − c¯ , a, c ∈ C, |a|2 + |c|2 = 1, z 7→ cz + a ¯ or of the equivalent form z−a z 7→ eiθ , θ ∈ R, a ∈ C∞ . 1+a ¯z The group of rotations of C∞ is a proper subgroup of M. Exercises. 1. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. Let F denote the family of all holomorphic functions f defined on D such that f (D) ⊆ Ω and f (0) = a. Set M = sup{|f ′ (0)| : f ∈ F}. Prove M < +∞ and that |f ′ (0)| = M if and only if f is a conformal map of D onto Ω with f (0) = a. Show M = 2/λΩ (a). 2. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. Let G denote the family of all holomorphic functions f defined on Ω such that f (Ω) ⊆ D. Set N = sup{|f ′ (a)| : f ∈ G}. Prove N < +∞ and that |f ′ (a)| = N if and only if f is a conformal map of Ω onto D with f (a) = 0. Show N = λΩ (a)/2. 3. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. Let H(Ω, a) denote the family of all holomorphic self-maps of Ω that fix a. Prove that {f ′ (a) : f ∈ H(Ω, a)} equals the closed unit disk.

7. Examples of the hyperbolic metric We give examples of simply connected regions and their hyperbolic metrics. These metrics are computed by using (6.2) in the following way: one finds an explicit conformal map f from the region Ω1 whose metric is sought onto a region


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Ω2 whose metric is known. Then (6.2) enables one to find an explicit expression for λΩ1 (z) for z in Ω1 . We omit almost all of the computations. The simplest instance of the Riemann Mapping Theorem is the fact that any disk or half-plane is Möbius equivalent to the unit disk. Because hyperbolic circles (disks) in D are Euclidean circle (disks) in D, we deduce that an analogous result holds for any disk or half-plane. Also, in any disk or half-plane hyperbolic geodesics are arcs of circles orthogonal to the boundary; in the case of a halfplanes we allow half-lines orthogonal to the edge of the half-plane. Example 7.1 (disk). As f (z) = (z − z0 )/R is a conformal map of the disk D = {z : |z − z0 | < R} onto D, we find λD (z)|dz| =

R2

2R |dz| . − |z − z0 |2

In particular, λD (z0 ) =

2 . R

Example 7.2 (half-plane). Let H be the upper half-plane {x+iy : y > 0}. Then g(H) = D, where g(z) = (z − i)/(z + i), so H = {x + iy : y > 0} has hyperbolic metric |dz| |dz| λH (z)|dz| = = . y Im z Similarly, the hyperbolic metric of the right half-plane K = {x + iy : x > 0} is |dz|/x. More generally, if H is any open half-plane, then λH (z)|dz| =

|dz| , d(z, ∂H)

where d(z, ∂H) denotes the Euclidean distance from z to ∂H. Theorem 7.3. If f : D → K is holomorphic and f (0) = 1, then (7.1) and (7.2)

1 − |z| 1 + |z| ≤ Re f (z) ≤ 1 + |z| 1 − |z| |Im f (z)| ≤

2|z| 1 − |z|2

Proof. This is an immediate consequence of the Schwarz-Pick Lemma after converting the conclusion into weaker Euclidean terms. Fix z ∈ D and set r = dD (0, z) = 2 tanh−1 |z| = log

1 + |z| . 1 − |z|

The Schwarz-Pick Lemma implies that f (z) lies in the closed hyperbolic disk ¯ K (1, r). The closed hyperbolic disk D ¯ K (1, r) has Euclidean center cosh r, EuD clidean radius sinh r and the bounding circle meets the real axis at e−r and er ;

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see the exercises. Therefore, f (z) lies in the closed Euclidean square {z = x+iy : e−r ≤ x ≤ er , |y| ≤ sinh r}. Since e−r = (7.1) is established. Finally,

1 − |z| 1 + |z|

and er =

sinh r = demonstrates (7.2).

1 + |z| , 1 − |z|

2|z| 1 − |z|2

Theorem 7.4. Suppose that H is any disk or half-plane. Then for all z and w in H, 1 sinh2 dH (z, w) = 41 |z − w|2 λH (z)λH (w). 2 Proof. It is easy to verify that for any Möbius map g we have 2 (7.3) g(z) − g(w) = (z − w)2 g ′ (z) g ′ (w).

Now take any Möbius map g that maps H onto D, and recall that g is an isometry from H to D if both are given their hyperbolic metrics. Then, using (2.8) and (7.3) 1 ′ 2 2 1 w = g(w) |g (z)| |g ′ (w)| z λ g(z) λ |z − w| λ |z − w| λ H D H D 4 4 = 41 |g(z) − g(w)|2 λD (g(z))λD (g(w)) 1 = sinh2 dD (g(z), g(w)) 2 1 = sinh2 dH z, w 2

There is another, less well known, version of the Schwarz-Pick Theorem available which is an immediate consequence of Theorem 7.4, and which we state in a form that is valid for all disks and half-planes. Theorem 7.5 (Modified Schwarz-Pick Lemma for Disks and Half-Planes). Suppose that Hj is any disk or half-plane, j = 1, 2, and that f : H1 → H2 is holomorphic. Then, for all z and w in H1 , |f (z) − f (w)|2 λH1 (z)λH1 (w) . ≤ 2 |z − w| λH2 f (z) λH2 f (w) Proof. By Theorem 7.4 and the Schwarz-Pick Lemma 1 1 |f (z) − f (w)|2 λH2 (f (z))λH2 (f (w)) = sinh2 dH2 (f (z), f (w)) 4 2 1 ≤ sinh2 dH1 (z, w) 2 1 = 4 |z − w)|2 λH1 z λH1 w .


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Observe that if w → z in Theorem 7.5, then we obtain (6.4) in the special case of disks and half-planes. We give an application of Theorem 7.5 to holomorphic functions. Example 7.6. Suppose that f is holomorphic in the open unit disk and that f has positive real part. Then f maps D into K, and we have 4 Re [f (z)] Re [f (w)] |f (z) − f (w)|2 ≤ . 2 |z − w| (1 − |z|2 )(1 − |w|2 )

This implies, for example, that if we also have f (0) = 1 then |f ′ (0)| ≤ 2. √ Example 7.7 (slit plane). Since f (z) = z maps P = C\{x ∈ R : x ≤ 0} onto K = {x + iy : x > 0}, the hyperbolic metric on P is λP (z) |dz| =

|dz| √ √ . 2| z| Re [ z]

This gives λP (z) =

1 1 ≥ , 2r cos(θ/2) 2|z|

where z = reiθ . Example 7.8 (sector). Let S(α) = {z : 0 < arg(z) < απ}, where 0 < α ≤ 2. Here, f (z) = z 1/α = exp α−1 log z is a conformal map of S(α) onto H, so S(α) has hyperbolic metric λS(α) (z) |dz| =

|z|1/α |dz|. α|z| Im[z 1/α ]

Note that this formula for the hyperbolic metric agrees (as it must) with the formula for λH in Example 7.2 (which is the case α = 1). The special case α = 2 is the preceding example. Example 7.9 (doubly infinite strip). S = {x + iy : |y| < π/2} has hyperbolic metric |dz| . λS (z) |dz| = cos y In this case we use the fact that ez maps S conformally onto K = {x + iy : x > 0}. Notice that λS (z) ≥ 1 with equality if and only if z lies on the real axis. In particular, the hyperbolic distance between points on R is the same as the Euclidean distance between the points. Theorem 7.10. Let S = {z : |Im(z)| < π/2}. Then for any a ∈ R and any holomorphic self-map f of S, |f ′ (a)| ≤ 1. Moreover, f ′ (a) = 1 if and only if f (z) = z + c for some c ∈ R and f ′ (a) = −1 if and only if f (z) = −z + c for some c ∈ R. In particular, for any interval [a, b] in R, the Euclidean length of the image f ([a, b]) is at most b − a.

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Proof. From Example 7.9 for z ∈ S λS (z) =

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1 ≥1 cos y

and equality holds if and only if Im(z) = 0. This observation together with the Schwarz-Pick Lemma gives |f ′ (a)| ≤ λS (f (a))|f ′ (a)| ≤ λS (a) = 1

and equality implies f (a) ∈ R. In this case, f −f (a)+a is a holomorphic self-map of S that fixes a and has derivative 1 at a, so it is the identity by the general form of Schwarz’s Lemma. Thus, from Corollary 6.5 f ′ (a) = 1 implies f (z) = z + c for some c ∈ R; the converse is trivial. If f ′ (a) = −1, then −f is a holomorphic self-map of S with derivative 1 at a, and so f (z) = −z − c for some c ∈ R. For a simply connected proper subregion Ω of C and a ∈ Ω, each hyperbolic disk DΩ (a, r) = {z ∈ Ω : dΩ (a, z) < r} is simply connected and the closed ¯ Ω (a, r) is compact. When Ω is a disk or half-plane, hyperbolic disks are disk D Euclidean disks since any conformal map of the unit disk onto a disk or half-plane is a Möbius transformation. Of course, this is no longer true when Ω is simply connected and not a disk or half-plane. For particular types of simply connected regions, more can be said about hyperbolic disks than just the fact that they are simply connected. Theorem 7.11. Suppose Ω is a convex hyperbolic region. Then for any a ∈ Ω and all r > 0 the hyperbolic disc DΩ (a, r) is Euclidean convex. Proof. Fix a ∈ Ω. Let h : D → Ω be a conformal mapping with h(0) = a. Since h(DD (0, r)) = DΩ (a, r), it suffices to show that h maps each disc DD (0, r) = D(0, tanh(r/2)) onto a convex set. Set R = tanh(r/2). Given b, c ∈ D(0, R) we must show (1 − t)h(b) + th(c) lies in h(D(0, R)) for t ∈ I. Choose S so that |b|, |c| < S < R and fix t ∈ I. The function cz bz g(z) = (1 − t)h + th S S

is holomorphic in D, g(0) = a and maps into Ω because Ω is convex. Therefore, f = h−1 ◦ g is a holomorphic self-map of D that fixes the origin and so f (D(0, R)) ⊆ D(0, R). Then (1 − t)h(b) + th(c) = g(S) = h(f (S)) lies in h(D(0, S)) because f (S) ∈ D(0, S). Therefore, h(D(0, S)) = DΩ (a, r) is Euclidean convex.

This result is effectively due to Study who proved that if f is convex univalent in D, then for any Euclidean disk D contained in D, f (D) is Euclidean convex, see [13]. The converse of Theorem 7.11 is elementary: If Ω is a simply connected proper subregion of C and there exists a ∈ Ω such that every hyperbolic disk DΩ (a, r) is Euclidean convex, then Ω is Euclidean convex since Ω = ∪{DΩ (a, r) : r > 0}, an increasing union of Euclidean convex sets. The radius of convexity √ for a univalent function on D is 2 − 3; see [13]. This implies that if Ω is simply


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connected, then for each a ∈ Ω and 0 < r < (1/2) log 3 the hyperbolic disk DΩ (a, r) is Euclidean convex. In a general simply connected region hyperbolic geodesics are no longer arcs of circles or segments of lines. It is possible to give a simple geometric property of hyperbolic geodesics in Euclidean convex regions that characterize convex regions, see [19] and [20]. Exercises. 1. Let K = {z = x + iy : x > 0}. For a > 0 and r > 0 verify that the ¯ K (1, r) is the Euclidean disk with Euclidean center closed hyperbolic disk D c = cosh r and Euclidean radius R = sinh r. This Euclidean disk meets the real axis at e−r and er . 2. Suppose f : D → K is holomorphic. Prove that (1 − |z|2 )|f ′ (z)| ≤ 2 Re f (z)

for all z ∈ D. When does equality hold? 3. Suppose f : K → D is holomorphic. Prove that

2|f ′ (z)|Re z ≤ 1 − |f (z)|2

for all z ∈ K. When does equality hold? 4. Suppose Ω is a simply connected proper subregion of C that is (Euclidean) starlike with respect to a ∈ Ω. This means that for each z ∈ Ω the Euclidean segment [a, z] is contained in Ω. For any r > 0 prove that the hyperbolic disk DΩ (a, r) is starlike with respect to a.

8. The Comparison Principle There is a powerful, and very general, Comparison Principle for hyperbolic metrics, which we state here only for simply connected plane regions. This Principle allows us to estimate the hyperbolic metric of a region in terms of other hyperbolic metrics which are known, or which can be more easily estimated. In general it is not possible to explicitly calculate the density of the hyperbolic metric, so estimates are useful. Theorem 8.1 (Comparison Principle). Suppose that Ω1 and Ω2 are simply connected proper subregions of C. If Ω1 ⊆ Ω2 then λΩ2 ≤ λΩ1 on Ω1 . Further, if λΩ1 (z) = λΩ2 (z) at any point z of Ω2 , then Ω1 = Ω2 and λΩ1 = λΩ2 . Proof. Let f (z) = z be the inclusion map of Ω1 into Ω2 . Then the Schwarz-Pick Lemma gives λΩ2 (z) ≤ λΩ1 (z). If equality holds at a point, then f is a conformal bijection of Ω1 onto Ω2 , that is, Ω1 = Ω2 . In other words, the Comparison Principle asserts that the hyperbolic metric on a simply connected region decreases as the region increases. The hyperbolic metric on the disk Dr = {z : |z| < r} is 2r|dz|/(r2 − |z|2 ) which decreases to zero as r increases to +∞.

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The Comparison Principle is used in the following way. Suppose that we want to estimate the hyperbolic metric λΩ of a region Ω. We attempt to find regions Ωj with known hyperbolic metrics (or metrics that can be easily estimated) such that Ω1 ⊆ Ω ⊆ Ω2 ; then λΩ2 ≤ λΩ ≤ λΩ1 . The next result is probably the simplest application of the Comparison Principle, and it gives an upper bound of the hyperbolic metric λΩ of a region Ω in terms of the Euclidean distance d(z, ∂Ω) = inf{|z − w| : w ∈ ∂Ω} of z to the boundary of Ω. The geometric significance of this quantity is that d(z, ∂Ω) is the radius of the largest open disk with center z that lies in Ω. Note, however, that d(z, ∂Ω) (which is sometimes denoted by δΩ (z) in the literature) is not conformally invariant. The metric |dz| |dz| = d(z, ∂Ω) δΩ (z) is called the quasihyperbolic metric on Ω. Example 7.2 shows that the quasihyperbolic metric for a half-plane is the hyperbolic metric. Theorem 8.2. Suppose that Ω is a simply connected proper subregion of C. Then for all z ∈ Ω (8.1)

λΩ (z) ≤

2 , d(z, ∂Ω)

and equality holds if and only if Ω is a disk with center z. Proof. Take any z0 in Ω, and let R = d(z0 , ∂Ω) and D = {z : |z − z0 | < R}. As D ⊆ Ω the Comparison Principle and Example 7.1 yield λΩ (z0 ) ≤ λD (z0 ) =

2 2 = , R d(z0 , ∂Ω)

which is (8.1). If λΩ (z0 ) = 2/d(z0 , ∂Ω) then λΩ (z0 ) = λD (z0 ) so, by the Comparison Principle, Ω = D. The converse is trivial. Theorem 8.2 gives an upper bound on the hyperbolic metric of Ω in terms of the Euclidean quantity d(z, ∂Ω). It is usually more difficult to obtain a lower bound on the hyperbolic metric. For convex regions it is easy to use geometric methods to obtain a good lower bound on the hyperbolic metric. Theorem 8.3. Suppose that Ω is convex proper subregion of C. Then for all z∈Ω 1 ≤ λΩ (z) (8.2) d(z, ∂Ω) and equality holds if and only if Ω is a half-plane. Proof. We suppose that Ω is convex. Take any z in Ω and let ζ be one of the points on ∂Ω that is nearest to z. Let H be the supporting half-plane of Ω at ζ;


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thus Ω ⊆ H, and the Euclidean line that bounds H is orthogonal to the segment from z to ζ. Thus, from the Comparison Principle, for any z ∈ Ω 1 1 λΩ (z) ≥ λH (z) = = |z − ζ| d(z, ∂Ω) which is (8.2). The equality statement follows from the Comparison Principle and Example 7.2.

Theorems 8.2 and 8.3 show that the hyperbolic and quasihyperbolic metrics are bi-Lipschitz equivalent on convex regions: 1 2 ≤ λΩ (z) ≤ . d(z, ∂Ω) d(z, ∂Ω) Lower bounds for the hyperbolic metric in terms of the quasihyperbolic metric are equivalent to covering theorems for univalent functions. Theorem 8.4. Suppose that f is holomorphic and univalent in D, and that f (D) is a convex region. Then f (D) contains the Euclidean disk with center f (0) and radius |f ′ (0)|/2. Proof. From Theorem 8.3 we have 2 = λD (0)

= λf (D) f (0) |f ′ (0)|

≥ We deduce that

|f ′ (0)| . d f (0), ∂f (D)

d f (0), ∂f (D) ≥ |f ′ (0)|/2, so that f (D) contains the Euclidean disk with center f (0) and radius |f ′ (0)|/2. There is an analogous covering theorem for general univalent functions on the unit disk, see [13]. Theorem 8.5 (Koebe 1/4–Theorem). Suppose that f is holomorphic and univalent in D. Then the region f (D) contains the open Euclidean disk with center f (0) and radius |f ′ (0)|/4. The Koebe 1/4–Theorem gives a lower bound on the hyperbolic metric in terms of the quasihyperbolic metric on a simply connected proper subregion of C. Theorem 8.6. Suppose that Ω is a simply connected proper subregion of C. Then for all z ∈ Ω 1 ≤ λΩ (z) (8.3) 2d(z, ∂Ω) and equality holds if and only if Ω is a slit-plane.

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Proof. Fix z ∈ Ω and let f : D → Ω be a conformal map with f (0) = z. Koebe’s 1/4–Theorem implies d(z, ∂Ω) ≥ |f ′ (0)|/4. Now 2 = λΩ (f (0))|f ′ (0)|

≤ 4λΩ (z)d(z, ∂Ω) which establishes (8.3). Sharpness follows from the sharp form of the Koebe 1/4–Theorem and Example 7.7. Theorems 8.2 and 8.6 show that the hyperbolic and quasihyperbolic metrics are bi-Lipschitz equivalent on simply connected regions: (8.4)

2 1 ≤ λΩ (z) ≤ . 2d(z, ∂Ω) d(z, ∂Ω)

Exercises. 1. (a) Suppose Ω is a simply connected proper subregion of C. Prove that limz→ζ λΩ (z) = +∞ for each boundary point ζ of Ω that lies in C. (b) Given an example of a simply connected proper subregion Ω of C that has ∞ as a boundary point and λΩ (z) does not tend to infinity as z → ∞. 2. Suppose Ω is starlike with respect to the origin; that is, for each z ∈ Ω the Euclidean segment [0, z] is contained in Ω. Use the Comparison Theorem to prove that (8.3) holds; do not use Theorem 8.6.

9. Curvature and the Ahlfors Lemma A conformal semimetric on a region Ω in C is ρ(z)|dz|, where ρ : Ω → [0, +∞) is a continuous function and {z : ρ(z) = 0} is a discrete subset of Ω. A conformal semimetric ρ(z)|dz| is a conformal metric if ρ(z) > 0 for all z ∈ Ω. The curvature of a conformal semimetric ρ(z)|dz| can be defined at any point where ρ is positive and of class C2 . Definition 9.1. Suppose ρ(z)|dz| is a conformal metric on a region Ω. If a ∈ Ω, ρ(a) > 0 and ρ(z) is of class C2 at a, then the Gaussian curvature of ρ(z)|dz| at a is △ log ρ(a) Kρ (a) = − , ρ2 (a) where △ is the usual (Euclidean) Laplacian, △=

∂2 ∂2 + . ∂x2 ∂y 2

Typically, we shall speak of the curvature of a conformal metric rather than Gaussian curvature. In computing the Laplacian it is often convenient to use △=4

∂2 ∂2 =4 , ∂z∂ z¯ ∂ z¯∂z


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where the complex partial derivatives are defined by 1 ∂ ∂ ∂ = −i , ∂= ∂z 2 ∂x ∂y 1 ∂ ∂ ∂ ¯ . = +i ∂= ∂ z¯ 2 ∂x ∂y Alternatively the Laplacian expressed can be expressed in terms of polar coordinates, namely ∂2 1 ∂ 1 ∂2 △= 2 + + 2 2. ∂r r ∂r r ∂θ One reason semimetrics play an important role in complex analysis is that they transform simply under holomorphic functions. Definition 9.2. Suppose ρ(w)|dw| is a semimetric on a region Ω and f : ∆ → Ω is a holomorphic function. The pull-back of ρ(w)|dw| by f is (9.1)

f ∗ (ρ(w)|dw|) = ρ(f (z))|f ′ (z)||dz|.

Since (ρ ◦ f )|f ′ | is a continuous nonnegative function defined on ∆, the pullback f ∗ (ρ(w)|dw|) of ρ(w)|dw| is a semimetric on ∆ provided f is nonconstant. Sometimes we write simply f ∗ (ρ) to denote the pull-back. However, the notation (9.1) is precise and indicates that the formal substitution w = f (z) converts ρ(w)|dw| into f ∗ (ρ(w)|dw|). The pull-back operation has several useful properties: (f ◦ g)∗ (ρ(w)|dw|) = g ∗ (f ∗ (ρ(w)|dw|)) and (f −1 )∗ = (f ∗ )−1 , when f is a conformal mapping. If f : Ω1 → Ω2 is a conformal mapping of simply connected proper subregions of C, then the conclusion of Theorem 6.3 in the pull-back notation is: f ∗ (λΩ2 ) = λΩ1 . In the context of complex analysis, a fundamental property of the curvature is its conformal invariance. More generally, curvature is invariant under the pull-back operation. Theorem 9.3. Suppose Ω and ∆ are regions in C, ρ(w)|dw| is a metric on Ω and f : ∆ → Ω is a holomorphic function. Suppose a ∈ ∆, f ′ (a) 6= 0, ρ(f (a)) > 0 and ρ is of class C2 at f (a). Then Kf ∗ (ρ) (a) = Kρ (f (a)). Proof. Recall that f ∗ (ρ(w)|dw|) = ρ(f (z))|f ′ (z)||dz|. Now, log (ρ(f (z))|f ′ (z)|) = log ρ(f (z)) + log |f ′ (z)| 1 1 = log ρ(f (z)) + log f ′ (z) + log f ′ (z), 2 2 so that

∂ log ρ 1 f ′′ (z) ∂ log (ρ(f (z))|f ′ (z)|) = (f (z))f ′ (z) + . ∂z ∂w 2 f ′ (z)

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Then ∂2 ∂ 2 log ρ log (ρ(f (z))|f ′ (z)|) = (f (z))f ′ (z)f ′ (z) ∂ z¯∂z ∂ w∂w ¯ ∂ 2 log ρ (f (z))|f ′ (z)|2 = ∂ w∂w ¯ gives △z [log (ρ(f (z))|f ′ (z)|)] = (△w log ρ) (f (z))|f ′ (z)|2 .

This is the transformation law for the Laplacian under a holomorphic function. Consequently, △z log(ρ(f (a))|f ′ (a)|) ρ2 (f (a))|f ′ (a)|2 (△w log ρ) (f (a))|f ′ (a)|2 =− ρ2 (f (a))|f ′ (a)|2 = Kρ (f (a)).

Kf ∗ (ρ) (a) = −

Theorem 9.4. The hyperbolic metric on a simply connected proper subregion of C has constant curvature −1. Proof. First, we establish the result for the unit disk. From λD (z) = we obtain

2 2 = 2 1 − |z| 1 − z z¯

∂2 2 ∂2 log =− log(1 − z z¯) ∂ z¯∂z 1 − z z¯ ∂ z¯∂z z¯ ∂ = ∂ z¯ 1 − z z¯ 1 = . (1 − z z¯)2 Consequently, KλD (z) = −1.

The general case of the hyperbolic metric on a simply connected proper subregion Ω of C follows from Theorem 9.3 since f ∗ (λD (w)|dw|) = λΩ (z)|dz| for any conformal map f : Ω → D. Ahlfors recognized that the Schwarz-Pick Lemma was a consequence of an extremely important maximality property of the hyperbolic metric in D. Theorem 9.5 (Maximality of the hyperbolic metric). Suppose ρ(z)|dz| is a C2 semimetric on a simply connected proper subregion Ω of C such that Kρ (z) ≤ −1 whenever ρ(z) > 0. Then ρ ≤ λΩ on Ω.


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Proof. First, we assume Ω = D. Given z0 in D choose any r satisfying |z0 | < r < 1. The hyperbolic metric on the disk Dr = {z : |z| < r} is 2r . λr (z) = 2 r − |z|2

Consider the function

ρ(z) λr (z) which is defined on the disk Dr . Then v(z) ≥ 0 when |z| < r, and v(z) → 0 as |z| → r, so that v attains its maximum at some point a in Dr . It suffices to show that v(a) ≤ 1 for then v(z) ≤ 1 on Dr and we have 2r ρ(z0 ) ≤ 2 . r − |z0 |2 v(z) =

By letting r → 1 we find that ρ(z0 ) ≤ λD (z0 ).

We now show that v(a) ≤ 1. If ρ(a) = 0, then v(a) = 0 < 1. Otherwise, ρ(a) > 0 and Kρ (a) ≤ −1. As a is a local maximum of v, it is also a local maximum of log v so that ∂ 2 log v (a) ≤ 0, ∂x2

We deduce that

∂ 2 log v (a) ≤ 0. ∂y 2

0 ≥ △ log v (a) = △ log ρ (a) − △ log λr (a)

= −Kρ (a)ρ(a)2 + Kλr (a)λr (a)2

≥ ρ(a)2 − λr (a)2 .

(9.2)

This implies that v(a) ≤ 1, and completes the proof in the special case Ω = D.

We now turn to the general case. Let h : D → Ω be a conformal mapping. Then h∗ (ρ(w)|dw|) := τ (z)|dz| is a C2 semimetric on D such that Kτ (z) ≤ −1 whenever τ (z) > 0. Hence, ρ(h(z))|h′ (z)| ≤ λD (z) = λΩ (h(z))|h′ (z)|,

and so ρ ≤ λΩ on Ω.

In fact, Ahlfors actually established a more general result (see [1] and [2]). The stronger conclusion that either ρ < λΩ or else ρ = λΩ is valid but less elementary. This sharp result was established by Heins [15]. Simpler proofs of the stronger conclusion are due to Chen [12], Minda [28] and Royden [32]. The Schwarz-Pick Lemma is a special case of Theorem 9.5. If f : Ω1 → Ω2 is a nonconstant holomorphic function, then f ∗ (λΩ2 (w)|dw|) is a semimetric on Ω1 with curvature −1 at each point where f ′ is nonvanishing, so is dominated by the hyperbolic metric λΩ1 (z)|dz|, or equivalently, (6.4) holds. The equality statement associated with (6.4) follows from the sharp version of Theorem 9.5.

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Theorem 9.6. There does not exist a C2 semimetric ρ(z)|dz| on C such that Kρ (z) ≤ −1 whenever ρ(z) > 0. Proof. Suppose there existed a semimetric ρ(z)|dz| on C such that Kρ (z) ≤ −1 whenever ρ(z) > 0. Theorem 9.5 applied to the restriction of this metric to the disk {z : |z| < r} gives 2r (9.3) ρ(z) ≤ λr (z) = 2 r − |z|2 for |z| < r. If we fix z and let r → +∞, (9.3) gives ρ(z) = 0 for all z ∈ C. This contradicts the fact that a semimetric vanishes only on a discrete set. Corollary 9.7 (Liouville’s Theorem). A bounded entire function is constant. Proof. Suppose f is a bounded entire function. There is no harm in assuming that |f (z)| < 1 for all z ∈ C. If f were nonconstant, then f ∗ (λD (z)|dz|) would be a semimetric on C with curvature at most −1, a contradiction. Theorem 9.3 provides a method to produce metrics with constant curvature −1. Loosely speaking, bounded holomorphic functions correspond to metrics with curvature −1. If f : Ω → D is holomorphic and locally univalent (f ′ does not vanish), then f ∗ (λD (z)|dz|) has curvature −1 on Ω. In fact, on a simply connected proper subregion of C every metric with curvature −1 has this form; see [36]. This reference also contains a stronger result that represents certain semimetrics with curvature −1 at points where the semimetric is nonvanishing by holomorphic (not necessarily locally univalent) maps of Ω into D. Theorem 9.8 (Representation of Negatively Curved Metrics). Let ρ(z)|dz| be a C3 conformal metric on a simply connected proper subregion Ω of C with constant curvature −1. Then ρ(z)|dz| = f ∗ (λD (w)|dw|) for some locally univalent holomorphic function f : Ω → D. The function f is unique up to post-composition with an isometry of the hyperbolic metric. Given a ∈ Ω the function f representing the metric is unique if f is normalized by f (a) = 0 and f ′ (a) > 0. Moreover, ρ(z)|dz| = f ∗ (λD (w)|dw|) is complete if and only if f is a conformal bijection; that is, the hyperbolic metric is the only conformal metric on Ω that has curvature −1 and is complete.

Exercises.

1. Determine the curvature of the Euclidean metric |dz| and of the spherical metric σ(z)|dz| = 2|dz|/(1 + |z|2 ). 2. Show that (1 + |z|2 )|dz| has negative curvature on C. 3. Determine the curvature of ex |dz| on C. 4. Determine the curvature of |dz|/|z| on C \ {0}. 5. Prove there does not exist a semimetric on C \ {0} with curvature at most −1. 6. Prove the following extension of Liouville’s Theorem: If f is an entire function and f (C) ⊆ Ω, where Ω is a simply connected proper subregion of C, then f is constant.


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10. The hyperbolic metric on a hyperbolic region In order to transfer the hyperbolic metric from the unit disk to nonsimply connected regions, a substitute for the Riemann Mapping Theorem is needed. For this reason we must understand holomorphic coverings. For the general theory of topological covering spaces the reader should consult [22]. A holomorphic function f : ∆ → Ω is called a covering if each point b ∈ Ω has an open neighborhood V such that f −1 (V ) = ∪{Uα : α ∈ A} is a disjoint union of open sets Uα such that f |Uα , the restriction of f to Uα , is a conformal map of Uα onto V . Trivially, a conformal mapping f : ∆ → Ω is a holomorphic covering. If Ω is simply connected, then the only holomorphic coverings f : ∆ → Ω are conformal maps of a simply connected region ∆ onto Ω. Example 10.1. The complex exponential function exp : C → C \ {0} is a holomorphic covering. Consider any w ∈ C \ {0} and let θ = arg w be any argument for w. Let V = C \ {−reiθ : r ≥ 0} be the complex S plane slit from the origin along the ray opposite from w. Then exp−1 (V ) = {Sn : n ∈ Z}, where Sn = {z : θ − nπ < Im z < θ + nπ}. Note that exp maps each horizontal strip Sn of width 2π conformally onto V . A region Ω is called hyperbolic provided C∞ \ Ω contains at least three points. The unit disk covers every hyperbolic plane region; that is, there is a holomorphic covering h : D → Ω for any hyperbolic region Ω. As a consequence we demonstrate that every hyperbolic region has a hyperbolic metric that is real-analytic with constant curvature −1. Theorem 10.2 (Planar Uniformization Theorem). Suppose Ω is a region in C. There exists a holomorphic covering f : D → Ω if and only if Ω is a hyperbolic region. Moreover, if a ∈ Ω, then there is a unique holomorphic universal covering h : D → Ω with h(0) = a and h′ (0) > 0. For a proof of the Planar Uniformization see [14] or [34]. If Ω is a simply connected hyperbolic region, then any holomorphic universal covering h : D → Ω is a conformal mapping. Therefore, the Riemann Mapping Theorem is a consequence of the Planar Uniformization Theorem. When Ω is a nonsimply connected hyperbolic region, then a holomorphic covering h : D → Ω is never injective. In fact, for each a ∈ Ω, h−1 (a) is an infinite discrete subset of D. If h : D → Ω is one holomorphic universal covering, then {h◦g : g ∈ A(D)} is the set of all holomorphic universal coverings of D onto Ω. The Planar Uniformization Theorem enables us to project the hyperbolic metric from the unit disk to any hyperbolic region. Theorem 10.3. Given a hyperbolic region Ω there is a unique metric λΩ (w)|dw| on Ω such that h∗ (λΩ (w)|dw|) = λD (z)|dz| for any holomorphic universal covering h : D → Ω. The metric λΩ (w)|dw| is real-analytic with curvature −1. Proof. We construct a metric with curvature −1 on any hyperbolic region. First we define the metric locally. For a hyperbolic region Ω, let h : D → Ω be

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a holomorphic covering. A metric is defined on Ω as follows. For any simply connected subregion U of Ω let H = h−1 denote a branch of the inverse that is holomorphic on U . Set λΩ (z) = λD (H(z))|H ′ (z)|. This defines a metric with curvature −1 on U . In fact, this defines a metric on Ω. Suppose U1 and U2 are simply connected subregions of Ω and U1 ∩ U2 is nonempty. Let Hj be a singlevalued holomorphic branch of h−1 defined on Uj . Then there is a g ∈ A(D) such that H2 = g ◦ H1 locally on U1 ∩ U2 . Hence, H2∗ (λD (z)|dz|) = (g ◦ H1 )∗ (λD (z)|dz|) = H1∗ (g ∗ (λD (z)|dz|)) = H1∗ (λD (z)|dz|) since each conformal automorphism of D is an isometry of the hyperbolic metric λD (z)|dz|. Therefore, λΩ (z) is defined independently of the branch of h−1 that is used and h∗ (λΩ ) = λD . Moreover, this metric is independent of the covering. Suppose k : D → Ω is another covering. Then k = h ◦ g for some g ∈ A(D), and so k ∗ (λΩ ) = (h ◦ g)∗ (λΩ ) = g ∗ (h∗ (λΩ ))

= g ∗ (λD ) = λD . That λΩ is real-analytic is clear from its construction. That the curvature is −1 follows from h∗ (λΩ ) = λD and Theorems 9.3 and 9.4. The unique metric λΩ (w)|dw| on a hyperbolic region Ω given by Theorem 10.3 is called the hyperbolic metric on Ω. The hyperbolic distance on a hyperbolic region is complete. The hyperbolic distance dΩ is defined by dΩ (z, w) = inf ℓΩ (γ), where the infimum is taken over all piecewise smooth paths γ in Ω that joining z and w. Unlike the case of simply connected regions, a holomorphic covering f : D → Ω onto a multiply connected hyperbolic region is not an isometry, but only a local isometry. That is, each point a ∈ Ω has a neighborhood U such that f |U is an isometry. In general, one can only assert that dΩ (f (z), f (w)) ≤ dD (z, w) for z, w ∈ D. When Ω is multiply connected, then f is not injective, so there exist distinct z, w ∈ D with f (z) = f (w). In this situation, dΩ (f (z), f (w)) = 0 < dD (z, w). In general, the hyperbolic metric is not just invariant under conformal mappings, but is invariant under holomorphic coverings. Theorem 10.4 (Covering Invariance). If f : ∆ → Ω is a holomorphic covering of hyperbolic regions, then f ∗ (λΩ (w)|dw|) = λ∆ (z)|dz|. In other words, every holomorphic covering of hyperbolic regions is a local isometry.


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Proof. Let h : D → ∆ be a holomorphic covering. Then k = f ◦ h : D → Ω is also a holomorphic covering, so λD = k ∗ (λΩ ) = h∗ (f ∗ (λΩ )). Thus, f ∗ (λΩ ) is a conformal metric on ∆ whose pull-back to the unit disk by a covering projection is λD , so f ∗ (λΩ ) is the hyperbolic metric on ∆ by Theorem 10.3. Theorem 10.4 implies that every h ∈ A(Ω) is an isometry of the hyperbolic metric, and more generally, each holomorphic self-covering h of Ω is a local isometry of the hyperbolic metric. A hyperbolic region can have self-coverings that are not conformal automorphisms, see Section 12. The maximality property of the hyperbolic metric given in Theorem 9.5 remains valid for hyperbolic regions. As noted after the proof of Theorem 9.5 this means that a version of the Schwarz-Pick Lemma holds for holomorphic maps between hyperbolic regions. In order to establish a sharp result, we provide an independent proof. Theorem 10.5 (Schwarz-Pick Lemma - general version). Suppose ∆ and Ω are hyperbolic regions and f : ∆ → Ω is holomorphic. Then for all z ∈ ∆, (10.1)

λΩ (f (z))|f ′ (z)| ≤ λ∆ (z).

If f : ∆ → Ω is a covering projection, then λΩ (f (z))|f ′ (z)| = λ∆ (z) for all z ∈ ∆. If there exists a point in ∆ such that equality holds in (10.1), then f is a covering. Proof. Let k : D → ∆ and h : D → Ω be holomorphic coverings. The function f ◦ k : D → Ω lifts relative to h to a holomorphic function F : D → D. Then f ◦ k = h ◦ F and the Schwarz-Pick Lemma for the unit disk give k ∗ (f ∗ (λΩ )) = (f ◦ k)∗ (λΩ ) = (h ◦ F )(λΩ ) = F ∗ (h∗ (λΩ )) = F ∗ (λD ) ≤ λD

= k ∗ (λ∆ ). Because k is a surjective local homeomorphism, the inequality k ∗ (f ∗ (λΩ )) ≤ k ∗ (λ∆ ) gives the inequality (10.1). Because h and k are coverings, f is a covering if and only if F is a covering. This observation establishes the sharpness. We need to establish a result about holomorphic self-coverings of a hyperbolic region that have a fixed point in order to obtain a good analog of Schwarz’s Lemma for hyperbolic regions.

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Theorem 10.6. A self-covering of a hyperbolic region that fixes a point is a conformal automorphism. In particular, if a hyperbolic region Ω is not simply connected and a ∈ Ω, then A(Ω, a) is isomorphic to the group of nth roots of unity for some positive integer n. Proof. Suppose Ω is a hyperbolic region, a ∈ Ω and f is a self-covering of Ω that fixes a. We prove f is a conformal automorphism. The result is trivial if Ω is simply connected since every covering of a simply connected region is a homeomorphism. Let h : D → Ω be a holomorphic universal covering with h(0) = a and h′ (0) > 0. Because a covering is surjective, it suffices to prove f is injective. Let f˜ be the lift of f ◦ h relative to h that satisfies f˜(0) = 0. Since h and f ◦ h are coverings, so is f˜. Because D is simply connected, f˜ is a conformal automorphism of D. Then f˜(z) = eiθ z for some θ ∈ R. Because Ω is not simply connected, the fiber h−1 (a) contains infinitely many points besides 0. As this fiber is a discrete subset of D, the nonzero elements of h−1 (a) have a minimum positive modulus r; say h−1 (a) ∩ {z : |z| = r} = {aj : j = 1, . . . , m}. From f˜(h−1 (a)) = h−1 (a), we conclude that f˜ induces a permutation of the set {aj : j = 1, . . . , m}. Therefore, there exists n ≤ m! such that fñ is the identity. Then f n ◦h = h◦ fñ = h and so f n is the identity. If n = 1, then f is the identity. If n ≥ 2, then f n = I, the identity, implies f is a conformal automorphism of Ω with inverse f n−1 . This argument shows that if Ω is not simply connected, then there is a nonnegative integer m such that for all f ∈ A(Ω, a), f m is the identity. Therefore, f ′ (a)m = 1, or f ′ (a) is an mth root of unity. Thus, f 7→ f ′ (a) defines a homomorphism of A(Ω, a) into the unit circle T and the image is a subgroup of the mth roots of unity. Hence, A(Ω, a) is a finite group isomorphic to the group of nth roots of unity for some positive integer n. Example 10.7. Let AR = {z : 1/R < |z| < R}, where R > 1. The group A(AR , 1) has order two; the only conformal automorphisms of AR that fix 1 are the identity map and f (z) = 1/z. Corollary 10.8 (Schwarz’s Lemma - General Version). Suppose Ω is a hyperbolic region, a ∈ Ω and f is a holomorphic self-map of Ω that fixes a. Then |f ′ (a)| ≤ 1 and equality holds if and only if f ∈ A(Ω, a), the group of conformal automorphisms of Ω that fix a. Moreover, f ′ (a) = 1 if and only if f is the identity. Proof. The Schwarz-Pick Lemma implies |f ′ (a)| ≤ 1 with equality if and only if f is a self-covering of Ω that fixes a. Each f ∈ A(Ω, a) is a covering, so |f ′ (a)| = 1. If f is a self-covering of Ω that fixes a, then f ∈ A(Ω, a) by Theorem 10.6. We use the proof of Theorem 10.6 to verify that f ′ (a) = 1 implies f is the identity. Let f˜ be the lift of f ◦ h as in the proof of Theorem 10.6. Then h ◦ f = f ◦ h gives 1 = f ′ (a) = f˜′ (0), so f˜ is the identity. This implies f is the identity. Picard established a vast generalization of Liouville’s Theorem.


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Theorem 10.9 (Picard’s Small Theorem). If an entire function omits two finite complex values, then f is constant. Proof. Suppose f is an entire function and f (C) ⊆ C\{a, b} := Ca,b , where a and b are distinct complex numbers. We derive a contradiction if f were nonconstant. The region Ca,b is hyperbolic; let λa,b (z)|dz| denote the hyperbolic metric on Ca,b . If f were nonconstant, then f ∗ (λa,b (z)|dz|) would be a semi-metric on C with curvature at most −1; this contradicts Theorem 9.6. Exercises. 1. Verify that f (z) = exp(iz) is a covering of the upper half-plane H onto the punctured disk D \ {0}. 2. Verify that for each nonzero integer n the function pn (z) = z n defines a holomorphic covering of the punctured plane C \ {0} onto itself. 3. Verify that for each positive integer n the function pn (z) = z n defines a holomorphic covering of the punctured disk D \ {0} onto itself. 4. Suppose Ω is a hyperbolic region and a ∈ Ω. Let F denote the family of all holomorphic functions f : D → Ω such that f (0) = a and set M = sup{|f ′ (0)| : f ∈ F}. Prove M is finite and for f ∈ F, |f ′ (0)| = M if and only if f is a holomorphic covering of D onto Ω. Conclude that M = 2/λΩ (a). 5. Suppose Ω is a hyperbolic region in C and a, b ∈ Ω are distinct points. If f is a holomorphic self-map of Ω that fixes a and b, prove f is a conformal automorphism of Ω with finite order. Give an example to show that f need not be the identity when Ω is not simply connected.

11. Hyperbolic distortion In Section 5 the hyperbolic distortion of a holomorphic self-map of the unit disk was introduced. We now define an analogous concept for holomorphic maps of hyperbolic regions. Definition 11.1. Suppose Ω and ∆ are hyperbolic regions and f : ∆ → Ω is holomorphic. The (local) hyperbolic distortion factor for f at z is f

∆,Ω

λΩ (f (z))|f ′ (z)| dΩ (f (z), f (w)) (z) := = lim . w→z λ∆ (z) d∆ (z, w)

If Ω = ∆, write f ∆ in place of f ∆,Ω . The hyperbolic distortion factor defines a mapping of ∆ into the closed unit disk by the Schwarz-Pick Lemma. If f is not a covering, then the hyperbolic distortion factor gives a map of ∆ into the unit disk. There is a Schwarz-Pick type of result for the hyperbolic distortion factor which extends Corollary 5.7 to holomorphic maps between hyperbolic regions.

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Theorem 11.2 (Schwarz-Pick Lemma for Hyperbolic Distortion). Suppose ∆ and Ω are hyperbolic regions and f : ∆ → Ω is holomorphic. If f is not a covering, then dD (f ∆,Ω (z), f ∆,Ω (w)) ≤ 2d∆ (z, w)

(11.1) for all z, w ∈ ∆.

Proof. Fix w ∈ Ω. Let h : D → ∆ and k : D → Ω be holomorphic coverings with h(0) = w and k(0) = f (w). Then there is a lift of f to a self-map f˜ of D such that k ◦ f˜ = f ◦ h. f˜ is not a conformal automorphism of D because f is not a covering of ∆ onto Ω. We begin by showing that f˜D (˜ z ) = f ∆,Ω (h(˜ z )) for ′ ˜ all z˜ in D. From k ◦ f = f ◦ h and λD (˜ z ) = λ∆ (h(˜ z ))|h (˜ z )| we obtain f ∆,Ω (h(˜ z )) = = =

λΩ (f (h(˜ z )))|f ′ (h(˜ z ))| λ∆ (h(˜ z )) λΩ (k(f˜(˜ z )))|k ′ (f˜(˜ z ))||f˜′ (˜ z )|

λ∆ (h(˜ z ))|h′ (˜ z )| λD (f˜(˜ z ))|f˜′ (˜ z )| λD (z) ˜D

= f (˜ z ). Now we establish (11.1). For z ∈ Ω there exists z˜ ∈ h−1 (z) with dD (0, z˜) = dΩ (w, z). Then dD (f ∆,Ω (z), f ∆,Ω (w)) = dD (f ∆,Ω (h(˜ z )), f ∆,Ω (h(0))) = dD (f˜D (˜ z ), f˜D (0)) ≤ 2dD (˜ z , 0)

= 2d∆ (z, w).

Corollary 11.3. Suppose ∆ and Ω are hyperbolic regions. Then for any holomorphic function f : ∆ → Ω, f ∆,Ω (z) ≤

(11.2)

f ∆,Ω (w) + tanh d∆ (z, w) . 1 + f ∆,Ω (w) tanh d∆ (z, w)

for all z, w ∈ ∆. Proof. Inequality (11.2) is trivial when f is a covering since both sides are identically one, Thus, it suffices to establish the inequality when f is not a covering of ∆ onto Ω. Then dD (0, f ∆,Ω (z)) ≤ dD (0, f ∆,Ω (w)) + dD (f ∆,Ω (z), f ∆,Ω (w)) ≤ dD (0, f ∆,Ω (w)) + 2d∆ (z, w)


gives f

∆,Ω

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1 ∆,Ω dD (0, f (z)) (z) = tanh 2 1 ∆,Ω ≤ tanh dD (0, f (w)) + d∆ (z, w) 2 f ∆,Ω (w) + tanh d∆ (z, w) = . 1 + f ∆,Ω (w) tanh d∆ (z, w)

Exercises. 1. For a holomorphic function f : D → K, explicitly calculate f D,K (z). 2. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. Let H(Ω, a) denote the set of holomorphic self-maps of Ω that fix a. Prove that {f Ω (a) : f ∈ H(Ω, a)} is the closed unit interval [0, 1].

12. The hyperbolic metric on a doubly connected region There is a simple conformal classification of doubly connected regions in C∞ . If Ω is a doubly connected region in C∞ , then Ω is conformally equivalent to exactly one of: (a) C∗ = C \ {0}, (b) D∗ = D \ {0}, or (c) an annulus A(r, R) = {z : r < |z| < R}, where 0 < r < R. In the first case Ω itself is the extended plane C∞ punctured at two points and so is not hyperbolic. In this section we calculate the hyperbolic metric for the punctured unit disk D∗ and for the annulus AR = {z : 1/R < |z| < R}, where R > 1. 12.1. Hyperbolic metric on the punctured unit disk. To determine the hyperbolic metric on D∗ we make use of a holomorphic covering from H onto D∗ and Theorem 10.4. The function h(z) = exp(iz) is a holomorphic covering from H onto D∗ . Therefore, the density of the hyperbolic metric on D∗ is 1 λD∗ (z) = . |z| log(1/|z|) For simply connected hyperbolic regions the only hyperbolic isometries of the hyperbolic metric are conformal automorphisms. For multiply connected regions there can be self-coverings that leave the hyperbolic metric invariant. For instance, the maps z 7→ z n , n = 2, 3, . . ., are self-coverings of D∗ that leave λD∗ (z)|dz| invariant. Up to composition with a rotation about the origin these are the only self-coverings of D∗ that are not automorphisms. Because each hyperbolic geodesic in D∗ is the image of a hyperbolic geodesic in H under h, every radial segment [reiθ , Reiθ ], where 0 < r < R < 1, is part of

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a hyperbolic geodesic. Since the density λD∗ is independent of θ, the hyperbolic length of a geodesic segment σr,R = [reiθ , Reiθ ] is independent of θ. Direct calculation gives Z Z R log R dt . = log ℓD∗ (σr,R ) = λD∗ (z)|dz| = t log t log r [r,R] r

As the formula shows, this length tends to infinity if either r → 0 or R → 1 which also follows from the completeness of the hyperbolic metric. The Euclidean circle Cr = {z : |z| = r}, where 0 < r < 1, is not a hyperbolic geodesic; it has hyperbolic length Z |dz| 2π ℓD∗ (Cr ) = = . log(1/r) |z|=r |z| log(1/|z|)

The hyperbolic length of Cr approaches 0 when r → 0 and ∞ when r → 1. The hyperbolic area of the annulus A(r, R) = {z : r < |z| < R} ⊂ D∗ is Z Z 1 aD∗ (A(r, R)) = dx dy 2 2 A(r,R) |z| log |z| Z R dt = 2π t log2 t r 1 1 = 2π − . log(1/R) log(1/r)

The hyperbolic area of A(r, R) tends to infinity when R → 1 and has the finite limit 2π/ log(1/R) when r → 0.

There is a Euclidean surface in R3 that is isometric to {z : 0 < |z| < 1/e} with the restriction of the hyperbolic metric on D∗ and makes it easy to see these curious results about length and area in a neighborhood of the puncture. If a tractrix is rotated about the y-axis and the resulting surface is given the geometry induced from the Euclidean metric on R3 , then this surface has constant curvature −1 and is isometric to {z : 0 < |z| < 1/e} with the restriction of the hyperbolic metric on D∗ . This picture provides a simple isometric embedding of a portion of D∗ into R3 . Radial segments correspond to rotated copies of the tractrix and these have infinite Euclidean length. At the same time the surface has finite Euclidean area. Recall that for a simply connected region Ω, the hyperbolic density λΩ and the quasihyperbolic density 1/δΩ are bi-Lipschitz equivalent; see (8.4). These two metrics are not bi-Lipschitz equivalent on D∗ because the behavior of λD∗ near the unit circle differs from its behavior near the origin. For 1/2 < |z| < 1, δD∗ (z) = 1 − |z| and so lim δD∗ (z)λD∗ (z) = 1. |z|→1

For 0 < |z| < 1/2, δ(z) = |z| and so

lim δD∗ (z)λD∗ (z) = 0.

|z|→1


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12.2. Hyperbolic metric on an annulus. Now we determine the hyperbolic metric on an annulus by using a holomorphic covering from a strip onto an annulus. In each conformal equivalence class of annuli we choose the unique representative that is symmetric about the unit circle. For 0 < r < R let A(r, R) = {z : r < |z| < R}. The number mod(A(r, R)) = log(R/r) is called the modulus of A(r, R). Two annuli A(rj , Rj ), j = 1, 2, are conformally (actually Möbius) equivalent if p and only if R1 /r1 = R2 /r2 ; that is, if and only if they have equal moduli. If S = (R/r), then AS = {z : 1/S < |z| < S} is the unique annulus conformally equivalent to A(r, R) that is symmetric about the unit circle. Here symmetry means AS is invariant under z 7→ 1/¯ z, reflection about the unit circle. Note that mod(AS ) = 2 log S. The function k(z) = exp(z) is a holomorphic universal covering from the vertical strip Sb = {z : |Im z| < b}, where b = log R, onto the annulus AR = {z : 1/R < |z| < R}. Therefore, the density of the hyperbolic metric of the annulus AR is 1 π . λR (z) = 2 log R |z| cos π log |z| 2 log R

Example 12.1. We investigate the hyperbolic lengths of the Euclidean circles Cr = {z : |z| = r} in AR . The hyperbolic length of Cr is Z |dz| π π2 = . ℓR (Cr ) = π log |z| π log r |z|=r 2 log R |z| cos (log R) cos 2 log R 2 log R

The symmetry of AR about the unit circle is reflected by the fact that two circles symmetric about the unit circle have the same hyperbolic length. Also, the hyperbolic length of Cr increases from π 2 / log R = 2π 2 /mod AR to ∞ as r increases from 1 to R. Hence, the hyperbolic lengths of the Euclidean circles Cr in AR have a positive minimum hyperbolic length. The Euclidean circle C1 is a hyperbolic geodesic; Cr is not a hyperbolic geodesic when r 6= 1. If γn (t) = exp(2πint), then I(γn , 0) = n, where I(δ, 0) denotes the index or winding number of a closed path δ about the origin, and ℓR (γn ) =

2π 2 |n| . mod A(R)

We now show that γn has minimal hyperbolic length among all closed paths in AR that wind n times about the origin. Theorem 12.2. Suppose γ is a piecewise smooth closed path in AR and I(γ, 0) = n 6= 0. Then (12.1)

2π 2 |n| ≤ ℓR (γ), mod A(R)

where ℓR (γ) denotes the hyperbolic length of γ. Equality holds in (12.1) if and only if γ is a monotonic parametrization of the unit circle traversed n times.

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Proof. Suppose γ : [0, 1] → AR is a closed path with I(γ, 0) = n 6= 0. Then Z ℓR (γ) = λR (z) |dz| γ

π = 2 log R π ≥ 2 log R since (12.2)

0 < cos

Z

1

0

Z

1

0

|γ ′ (t)| dt |γ(t)| |γ(t)| cos π log 2 log R |γ ′ (t)| |γ(t)|

π log |γ(t)| 2 log R

≤1

and equality holds if and only if |γ(t)| = 1 for t ∈ [0, 1]. Next, Z Z 1 ′ |γ (t)| 1 γ ′ (t) (12.3) ≥ dt 0 γ(t) 0 |γ(t)| Z dz = z γ

= |2πiI(γ, 0)|

= 2π|n|.

Hence, π 2 |n| 2π 2 |n| = . log R mod AR It is straightforward to verify that if γn (t) = exp(2πint), t ∈ [0, 1], then equality holds in (12.1). Conversely, suppose γ is a path for which equality holds. Then equality holds in (12.2), so |γ(t)| = 1 for t ∈ [0, 1]. Let δ : [0, 1] → C be a lift of γ relative to the covering exp : C → C∗ . From I(γ, 0) = n, we obtain δ(1) − δ(0) = 2πni. The condition |γ(t)| = 1 implies δ(t) ∈ iR for t ∈ [0, 1]. The function h(t) = (δ(t) − δ(0))/2πi is real-valued, h(0) = 0 and h(1) = n. Also, γ(t) = exp(2πih(t) + δ(0)). Equality must hold in (12.3) and this means γ ′ (t)/γ(t) = 2πih′ (t) has constant argument. Hence, h′ (t) is either positive or negative, so t 7→ exp(2πih(t) + δ(0) is a parametrization of the unit circle starting at γ(0) = exp δ(0) and the unit circle is traversed either clockwise or counterclockwise. ℓR (γ) ≥

Exercises. 1. Consider the metric |dz|/|z| on C \ {0} := C∗ and let ℓC∗ (γ) denote the length of a path γ in C∗ relative to this metric. If γ is a closed path in C∗ , prove ℓC∗ (γ) ≥ 2π|I(γ, 0)|. When does equality hold? 2. Suppose f is holomorphic on D and f (D) ⊆ D \ {0}. Prove that |f ′ (0)| ≤ 2/e. Determine when equality holds.


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13. Rigidity theorems If a holomorphic mapping of hyperbolic regions is not a covering, then strict inequality holds in the general version of the Schwarz-Pick Lemma. It is often possible to provide a quantitative version of this strict inequality that is independent of the holomorphic mapping for multiply connected regions. We begin by establishing a refinement of Schwarz’s Lemma. Lemma 13.1. Suppose 0 6= a ∈ D and 0 < t < 1. If f is a holomorphic self-map of D, f (0) = 0 and |f (a)| ≤ t|a|, then t + |a| < 1. (13.1) |f ′ (0)| ≤ 1 + t|a| Proof. The Three-point Schwarz-Pick Lemma (Theorem 4.4) with z = 0 = v and w = a gives dD (f ′ (0), f (a)/a) = dD (f ∗ (0, 0), f ∗ (a, 0)) ≤ dD (0, a).

Hence,

dD (0, |f ′ (0)|) = dD (0, f ′ (0))

≤ dD (0.f (a)/a) + dD (0, a) ≤ dD (0, t) + dD (0, −|a|)

= d(−|a|, t), which is equivalent to (13.1).

Theorem 13.2. Suppose ∆ and Ω are hyperbolic regions with a ∈ ∆, b ∈ Ω and ∆ is not simply connected. There is a constant α = α(a, ∆; b, Ω) ∈ [0, 1) such that if f : ∆ → Ω is any holomorphic mapping with f (a) = b that is not a covering, then (13.2)

λΩ (f (a))|f ′ (a)| ≤ αλ∆ (a);

or equivalently, f ∆,Ω (a) ≤ α. Moreover, for all z ∈ ∆ α + tanh d∆ (a, z) < 1. (13.3) f ∆,Ω (z) ≤ 1 + α tanh d∆ (a, z) Proof. Let h : D → ∆ and k : D → Ω be holomorphic coverings with h(0) = a and k(0) = b. Because ∆ is not simply connected, the fiber h−1 (a) is a discrete subset of D and contains infinitely many points in addition to 0. Let 0 < r = min{|z| : z ∈ h−1 (a), z 6= 0} < 1. The set h−1 (a) ∩ {z : |z| = r} is finite, say a ˜j , 1 ≤ j ≤ m. Next, {|z| : z ∈ k −1 (b)} is a discrete subset of [0, 1), so this set contains finitely many values in the interval [0, r). Let s be the maximum value of the finite set {|z| : z ∈ k −1 (b)} ∩ [0, r). Suppose f : ∆ → Ω is any holomorphic mapping with f (a) = b and f is not a covering. Let f˜ be the unique lift of f ◦ h relative to k that satisfies f˜(0) = 0. Then |f˜′ (0)| = f ∆,Ω (a). Because f is not a covering, f˜ is not a rotation about the origin. From f ◦ h = k ◦ f˜ we deduce

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that f˜ maps h−1 (a) into k −1 (b). In particular, |f˜(˜ a)| ≤ s = t|˜ a|, where a ˜=a ˜1 and t = s/r < 1. Lemma 13.1 gives |f˜′ (0)| ≤

(s/r) + r (s/r) + |˜ a| = = α < 1. 1 + (s/r)|˜ a| 1+s

Since |f˜′ (0)| = f ∆,Ω (a), this establishes (13.2). Inequality (13.3) follows immediately from Corollary 11.3. The pointwise result (13.2) is due to Minda [24] and was motivated by the Aumann-Carathéodory Rigidity Theorem [4] which is the special case when Ω = ∆ and a = b. The global result (13.3) is due to the authors [9]. The AumannCarathéodory Rigidity Theorem asserts there is a constant α = α(a, Ω) ∈ [0, 1) such that |f ′ (a)| ≤ α for all holomorphic self-maps of Ω that fix a and are not conformal automorphisms. The exact value of the Aumann-Carathéodory rigidity constant for an annulus was determined in [23]. The following extension of the Aumann-Carathéodory Rigidity Theorem to a local result is due to the authors [9]. The corollary is given in Euclidean terms and asserts that holomorphic self-maps with a fixed point are locally strict Euclidean contractions if they are not conformal automorphisms. Corollary 13.3 (Aumann-Carathéodory Rigidity Theorem - Local Version). Suppose Ω is a hyperbolic region, a ∈ Ω and Ω is not simply connected. There is a constant β = β(a, Ω) ∈ [0, 1) and a neighborhood N of a such that if f is a holomorphic self-map of Ω that fixes a and is not a conformal automorphism of Ω, then |f ′ (z)| ≤ β for all z ∈ N . Proof. From the theorem |f ′ (z)| ≤

λΩ (z) α + tanh dΩ (a, z) . λΩ (f (z)) 1 + α tanh dΩ (a, z)

Set M (r) = max{λΩ (z) : dΩ (a, z) ≤ r} and m(r) = min{λΩ (z) : dΩ (a, z) ≤ r}. Since f (DΩ (a, r)) ⊆ DΩ (a, r), we have |f ′ (z)| ≤

M (r) α + tanh dΩ (a, z) . m(r) 1 + α tanh dΩ (a, z)

The right-hand side of the preceding equality is independent of f and tends to α as r → 1. Therefore, for α < β < 1 there exists r > 0 such that M (r) α + tanh dΩ (a, z) ≤β m(r) 1 + α tanh dΩ (a, z)

for dΩ (a, z) ≤ r. Then |f ′ (a)| ≤ β holds in DΩ (a, r). Our final topic is a rigidity theorem for holomorphic maps between annuli. The original results of this type are due to Huber [17]. Marden, Richards and Rodin [21] presented an extensive generalization of Huber’s work to holomorphic self-maps of hyperbolic Riemann surfaces.


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Definition 13.4. Suppose 1 < R, S ≤ +∞ and f : AR → AS is a continuous function. The degree of f is the integer deg f := I(f ◦ γ, 0), where γ(t) = exp(2πit). The positively oriented unit circle γ generates the fundamental group of both AR and AS . Therefore, for any continuous map f : AR → AS , f ◦ γ ≈ γ n for the unique integer n = deg f , where ≈ denotes free homotopy. Algebraically, f induces a homomorphism f∗ from π(AR , a) ∼ = Z and the = Z to π(AS , f (a)) ∼ image of 1 is an integer n; here π(AR , a) denotes the fundamental group of AR with base point a. The reader should verify that deg(f ◦ g) = (deg f ) (deg g).

Since the degree of the identity map is one, this implies that deg f = ±1 for any homeomorphism f . If f is holomorphic, then Z ′ f (z) 1 dz. deg f = 2πi γ f (z)

Suppose f, g : AR → AS are continuous functions. Then deg f = deg g if and only if f and g are homotopic maps of AR into AS .

Example 13.5. For any integer n the holomorphic self-map pn (z) = z n of C∗ has degree n. Given annuli AR and AS with 1 < R, S < +∞, it is easy to construct a continuous map of AR into AS with degree n; for example, the function z 7→ (z/|z|)n has degree n. For R = S each conformal automorphism has degree ±1. In fact, the rotations z 7→ eiθ z have degree 1 and the maps z 7→ eiθ /z have degree −1. Constant self-maps of AR have degree 0. Can you find a holomorphic self-map of AR with degree n 6= 0, ±1? Surprisingly, the answer is negative! Holomorphic mappings of proper annuli are very rigid. The moduli of the annuli provide sharp bounds for the possible degrees of a holomorphic mapping of one annulus into another. Theorem 13.6. If f : AR → AS is a holomorphic mapping, then mod AS (13.4) | deg f | ≤ . mod AR For n = deg f 6= 0 equality holds if and only if S = R|n| and f (z) = eiθ z n for some θ ∈ R. Proof. Let γ(t) = exp(2πit) for t ∈ [0, 1]. If n = deg f , then I(f ◦ γ, 0) = n, so Theorem 12.2 gives 2π 2 |n| ≤ ℓS (f ◦ γ). mod AS Since holomorphic functions are distance decreasing relative to the hyperbolic metric, 2π 2 ℓS (f ◦ γ) ≤ ℓR (γ) = . mod AR The preceding two inequalities imply (13.4). Suppose equality holds. Then f is a covering of AR onto AS that maps the unit circle onto itself. By post-composing

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f with a rotation about the origin, we may assume that f fixes 1. Equality in (13.4) implies S = R|n| , where n = deg f . The covering f lifts relative to pn (z) = z n to a holomorphic self-covering f˜ of AR that fixes 1. Theorem 10.6 implies that f˜ is the identity and so f (z) = z n . Corollary 13.7 (Annulus Theorem). Suppose f is a holomorphic self-map of AR . Then | deg f | ≤ 1 and equality holds if and only if f ∈ A(AR ). Proof. The inequality follows immediately from Theorem 13.6. If f ∈ A(AR ), then | deg f | = 1 since this holds for any homeomorphism. It remains to show that if | deg f | = 1, then f ∈ A(AR ). Equality implies f maps the unit circle into itself. By post-composing f with a rotation about the origin, we may assume f fixes 1. By Theorem 10.6 if a self-covering of a hyperbolic region has a fixed point, it is a conformal automorphism. Note that if f is any holomorphic self-map of AR that is not a conformal automorphism, then deg f = 0. A result analogous to Corollary 13.7 is not valid for a punctured disk. For each integer n ≥ 0 the function z 7→ z n is a holomoprhic self-map of D∗ with degree n. Exercises. 1. Show that Theorem 13.2 is false when ∆ is simply connected. Hint: Suppose ∆ = D and a = 0. For any number r ∈ [0, 1) show there exists a holomorphic function f : D → Ω that is not a covering and f D,Ω (0) = r. 2. Suppose f is a holomorphic self-map of C \ {0}. Prove that deg f = 0 if and only if f = exp ◦g for some holomorphic function g defined on C \ {0}.

14. Further reading There are numerous topics involving the hyperbolic metric and geometric function theory that are not discussed in these notes. The subject is too extensive to include even a reasonably complete bibliography. We mention selected books and papers that the reader might find interesting. Anderson [3] gives an elementary introduction to hyperbolic geometry in two dimensions. Krantz [18] provides an elementary introduction to certain aspects of the hyperbolic metric in complex analysis. Ahlfors introduced the powerful method of ultrahyperbolic metrics [1]. A discussion of this method and several applications to geometric function theory, including a lower bound for the Bloch constant, can be found in [2]. Ahlfors’ method can be used to estimate various types of Bloch constants, see [25], [26], [27]. In a long paper Heins [15] treates the general topic of conformal metrics on Riemann surfaces. He gives a detailed treatment of SK-metrics, a generalization of ultrahyperbolic metrics. Roughly speaking, SK-metrics are to metrics with curvature −1 as subharmonic functions are to harmonic functions. The circle of ideas surrounding the theorems of Picard, Landau and Schottky and Montel’s normality criterion all involve three omitted values. Theorems of


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this type follow immediately from the existence of the hyperbolic metric on C∞ punctured at three points. Interestingly, only a metric with curvature at most −1 on a thrice punctured sphere is needed to establish these results. An elementary construction of such a metric, based on earlier work of R. M. Robinson [31], is given in [30]. Hejhal [16] obtained a Carathéodory kernel-type of theorem for coverings of the unit disk onto hyperbolic regions. This result implies that the hyperbolic metric depends continuously on the region. The method of polarization was extended by Solynin to apply to the hyperbolic metric, see [10]. It include earlier work of Weitsman [35] on symmetrization and Minda [29] on a reflection principle for the hyperbolic metric. For the role of hyperbolic geometry in the study of discrete groups of Möbius transformations, see [5]. This reference includes a brief treatment of hyperbolic trigonometry.

References 1. Ahlfors, L.V., An extension of Schwarz’s Lemma, Trans. Amer. Math. Soc., 43 (1938), 259-264. 2. Ahlfors, L.V., Conformal invariants, McGraw-Hill, 1973. 3. Anderson, J.W., Hyperbolic geometry, 2nd. ed., Springer Undergraduate Mathematics Series, Springer, 2005. 4. Aumann, G. and Carathéodory, C., Ein Satz u ¨ber die konforme Abbildung mehrfach zusammenhängender Gebiete, Math. Ann., 109 (1934), 756-763. 5. Beardon, A.F., The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York 1983. 6. Beardon, A.F., The Schwarz-Pick Lemma for derivatives, Proc. Amer. Math. Soc., 125 (1997), 3255-3256. 7. Beardon, A.F. and Carne, T.K., A strengthening of the Schwarz-Pick Inequality, Amer. Math. Monthly, 99 (1992), 216-217. 8. Beardon, A.F. and Minda, D., A multi-point Schwarz-Pick Lemma, J. d’Analyse Math., 92 (2004), 81-104. 9. Beardon, A.F. and Minda, D., Holomorphic self-maps and contractions, submitted. 10. Brock, F. and Solynin, A.Yu., An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352 (2000), 1759-1796. 11. Carathéodory, C., Theory of functions of a complex variable, Vol. II, Chelsea, 1960. 12. Chen, Huaihui, On the Bloch constant, Approximation, complex analysis and potential theory (Montreal, QC 2000), NATO Sci. Ser. II Math. Phys. Chem., 37, Kluwer Acad. Publ., Dordrecht, 2001. 13. Duren, P.L., Univalent functions, Springer-Verlag, New York 1983. 14. Goluzin, G.M., Geometric theory of functions of a complex variable, Amer. Math. Soc., 1969. 15. Heins, M., On a class of conformal metrics, Nagoya Math. J., 21 (1962), 1-60. 16. Hejhal, D., Universal covering maps for variable regions, Math. Z., 137 (1974), 7-20. ¨ 17. Huber, H., Uber analytische Abbildungen von Ringgebieten in Ringgebiete, Compositio Math., 9 (1951), 161-168. 18. Krantz, S.G., Complex analysis: the geometric viewpoint, 2nd. ed., Carus Mathematical Monograph 23, Mathematical Association of America, 2003.

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19. Ma, W. and Minda, D., Geometric properties of hyperbolic geodesics, Proceedings of the International Workshop on Quasiconformal Mappings and their Applications, pp ??. 20. Ma, W. and Minda, D., Euclidean properties of hyperbolic polar coordinates, submitted 21. Marden, A., Richards, I. and Rodin, B., Analytic self-mappings of Riemann surfaces, J. Analyse Math., 18 (1967), 197-225. 22. Massey, W.S., Algebraic topology: an introduction, Springer-Verlag, 1977. 23. Minda, C.D., The Aumann-Carathéodory rigidity constant for doubly connected regions, Kodai Math. J. 2 (1979), 420-426. 24. Minda, C.D., The hyperbolic metric and coverings of Riemann surfaces, Pacific J. Math. 84 (1979), 171-182. 25. Minda, C.D., Bloch constants, J. Analyse Math. 41 (1982), 54-84. 26. Minda, C.D., Lower bounds for the hyperbolic metric in convex regions, Rocky Mtn. J. Math. 13 (1983), 61-69. 27. Minda, C.D., The hyperbolic metric and Bloch constants for spherically-convex regions, Complex Variables Theory Appl. 5 (1986), 127-140. 28. Minda, C.D., The strong form of Ahlfors’ Lemma, Rocky Mtn. J. Math. 17 (1987), 457-461. 29. Minda, C.D., A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129-144. 30. Minda, D. and Schober, G., Another elementary approach to the theorems of Landau, Montel, Picard, and Schottky, Complex Variables Theory Appl. 2 (1983), 157-164. 31. Robinson, R.M, A generalization of Picard’s and related theorems, Duke Math. J. 5 (1939), 118-132. 32. Royden, H., The Ahlfors-Schwarz Lemma: the case of equality, J. Analyse Math. 46 (1986), 261-270. ¨ 33. Pick, G., Uber eine Eigenschaft der konformen Abbildung kreisf¨ ormiger Bereiche, Math. Ann. 77 (1915), 1-6. 34. Veech, W.A., A second course in complex analysis, W.A. Benjamin, 1967. 35. Weitsman, A., Symmetrization and the Poincaré metric, Ann. of Math.(2) 124 (1986), 159-169. 36. Yamada, A., Bounded analytic functions and metrics of constant curvature, Kodai Math. J. 11 (1988), 317-324. A.F. Beardon E-mail: [email protected] Address: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England D. Minda E-mail: [email protected] Address: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA


Isometries of relative metrics Peter H¨ ast¨ o Abstract. This survey contains a review of the basics of Möbius mappings and some basic metrics used in relations with quasiconformal mappings, the jG metric and the quasihyperbolic metric. The second and third sections are devoted to the study of the isometries of these two metrics. Keywords. Relative metrics, isometries, distance ratio metric, Seittenranta’s metric. 2000 MSC. 30F45 (primary), 30C65 (secondary).

Contents 1. Overview 2. Möbius mappings 3. The jG metric 3.1. Isometries of j-type metrics 3.2. Other properties of j-type metrics 4. The quasihyperbolic metric Additional notation 4.1. Isometries which are Möbius 4.2. Curvature of the quasihyperbolic metric 4.3. Isometries of the quasihyperbolic metric References

57 58 61 63 66 66 68 68 71 73 76

1. Overview These lecture notes consist of three parts: In the first part the basic theory of Möbius mappings is reviewed. Particular emphasis will be given to concrete calculations within the context of a single mapping in Euclidean space. Although this presentation is perhaps not the most elegant one possible, it has the advantage that it does a good job in preparing us for the isometry questions that come up later. For a more detailed exposition of the basics of Möbius mappings see e.g. [2, 31]. Supported by the Academy of Finland.

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The latter two parts deal with the problem of characterizing isometries of two metrics which have turned out to be very important in the theory of quasiconformal mappings, namely, the jG and the quasihyperbolic metric. Specifically, in the second part we deal with the jG metric — this part is based on joint work with Z. Ibragimov and H. Lindén [18] in Computational Methods and Function Theory. The third part reproduces parts of my recent manuscript [15], which deals with isometries of the quasihyperbolic metric. Characterizing isometries of a metric can in some sense be thought of as solving a (system of) functional equation(s): we know that df (G) (f (x), f (y)) = dG (x, y) for all x, y ∈ G and we want to determine f . However, the fact that we have at our disposal a continuum of functional equations implies that the methods used to approach this problem are somewhat different than those usually found when dealing with functional equations. Thus our methods will often be based on some geometric considerations: we will employ geodesics (locally and globally), intrinsic curvature, as well as limiting behavior of the metric in infinitesimal regions more generally. Many other properties of these and related metrics have also been studied. A review of some of these results is presented in the chapter by H. Lindén in these notes.

2. M¨ obius mappings We denote by Rn = Rn ∪{∞} the one-point compactification of Rn , so its open balls are the open balls of Rn , complements of closed balls in Rn and half-spaces. If D ⊂ Rn we denote by ∂D and D its boundary and closure, respectively, all with respect to Rn . By B n (x, r) and S n−1 (x, r) we denote the open ball centered at x ∈ Rn with radius r > 0, and its boundary, respectively. For x ∈ D ( Rn we denote δ(x) = d(x, ∂D) = min{|x − z| : z ∈ ∂D}. By [x, y], (x, y] we denote the closed and half-open segment between x and y, respectively. The (absolute) cross-ratio of four distinct points is defined by |a, b, c, d| =

|a − c| |b − d| , |a − b| |c − d|

= 1 for all x, y ∈ Rn . A homeomorphism with the understanding that |∞−x| |∞−y| f : Rn → Rn is a Möbius mapping if |f (a), f (b), f (c), f (d)| = |a, b, c, d|

for every quadruple of distinct points a, b, c, d ∈ Rn . A mapping of a subdomain of Rn is Möbius, if it is a restriction of a Möbius mapping defined on Rn . Although the previous definition is very compact and brings out one important aspect of Möbius mappings, it does not tell us what the behavior of a Möbius mapping is in terms of geometry. However, it is not so difficult to get some

Isometries of relative metrics

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results in this direction: Let us regard three points a, b, c as fixed and a fourth point x as variable. Then the cross-ratio equation reads |a − c| |b − x| |a′ − c′ | |b′ − x′ | = ′ , |a − b| |c − x| |a − b′ | |c′ − x′ |

where a′ is the image of a under the Möbius mapping. We can rewrite this as |b′ − x′ | |b − x| =C ′ , |c − x| |c − x′ |

where C is a constant not depending on x. However, for fixed b, c and C > 0 the set n o |b − x| x ∈ Rn : =C |c − x| is a sphere. Thus the previous equation implies that the Möbius mapping maps spheres to spheres. The converse of this statement is also true, see [4]. It is also possible to take a more constructive approach to Möbius mappings. Let us first of all make the trivial observation that a mapping which preserves Euclidean distances is Möbius. Second, we note that mappings preserving ratios of Euclidean distances (so-called similarity mappings) are Möbius. These mappings are: • • • •

translations; reflections; rotations; and dilatations.

Are there any other Möbius mappings? From the definition it is clear that the set of Möbius mappings is closed under composition (in fact, the set is a group under composition). Thus we may employ a very useful trick in trying to identify any other Möbius mappings, namely, we normalize by mappings that we already know are Möbius. This means that we consider the mapping g = s1 ◦ f ◦ s2 , where f is our original Möbius mapping and s1 and s2 are similarities. Suppose first that f is such that f (∞) = ∞. Inserting d = ∞ in the definition implies that |f (a) − f (c)| |a − c| = |a, b, c, ∞| = |f (a), f (b), f (c), ∞| = , |a − b| |f (a) − f (b)|

so f is a similarity. Otherwise there exists a finite point a such that f (a) = ∞. By an auxiliary similarity we may assume that a = 0 (i.e. we choose s2 (x) = x+a above). Similarly, f (∞) = b 6= ∞, and if we choose s1 (x) = x − b, then g is a Möbius mapping which swaps 0 and ∞. Using this in the equation gives |b| |g(c)| = |∞, b, c, 0| = |0, g(b), g(c), ∞| = . |c| |g(b)|

From this we see that |x| |g(x)| is a constant. By another similarity we may assume that this constant equals 1, so that |g(x)| = |x|−1 for every x ∈ Rn . To

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get a grip of the non-radial action of g we use the equation inserting 0 and ∞ in other places: |b − d| |g(b) − g(d)| = |∞, b, 0, d| = |0, g(b), ∞, g(d)| = |d| |g(b)| . Using the previous formula for |g(b)| this gives |g(b) − g(d)| =

(2.1)

|b − d| , |b| |d|

which is a central formula for calculating how a Möbius mapping affects distances. We can rewrite (2.1) as |g(b) − g(d)|2 |b − d|2 = . |g(b)| |g(d)| |b| |d| Using the cosine formula c , |b − d|2 = |b|2 + |d|2 − 2|b| |d| cos b0d

c stands for the angle between the vectors b − 0 and d − 0, and similarly where b0d for |g(b)−g(d)|2 we see that g preserves angles at the origin and lines through the origin. Thus, up to additional normalization by a reflection and/or a rotation, we see that g(x) = x |x|−2 . A Möbius mapping which swaps ∞ and with a point of Rn and which maps every line through this point to itself is called an inversion. Note that every inversion is an involution, i.e. it is its own inverse. The point which is mapped to ∞ is called the center of inversion. We have shown that every inversion equals x 7→ x |x|−2 , up to similarity. In particular, every Möbius mapping can be written as s ◦ i, where s is a similarity and i is an inversion or the identity. Now that we have identified the Möbius mappings we can proceed to show the following basic property: given two ordered triples of distinct points in Rn , (a, b, c) and (a′ , b′ , c′ ), there exists a Möbius map f with f (a) = a′ , f (b) = b′ and f (c) = c′ . It is clearly sufficient to show this claim in the case when a′ , b′ and c′ are the vertices of an equilateral triangle. Let us first find a point x such that |b − c| |a − x| |a − c| |b − x| = = 1. |a − b| |c − x| |a − b| |c − x| The easiest way to see that such a point x exists is to use an inversion i with center a. Then the equations to satisfy become |i(b) − z| |i(b) − i(c)| = = 1. |i(c) − z| |i(c) − z| We see that the first fraction describes a hyperplane which is the perpendicular bisector of the segment [i(b), i(c)] and the second fraction the sphere with center i(c) and radius |i(b) − i(c)|. Since these objects clearly intersect, we can find a


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suitable z, and then our x is given by i(z). Let ˜ı be an inversion with center x. The choice of x implies that |˜ı(a) − ˜ı(c)| |˜ı(b) − ˜ı(c)| = = 1, |˜ı(a) − ˜ı(b)| |˜ı(a) − ˜ı(b)|

so ˜ı(a), ˜ı(b), ˜ı(c) are the vertices of an equilateral triangle. These can be mapped to the given points (a′ , b′ , c′ ) by a similarity transform s, so our final Möbius map is then s ◦ ˜ı.

3. The jG metric This section reproduces parts of the article [18] on isometries of some relative metrics. The term “relative metric” implies that the metric is evaluated in a proper subdomain of Rn relative to its boundary. More precisely, we want the metric to blow up towards the boundary of the domain, i.e., we want the boundary to be at infinity intrinsically. Let D ( Rn be a domain containing the points x and y. The well-known distance ratio metric is defined by |x − y| , jD (x, y) = log 1 + min{δ(x), δ(y)} where δ(·) = dist( · , ∂D) denotes distance to the boundary. It was used, for instance, by Gehring and Osgood [11] to characterize uniform domains (namely, in such domains the jD metric is quasiconvex). Note that this metric has sometimes been called simply “the relative metric”, and will be used in this meaning. To see how these metrics fit into a larger framework we recall the concept of an inner metric. Let d be a metric in D and γ be a path in D (i.e. a continuous mapping from an interval I to D). The length (or, more explicitly, d-length) of γ is defined as k−1 X d γ(ti ), γ(ti+1 ) , d(γ) = sup i=1

where the supremum is taken over k and all increasing sequences (ti )ki=1 of points in I. Then the inner or intrinsic metric of d is defined by ˜ y) = inf d(γ), d(x, γ

where the infimum is taken over all paths γ connecting x and y in D (note that this need not be finite, unless D is rectifiably connected). It is clear that ˜ y) and that d(γ) = d(γ) ˜ d(x, y) ≤ d(x, for any metric and path. The theory of length-metrics, including in particular intrinsic metrics, is presented e.g. in [5, 6]. Suppose now that D ⊂ Rn and d is a metric in D. If ¯ = lim d(x, y) d(x) y→x |x − y|

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exists for all x ∈ D and is continuous, then we can express the inner metric of d by Z ¯ |dz|, ˜ d(x, y) = inf d(z) γ

γ

where |dz| represents integration with respect to d-arclength, and the infimum is taken over rectifiable curves with end points x and y. In this case d˜ is called a conformal metric. We easily see that the inner metric of jD is the quasihyperbolic metric, Z |dz| ˜jD (x, y) = kD (x, y) = inf . γ γ d(z, ∂D) Length-metrics are interesting from a geometric point of view, but for getting explicit estimates they are often of little use. The role of point-distance functions, like the jD metric, is that they share features with their inner metrics, but are much more explicit.

In this paper we want to consider not only the jD metric, but all metrics which resemble them in the very small and very large scale. The small scale equivalence implies that the metrics have the same inner metrics, whereas the large scale equivalence allows us to get a hold of the boundary behavior and thus start unraveling the isometry story. Remark 3.1. Note that jD is really families of metrics, namely for every domain D we have one metric. We will continue to use this convention when talking about this and other metrics in this paper. Definition 3.2. We say that d is a j-type metric if the following three conditions hold on every domain D ( Rn : 1. dD is a metric on D. 2. For each y ∈ D and for each sequence (xi ) with jD (xi , y) → 0 we have dD (xi , y) = 1. i→∞ jD (xi , y) lim

3. For each y ∈ D and for each sequence (xi ) with jD (xi , y) → ∞ we have lim dD (xi , y) − jD (xi , y) = 0. i→∞

The fact that y can be any interior point in (2) means that being a j-type metric is quite a strong condition; for instance, if d and f ◦ d are j-type metrics, then f = id (Corollary 3.12). It seems to be quite difficult to construct other natural metrics of j-type. The main purpose of our more abstract treatment is to highlight the features that are crucial, which in turn indicates that these techniques might be relevant also for handling the isometries of the corresponding inner metrics. In this part we characterize the isometries of j-type metrics. We start by collecting some basic properties of these metrics. In Section 3.1 we solve the


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isometry problem for j-type metrics using a boundary rigidity result, and in Section 3.2 we list some additional properties whose proofs can be found in [18]. The following result is more or less restatements of the definition. However, it directly implies that we may restrict our focus very much without losing any isometries. Proposition 3.3. If d is a j-type, then |x − y| lim =0 dD (x, y) − log y→∂D\{∞} δ(y)

and

1 dD (x, y) = y→x |x − y| δ(x) lim

for every x ∈ D. The inner metric of a j-type metrics is the quasihyperbolic metric kD . In particular, every isometry of a j-type metric is an isometry of kD . Every isometry of kD is a conformal mapping [29, Theorem 2.6]. Hence we conclude: Corollary 3.4. Every isometry of a j or δ-type metric is conformal. In particular, if n ≥ 3, then such an isometry is M¨ obius. We plunge right into the main result of this section, a characterization of the isometries of j-type metrics. In Section 3.2 we derive some miscellaneous results, which give a clearer picture of j-type metrics. 3.1. Isometries of j-type metrics. The proof of the following theorem is partly based on ideas from [17]. Incidentally, it is possible to give a much simpler proof for the particular case of the jD metric itself, since in this case we can cancel the logarithm and the 1+ terms. Thus f is a jD isometry if and only if |f (x) − f (y)| |x − y| = , min{δ(x), δ(y)} min{δ ′ (f (x)), δ ′ (f (y))} where, as usual, δ ′ denotes the distance to the boundary in the image domain f (D). The reader is challenged to find the very short argument which shows that this implies that the isometry is a similarity. In the general case of j-type metrics we have less information about the metric, so we have to look at what happens at the boundary. In this case we can nevertheless prove the following theorem, whose proof is reproduced from [18]. Theorem 3.5. Let d be a j-type metric, D ( Rn and f : D → Rn be a disometry. Then either 1. f is a similarity, or 2. D = Rn \ {a} and, up to similarity, f is an inversion in a sphere centered at a. Proof. Denote D′ = f (D) and δ ′ (x) = d(x, ∂f (D)). Fix z ∈ ∂D \ {∞} and let (zi ) be a sequence of points in D tending to z. We first assume that there exists a subsequence, which we also denote by (zi ), such that f (zi ) converges to some

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point w1 ∈ Rn . Since d is a j-type metric we see, using Proposition 3.3 for the third equality, that for every x ∈ D we have that 0 = lim dD′ (f (x), f (zi )) − dD (x, zi ) i→∞ 1 1 = lim dD′ (f (x), f (zi )) − log ′ − lim dD (x, zi ) − log + i→∞ i→∞ δ (f (zi )) δ(zi ) δ(zi ) + lim log ′ i→∞ δ (f (zi )) |f (x) − w1 | δ(zi ) = log + lim log ′ . i→∞ |x − z| δ (f (zi )) Taking exponentials gives (3.6)

|f (x) − w1 | δ ′ (f (zi )) = < ∞. lim i→∞ δ(zi ) |x − z|

Suppose now that (ˆ zi ) is a second sequence of points in D tending to z, but that this time f (ˆ zi ) → w2 ∈ Rn \ {w1 }. Using x = zˆj for every j = 1, 2, . . . in (3.6) gives δ ′ (f (zi )) |f (ˆ zj ) − w1 | lim = →∞ i→∞ δ(zi ) |ˆ zj − z|

as j → ∞, which is a contradiction. In other words, f (ˆ zi ) → w1 for every sequence of points (ˆ zi ) → z, so we may extend f continuously to D by defining f (z) = limi→∞ f (zi ). Therefore we conclude from (3.6), since the left-hand side of this equation does not depend on x, that |f (x) − f (z)| = hf (z)|x − z|

for some function hf : ∂D → (0, ∞). This means that for z, w ∈ ∂D we have hf (z)|w − z| = |f (w) − f (z)| = hf (w)|w − z|, so hf is in fact a constant. Therefore f acts as a similarity, say g, on the boundary. We then extend f to all of Rn by setting f (x) = g(x) outside the original domain of definition. Then it is clear that (3.7)

|f (x) − f (z)| = hf |x − z|

for every point x ∈ Rn , i.e. the sphere S n−1 (z, r) is mapped to S n−1 (f (z), hf r). This clearly implies that the conformal mapping is Möbius, and a Möbius mapping satisfying (3.7) is a similarity. We still have one assumption to consider. In the beginning of the proof we assumed that we can find a boundary point z and a sequence (zi ) of points in D tending to z such that f (zi ) tends to a finite limit. So we suppose now that no such sequence can be found, i.e. that for every sequence (zi ) of points in D tending to a boundary point z the sequence f (zi ) tends to ∞. As before we


conclude that

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0 = lim dD′ (f (x), f (zi )) − dD (x, zi ) i→∞ = lim jD′ (f (x), f (zi )) − jD (x, zi ) i→∞ δ(zi ) |f (x) − f (zi )| . = lim log i→∞ min{δ ′ (f (zi )), δ ′ (f (x))} |x − z|

So it follows that

|f (x) − f (zi )|δ(zi ) = |x − z|. i→∞ min{δ ′ (f (zi )), δ ′ (f (x))} lim

Since f (zi ) → ∞, we see that we can replace |f (x)−f (zi )| by |f (zi )| in the above formula. Since the right-hand-side depends on x (which lies in an open set) we see that the left-hand-side must do so, too, hence we have to choose the second term in the minimum. Taking this into account we have gf (z) = lim |f (zi )| δ(zi ) = |x − z| δ ′ (f (x)), i→∞

where gf : ∂D → (0, ∞). Suppose that D has at least two finite boundary points, and let a, b ∈ ∂D be such that the open segment (a, b) is contained in D. Now if we first consider x (in the previous equation) to be the mid-point x of (a, b), then we conclude that gf (a) = |x − a| δ ′ (f (x)) = |x − b| δ ′ (f (x)) = gf (b). But if we take some other point on the segment, then we get gf (a) 6= gf (b), a contradiction. So only the case when D has a single boundary point remains to consider. Then we have lim |f (zi )| |zi − a| = |x − a| |f (x) − b|

i→∞

(for D = Rn \ {a} and D′ = Rn \ {b}) and we directly see that x 7→ f (x) + b − a is an inversion, which concludes the proof. Corollary 3.8. Let d be a similarity invariant j-type metric and let D ( Rn . Then f : D → Rn is a d-isometry if and only if 1. f is a similarity, or 2. D = Rn \ {a} and, up to similarity, f is the inversion in a sphere centered at a. Proof. The previous proposition established that every d-isometry is of the given kind. If f is a similarity, then it is an isometry by assumption. So it remains (after normalization) to consider the case D = Rn \ {0}. In thiscase we see that similarity invariance implies that dD (x, y) depends only on max |x| , |y| and the |y| |x| d On the other hand, an inversion in a sphere about the origin swaps angle x0y. d invariant, so we see that it is an isometry. |x|/|y| and |y|/|x| and leaves x0y

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3.2. Other properties of j-type metrics. We said before that every j-type metric has an upper bound in terms of the quasihyperbolic metric. Surprisingly, it is also possible to get a universal lower bound by a metric, the so-called halfapollonian metric [19]. For a domain D ( Rn this metric is defined by |x − z| ηD (x, y) = sup log . |y − z| z∈∂D The metric ηD is similarity invariant, and every Möbius mapping is bilipschitz. The proofs of the following results can be found in [18].

Proposition 3.9. For every j-type metric d and every D ( Rn we have dD ≥ ηD . Using the previous proposition and the quasihyperbolic upper bound we can squeeze in j-type metrics to get the exact value on some subset of the domain: Corollary 3.10. Let w ∈ D and z ∈ ∂D ∩ S n−1 (w, δ(w)). Then dD (x, y) = jD (x, y) for every x, y ∈ [w, z). The following is easily checked by a direct computation using the definition of the j-metric, but also follows from Corollary 3.10 and the fact that line segments are also geodesic rays for the ηD -metric ([19, Example 3.4]). Corollary 3.11. Let w ∈ D and z ∈ ∂D ∩ S n−1 (w, δ(w)). Then [w, z) is a geodesic ray for the j-type metric d, i.e. for every x, ξ, y ∈ [w, z) in this order we have dD (x, y) = dD (x, ξ) + dD (ξ, y). In general, if we have a metric d and a subadditive function f : [0, ∞) → [0, ∞) for which f (x) = 0 if and only if x = 0, then f ◦ d is also a metric. It turns out that the conditions for begin a j-type metric are so rigid, that this transformation is never possible in this context: Corollary 3.12. Let d be a j-type metric and fD : [0, ∞) → [0, ∞) be a family of arbitrary functions. If f ◦ d is a j-type metric, then fD = id for all relevant D.

4. The quasihyperbolic metric The remainder of this article is reproduced with minor modifications from [15]. Let D ( R2 be an open set and denote δ(x) = d(x, ∂D), the distance to the boundary. The quasihyperbolic metric in D is the conformal metric with the density δ(x)−1 , in other words, the metric is given by Z ds(z) , kD (x, y) = inf γ γ δ(z) where the infimum is taken over paths γ connecting x and y in D and ds represents integration with respect to arc-length.


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The quasihyperbolic metric was first introduced in the seventies, and since then it has found innumerable applications, especially in the theory of quasiconformal mappings, see, e.g. [11, 12, 22, 28, 29]; new connections are still being made, for instance P. Jones and S. Smirnov [24] recently gave a criterion for removability of a set in the domain of definition of a Sobolev space in terms of the integrability of the quasihyperbolic metric, see also [25], and Z. Balogh and S. Buckley [1] used the metric in a geometric characterization of Gromov hyperbolic spaces. Despite the prominence of the quasihyperbolic metric, there have been almost no investigations of its geometry. Three exceptions are the papers by G. Martin [28] and Martin and B. Osgood [29], the second of which was the main motivation for the approach presented in this paper, and the thesis by H. Lindén [27]. Part of the reason for this lack of geometrical investigations is probably that the density of the quasihyperbolic metric is not differentiable in the entire domain, which places the metric outside the standard framework of Riemanian metrics. At least two modifications of the quasihyperbolic metric have been proposed which do not suffer from this problem. J. Ferrand [10] suggested replacing the density δ −1 by |a − b| . σD (x) = sup a,b∈∂D |a − x| |b − x| Note that δ(x)−1 ≤ σD (x) ≤ 2δ(x)−1 , so the Ferrand metric and the quasihyperbolic metric are bilipschitz equivalent. Moreover, the Ferrand metric is Möbius invariant, whereas the quasihyperbolic metric is only Möbius quasi-invariant. A second variant was proposed more recently by R. Kulkarni and U. Pinkall [26], see also [23]. The K–P metric is defined by the density o n 2r : x ∈ B(z, r) ⊂ D . µD (x) = inf (r − |x − z|)2 Equivalently, the infimum is taken over the hyperbolic densities of x in balls contained in D. This density satisfies the same estimate as Ferrand’s density, i.e. δ(x)−1 ≤ µD (x) ≤ 2δ(x)−1 , and the K–P metric is also Möbius invariant. Although the Ferrand and K–P metrics are in some sense better behaved than the quasihyperbolic metric, they suffer from the short-coming that it is very difficult to get a grip even of the density, even in simple domains. Despite this, D. Herron, Z. Ibragimov and D. Minda [21] recently managed to solve the isometry problem of the K–P metric in most cases. By the isometry problem of the metric d we mean characterizing mappings f : D → R2 with dD (x, y) = df (D) (f (x), f (y))

for all x, y ∈ D. Notice that in some sense we are here dealing with two different metrics, due to the dependence on the domain. Hence the usual way of approaching the isometry problem is by looking at some intrinsic features of the metric which are then preserved under the isometry. Since irregularities (e.g. cusps) in the domain often lead to more distinctive features, this implies that the problem is often easier for more complicated domains.

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The work by Herron, Ibragimov and Minda [21] bears out this heuristic – they were able to show that all isometries of the K–P metric are Möbius mappings except in simply and doubly connected domains. Their proof is based on studying the curvature of the metric. For the quasihyperbolic metric, formulae for the curvature were worked out already in [29] (see Section 4.2, below), and were used in that paper to prove that all the isometries of the disc are similarity mappings. These will be our main tool in this paper. The other source of the ideas used below are the papers [16, 17, 18, 19] on isometries of some other similarity and Möbius invariant metrics. There are three steps in characterizing quasihyperbolic isometries: 1. show that they are conformal; 2. show that they are Möbius; and 3. show that they are similarities. The first step has been carried out by Martin and Osgood [29, Theorem 2.6] for completely arbitrary domains, so there is no more work to do there. In Section 4.3 we will use the results from [29] on the curvature of the quasihyperbolic metric, and some new ideas to prove that the conformal isometries are Möbius (second step). For this we need to assume that the boundary of the domain is at least C 3 -smooth. In Section 4.1 we will work on the third step – we show that Möbius isometries are similarities provided the boundary is C 1 . In Section 4.2 we study the Gaussian curvature of the quasihyperbolic metric, and the gradient of the curvature. Additional notation. We employ some additional conventions in this section: We tacitly identify R2 with C, and speak about real and imaginary axes, etc. We will often work with a mapping f : D → R2 . In such cases we will use a prime to denote quantities on the image side, e.g. x′ = f (x), D′ = f (D) and δ ′ (x) = d(x, ∂D′ ), and so on. 4.1. Isometries which are M¨ obius. Let D be a domain and ζ ∈ ∂D. We say that ζ is circularly accessible, if there exists a disc B ⊂ D such that ζ ∈ ∂B.

Lemma 4.1. Let D ( R2 be a Jordan domain with circularly accessible boundary, and let f : D → R2 be a quasihyperbolic isometry which is also M¨ obius. Then, up to composition by similarity mappings, f is the identity or the inversion in a circle centered at a boundary point.

Proof. Assume that f is not a similarity. Since f is a Möbius map, it is an inversion, up to similarities, which are always isometries of the quasihyperbolic metric. Thus it suffices to consider the case when f is an inversion in a unit sphere. Let us denote the center of this sphere by w. Suppose first that w 6∈ D and let ζ ∈ ∂D be the closest boundary point to w. For simplicity we normalize the situation so that ζ lies on the positive real axis and w = 0. Since ζ is circularly accessible, we find a disc B(z, r) ⊂ D which contains ζ in its closure. Since ζ is the closest boundary point to w, we see that


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z has to lie on the positive real axis, as well. Let x and y be points satisfying . The right-hand inequality ensures that ζ is the closest ζ < x < y ≤ ζ(ζ+2r) ζ+r boundary point to [x, y], and that ζ ′ is the closest boundary point to [x′ , y ′ ]. Thus we find that kD (x, y) = log

|x′ − ζ ′ | |x − ζ| and kD′ (x′ , y ′ ) = log ′ . |y − ζ| |y − ζ ′ |

Since f is the inversion in the unit sphere, we have |x′ − ζ ′ | =

|x − ζ| , |x| |ζ|

and similarly for y. Then the equation exp kD (x, y) = exp kD′ (x′ , y ′ ) gives us |x − ζ| |x − ζ| |y| |ζ| = , |y − ζ| |x| |ζ| |y − ζ| i.e. |x| = |y|. This contradiction shows that w ∈ D. Since f maps D into R2 , it is clear that w 6∈ D, so it follows that w is a boundary point. We call D a C k domain, if ∂D is locally the graph of a C k function. Note that if D is a C 1 domain, then certainly every boundary point is circularly accessible. Proposition 4.2. Let D ( R2 be a C 1 domain, and let f : D → R2 be a quasihyperbolic isometry which is also M¨ obius. If D is not a half-plane, then f is a similarity. Proof. We assume that f is not a similarity map. By the previous lemma we see that there is no loss of generality in considering only the case when f is the inversion centered at a boundary point. For simplicity of exposition, we normalize so that the origin is this center. Let ζ be a boundary point of D distinct from 0 and let u be the inward pointing unit normal at ζ. For all sufficiently small t > 0, the point xt = ζ + tu lies in D and its closest boundary point is ζ. For such s < t, we have t kD (xt , xs ) = log . s To estimate the distance of the image points, we use the inequality |x′ − y ′ | ′ ′ ≤ kD′ (x′ , y ′ ), jD′ (x , y ) = log 1 + min{δ ′ (x′ ), δ ′ (y ′ )} which is always valid (since kD′ is the inner metric of jD′ , e.g. [12, Lemma 2.1]). We also need the formula |x − y| |x′ − y ′ | = |x| |y|

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for the length distortion of an inversion. Using these facts and the estimate δ ′ (x′ ) ≤ |x′ − ζ ′ |, we derive the inequality |x′ − y ′ | kD′ (x′ , y ′ ) ≥ log 1 + min{δ ′ (x′ ), δ ′ (y ′ )} |x − y|/(|x| |y|) ≥ log 1 + min{|x′ − ζ ′ |, |y ′ − ζ ′ |} |x − y| |ζ| = log 1 + |x| |y| min{|x − ζ|/|x|, |y − ζ|/|y|} |x − y| |ζ| . = log 1 + min{|y| |x − ζ|, |x| |y − ζ|} Applying this inequality to the points xt and xs as defined before, we have (t − s) |ζ| kD′ (x′t , x′s ) ≥ log 1 + . min{t |xs |, s |xt |}

Let us choose t = 2s. Since |x2s | and |xs | both tend to |ζ| as s → 0, we see that the second term in the minimum is smaller. Since the inversion is supposed to be an isometry, we can use the formula for kD (xt , xs ) from before with the previous inequality to conclude that 2s (2s − s) |ζ| log ≥ log 1 + . s s |x2s | Taking the exponential function gives |x2s | ≥ |ζ|. Since xs = ζ + su, this implies that hζ − 0, ui ≥ 0 as s → 0, where h, i denotes the scalar product.

Applying the same argument, but starting with points on the image side, we conclude that the opposite inequality is also valid. (There is actually a slight asymmetry here: the domain D′ need not have circularly accessible boundary at the origin. However, it is clear that this does not affect the argument so far.) Thus it follows that hζ − 0, ui = 0 for all boundary points. But since the boundary is assumed to be C 1 , this implies that the domain is a half-plane.

From [29, Theorem 2.8] we know that if f : D → R2 is a quasihyperbolic isometry, then f is conformal in D. In dimensions three and higher every conformal mapping is Möbius. It is easy to see that the proofs in this section work also in the higher dimensional case. Therefore, we have proved the following result: Corollary 4.3. Let D be a C 1 domains in Rn , n ≥ 3, which is not a half-space. Then every quasihyperbolic isometry is a similarity mapping. Example 4.4. Note that if we do not assume C 1 boundary, then there are some further domains with non-trivial isometries: the punctured planes R2 \ {a} and sector domains (i.e. domains whose boundary consists of two rays). In both cases inversions centered at the distinguished boundary point (a or the vertex of the sector). The previous proposition strongly suggests that these are all the examples.


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4.2. Curvature of the quasihyperbolic metric. Let D be a domain in R2 . We call a disc B ⊂ D maximal, if it is not contained in any other disc contained in D. The set consisting of the centers of all maximal discs in D is called the medial axis of D and denoted by MA(D). The medial axis and differentiability properties of the distance-to-the-boundary function have been studied e.g. in [7, 8, 9]. In a general domain the Gaussian curvature of the quasihyperbolic metric is not defined, since the distance-to-the-boundary function is not C 2 . M. Heins [20] considered this situation for a quite general class of metric, and defined the notions of upper and lower curvature. Martin and Osgood worked with these curvatures in the context of the quasihyperbolic metric, see [29, Section 3] for details. However, if our domain is sufficiently regular (say C 2 ), and we are considering points not on the medial axis, then the upper and lower curvature agree, and define the curvature. In this case the curvature of kD is given by KD (z) = −δ(z)2 △ log δ(z),

[20, (1.3)] or [29, (3.1)]. On the medial axis this formula does not make sense, but the upper and lower curvatures still agree, and both equal −∞, by [29, Corollary 3.12]. The next lemma is a specialization of Lemma 3.5, [29] to the case there the upper and lower curvatures agree. ˜ be C 2 domains such that B(z, r) ⊂ Lemma 4.5 (Lemma 3.5, [29]). Let G and G ˜ and ζ ∈ (∂G) ∩ (∂ G) ˜ ∩ (∂B(z, r)). If there is a neighborhood U of ζ such G∩G ˜ ˜ \ U ) > d(z, ∂ G), ˜ then KG (z) ≤ K ˜ (z). that G ∩ U ⊂ G ∩ U and d(z, ∂ G G Using this lemma we can derive the following very plausible statement, which says that the Gaussian curvature of the quasihyperbolic metric depends only on the curvature of the boundary at the closest boundary point. We sill need some more notation. Let B be a disc with ζ ∈ (∂B) ∩ (∂D). Then we call B the osculating disc at ζ if ∂B and ∂D have second order contact at ζ. Let D be at least a C 2 domain. Then there exists an osculating disc at every boundary point ζ. If this disc has radius r, then we define Rζ to be r if the disc lies in the direction of the interior of D, and −r otherwise. Note that the function ζ 7→ 1/Rζ is C k−2 in a C k domain, k ≥ 2. Proposition 4.6. Let D ( R2 be a C 2 domain and z ∈ D \ MA(D) have closest boundary point ζ ∈ ∂D. Then 1 Rζ =− . KD (z) = − Rζ − δ(z) 1 − δ(z)/Rζ

If z lies on the medial axis, then KD (z) = −∞.

Proof. The medial axis consists of points equidistant to two or more nearest boundary points, and of centers of osculating circles. For the former, the claim

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that KD (z) = −∞ follows from [29, Corollary 3.12]. So we assume that z has a unique nearest boundary point, ζ. We suppose further that Rζ > 0, the other case begin similar. Let B(w, Rζ ) be the osculating disc at ζ. We define Bt = B(w +

w−ζ t, Rζ Rζ

+ t),

and note that ∂Bt contains ζ for all t > −Rζ . We have the formula r r =− KB(0,r) (x) = − |x| r − d(x, ∂B(0, r))

for the curvature of the quasihyperbolic metric in a ball [29, Lemma 3.7], so we can calculate KBt (z) explicitly. ˜ = Bt for t > 0 gives KD (z) ≤ Using the previous lemma with G = D and G KBt (z). If z is the center of B0 , then right-hand-side of this inequality tends to −∞ as t → 0, which completes the proof of the claim regarding the medial axis. So we assume that z is not the center of B0 , and then we can apply the ˜ = D to get Lemma 4.5 with G = Bt for t < 0 (sufficiently close to 0) and G KBt (z) ≤ KD (z). Thus we have KB−t (z) ≤ KD (z) ≤ KBt (z)

for small t > 0. Since KBt is continuous in t, we get KD (z) = KB0 (z) as we let t → 0. The proof is completed by applying the aforementioned formula for the curvature to the ball B0 = B(w, Rζ ). Let f : D → R2 be a C 1 mapping. By ∇f we denote the gradient of f , ˜ (z) we denote δ(z)∇f (z). The reason for i.e. the vector (∂1 f, ∂2 f ), and by ∇f multiplying by δ(x) is that |x − y| , x→y kD (x, y)

δ(y) = lim

˜ operator is more natural in the setting where the quasihyperbolic so that the ∇ but not the Euclidean distance is preserved (see (4.9), below). ˜ D . For this need a mapping which We next present an explicit formula for ∇K associates to every point in D \ MA(D) its closest boundary point. We call this mapping ζ = ζ(z). Lemma 4.7. Let D ( R2 be a C 3 domain. Then ˜ D (z) = (KD (z) + 1) KD (z)∇δ(z) − (KD (z) + 1)∇Rζ(z) ∇K

for every z off the medial axis, where all differentiation is with respect to the variable z. Proof. We use the formula from Proposition 4.6. Thus ∇KD (z) = −∇

δ(z) KD (z)2 1 = KD (z)2 ∇ = (Rζ ∇δ(z) − δ(z)∇Rζ ), 1 − δ(z)/Rζ Rζ Rζ2


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where we understand ζ as a function of z. Note that Rζ and δ are C 1 , since D is C 3 and we are not on the medial axis. From Proposition 4.6 we also get KD (z) + 1 δ(z) = . Rζ KD (z)

Thus we continue the equation by ˜ D (z) = KD (z)2 δ(z) ∇δ(z) − δ(z) ∇Rζ ∇K Rζ Rζ

= (KD (z) + 1) KD (z)∇δ(z) − (KD (z) + 1)∇Rζ .

˜ is an intrinsic quantity of the quasihyperbolic metric. We next show that |∇K| Lemma 4.8. Let D be a C 3 domain. If f : D → R2 is a quasihyperbolic isometry, ˜ D (z)| = |∇K ˜ f (D) (f (z))| for every z ∈ D. then |∇K Proof. We know that f is conformal. For a unit vector u we find that

˜ D (z), u = lim KD (z + εu) − KD (z) ∇K ε→0 kD (z + εu, z) (4.9) Kf (D) (f (z + εu)) − Kf (D) (f (z)) = lim . ε→0 kf (D) (f (z + εu), f (z)) Next we note that f (z + εu) = f (z) + εf ′ (z)u + O(ε2 ). Here f ′ (z)u is understood ′ (z) u. Then as complex multiplication. Let us define another unit vector u˜ = |ff ′ (z)| we continue the previous equation by

Kf (D) (f (z) + εf ′ (z)u) − Kf (D) (f (z)) ˜ ∇KD (z), u = lim ε→0 kf (D) (f (z) + εf ′ (z)u, f (z)) ε|f ′ (z)|h∇Kf (D) (f (z)), u˜i = lim ε→0 ε|f ′ (z)|δ ′ (f (z))−1

˜ f (D) (f (z)), u˜ . = ∇K ˜ D (z)| = |∇K ˜ f (D) (f (z))|. Since u was an arbitrary unit vector, we see that |∇K

4.3. Isometries of the quasihyperbolic metric. We know that similarities are always quasihyperbolic isometries, and we want to show that in most cases these are the only ones. In view of the results in Section 4.1, it suffices for us to show that a quasihyperbolic isometry is a Möbius mapping, so this will be what we aim at in the proofs of this section. A curve γ in D is a (quasihyperbolic) geodesic if kD (x, y) = kD (x, z) + kD (z, y) for all x, z, y ∈ γ in this order. It is clear from this definition that geodesics are preserved by isometries. A geodesic ray is a geodesic which is isometric to R+ . For every z ∈ D we easily find one geodesic ray, namely [z, ζ(z)), which also happens to be a Euclidean line segment. The idea is to show that this geodesic

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is somehow special (from a quasihyperbolic point-of-view), so that it would map to a geodesic ray of the same kind. Lemma 4.10. Let D ( R2 be a C 2 domain with a boundary point ξ such that 1/Rξ = 0. Then every isometry f : D → R2 of the quasihyperbolic metric is M¨ obius. Proof. Let B ⊂ D be a non-maximal disc whose boundary contains ξ and let z denote the center of B. By Proposition 4.6 we find that KD ≡ −1 on the segment γ = [z, ξ). Thus Kf (D) ≡ −1 on γ ′ , so 1/Rζ′ ′ (z′ ) = 0 for every point z ′ on this curve. We consider two cases: either ζ ′ (z ′ ) is just a single point for all z ′ ∈ γ ′ , or it sweeps out a non-degenerate subcurve of the boundary ∂D′ as z ′ varies over γ ′ . (There is no third possibility, since ζ ′ is a continuous function on γ ′ .) In the single-point case we see that γ ′ has to be a line segment, since the boundary does not have corners. In this case we find that ′ ′ kD (x, y) = log |x−ξ| and kD′ (x′ , y ′ ) = log |x′ −ξ′ | , ′

|y−ξ|

|y −ξ |

where ξ is the closest boundary point to the every point on γ ′ . But this easily implies that f is Möbius on γ. Since f is conformal it follows by uniqueness of analytic extension that f is a Möbius mapping on all of D. So we consider the second case, that ζ ′ (z ′ ) sweeps out a non-degenerate subcurve of the boundary ∂D′ . Since the curvature of the boundary at all these points is zero, it follows that the piece of the boundary is a line segment, L′ . Let U ′ ⊂ D′ be an open set such that (∂U ′ ) ∩ (∂D′ ) = L′ and the nearest boundary point of every point in U ′ lies in L′ . Then the geometry of the quasihyperbolic metric in U is the same as in a half-plane, in particular KD′ ≡ −1 on U ′ . Then KD ≡ −1 on U = f −1 (U ′ ), so it follows that (∂U ) ∩ (∂D) = L, for some line segment L. So it follows that f |U is the restriction of a quasihyperbolic isometry of the half-plane. But these are only the Möbius mappings. Then we again conclude from the uniqueness of analytic extension that f is a Möbius mapping on all of D. Let us call a domain strictly concave, if its complement is strictly convex. Corollary 4.11. Let D ( R2 be a C 2 domain which is not a half-plane, strictly convex or strictly concave. Then every quasihyperbolic isometry is a similarity mapping. Proof. Suppose that 1/Rζ 6= 0 for all boundary points. Since 1/Rζ is continuous by assumption, this implies that it is either everywhere positive, or everywhere negative. In these cases we have a strictly convex and strictly concave domain, respectively, which was ruled out by assumption. So we find some point at which 1/Rζ = 0. Then it follows from Lemma 4.10 that the isometry is Möbius and from Lemma 4.2 that it is a similarity. So we are left with only two types of domains that we cannot handle: strictly convex and strictly concave ones. As usual when working with isometries, the


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nicest domains turn out to be the most difficult. Unfortunately, we need to assume more regularity of the boundary in order to take care of these cases. Theorem 4.12. Let D ( R2 be a C 3 domain, which is not a half-plane. Then every isometry f : D → R2 of the quasihyperbolic metric is a similarity mapping. Proof. In view of Corollary 4.11, we may restrict ourselves to the case when KD (z) 6= −1 for all z ∈ D. Let z ∈ D \ MA(D) and ζ be its nearest boundary point. We note that ∇δ(z) and ∇Rζ are perpendicular – first of all, ∇δ(z) is parallel to z − ζ; second, Rζ is a constant in the direction of z − ζ, since ζ is the closest boundary point to all points on this line (near z). If D is bounded, then it is clear that Rζ has a critical point. If D is unbounded, then we note that 1/Rζ cannot have any other limit than 0 at ∞ (although a limit need not exist, of course). Thus we see that Rζ has a critical point in the unbounded case as well. Let ζ be a critical point of ξ 7→ Rξ and fix a point z ∈ D with KD (z) 6= −∞ whose nearest boundary point is ζ. Of course, ∇Rζ = 0 at the critical point ζ. Then it follows from Lemma 4.7 that ˜ D (z) = (KD (z) + 1)KD (z)∇δ(z). ∇K Since the curvature is intrinsic to the metric, we have KD′ (z ′ ) = KD (z). Also, ˜ D′ (z ′ )| = |∇K ˜ D (z)| by Lemma 4.8, so we have |∇K (KD (z) + 1)KD (z)∇δ(z) = (KD (z) + 1) KD (z)∇δ ′ (z ′ ) − (KD (z) + 1)∇Rζ′ ′ (z′ ) .

We know that KD (z) 6= −1 and that ∇δ ′ (z ′ ) and ∇Rζ′ ′ (z′ ) are orthogonal. Thus the previous equation simplifies to 2 2 2 KD (z)|∇δ(z)| = KD (z)|∇δ ′ (z ′ )| + (KD (z) + 1) ∇Rζ′ ′ (z′ ) . Since |∇δ| = 1 off the medial axis for every domain, this equation implies that ∇Rζ ′ = 0.

So for our point z, ∇KD (z) and ∇KD′ (z ′ ) point to the nearest boundary point of z and z ′ , respectively. Let γ = [z, ζ). Note that γ is a geodesic of the quasihyperbolic metric. Also, ∇KD (z) and γ are parallel at z. Now γ is mapped to some geodesic ray γ ′ , and since f is a conformal mapping, γ ′ is parallel to ∇KD′ (z ′ ) at z ′ . But [z ′ , ζ ′ ) is a geodesic parallel to ∇KD′ (z ′ ) at z ′ , and since geodesics are unique (when the density is C 2 , i.e. except possibly on the medial axis) we see that γ ′ = [z ′ , ζ ′ ). So we have shown that f ([z, ζ)) = [z ′ , ζ ′ ). Moreover, we have |x−ζ| |x′ −ζ ′ | ′ ′ kD (x, y) = log |y−ζ| and kD′ (x , y ) = log |y′ −ζ ′ |

for x, y ∈ [z, ζ). Thus we see that f is just a similarity on [z, ζ). But f is a conformal map, so this implies that f is a similarity in all of D. Acknowledgment. I would like to thank Zair Ibragmov for several discussions about the isometries of this and related metrics and Swadesh Sahoo for some comments on this manuscript.

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References [1] Z. Balogh and S. Buckley: Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003), no. 2, 261–301. [2] A. Beardon: Geometry of Discrete Groups, Springer-Verlag, New York, 1983; corrected reprint, 1995. : The Apollonian metric of a domain in Rn , pp. 91–108 in Quasiconformal mappings [3] and analysis (P. Duren, J. Heinonen, B. Osgood and B. Palka (eds.)), Springer-Verlag, New York, 1998. [4] A. Beardon and D. Minda: Sphere-preserving maps in inversive geometry, Proc. Amer. Math. Soc. 130 (2002), 987–998. [5] L. Blumenthal: Distance Geometry. A study of the development of abstract metrics. With an introduction by Karl Menger, Univ. of Missouri Studies Vol. 13, No. 2, Univ. of Missouri, Columbia, 1938. [6] D. Burago, Yu. Burago and S. Ivanov: A course in metric geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc., Providence, RI, 2001. [7] L. Cafarelli and A. Friedman: The free boundary for elastic-plastic torsion problems, Trans. Amer. Math. Soc. 252 (1979), 65–97. [8] H. I. Choi, S. W. Choi and H. P. Moon: Mathematical theory of medial axis transform, Pacific J. Math. 181 (1997), no. 1, 57–88. [9] J. Damon: Smoothness and geometry of boundaries associated to skeletal structures. I. Sufficient conditions for smoothness, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1941–1985. [10] J. Ferrand: A characterization of quasiconformal mappings by the behavior of a function of three points, pp. 110–123 in Proceedings of the 13th Rolf Nevalinna Colloquium (Joensuu, 1987; I. Laine, S. Rickman and T. Sorvali (eds.)), Lecture Notes in Mathematics Vol. 1351, Springer-Verlag, New York, 1988. [11] F. Gehring and B. Osgood: Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74. [12] F. Gehring and B. Palka: Quasiconformally homogeneous domains, J. Anal. Math. 30 (1976), 172–199. [13] P. Hästö: A new weighted metric: the relative metric II, J. Math. Anal. Appl. 301 (2005), no. 2, 336–353. [14] : Gromov hyperbolicity of the jG and ˜G metrics, Proc. Amer. Math. Soc. 134 (2006), 1137–1142. [15] : Isometries of the quasihyperbolic metric, submitted. [16] P. Hästö and Z. Ibragimov: Apollonian isometries of planar domains are Möbius mappings, J. Geom. Anal. 15 (2005), no. 2, 229–237. [17] : Apollonian isometries of regular domains are Möbius mappings, Ann. Acad. Sci. Fenn. Math., to appear. [18] P. Hästö, Z. Ibragimov and H. Lindén: Isometries of relative metrics, Comput. Methods Funct. Theory 6 (2006), no. 1, 15–28. [19] P. Hästö and H. Lindén: Isometries of the half-apollonian metric, Complex Var. Theory Appl. 49 (2004), 405–415. [20] M. Heins: On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60. [21] David Herron, Zair Ibragimov and David Minda: Geometry of the K-P metric, preprint (2005). [22] D. Herron and P. Koskela: Conformal capacity and the quasihyperbolic metric, Indiana Univ. Math. J. 45 (1996), no. 2, 333–359. [23] D. Herron, W. Ma and D. Minda: A Möbius invariant metric for regions on the Riemann sphere, pp. 101–118 in Future Trends in Geometric Function Theory (RNC Workshop, Jyv¨ askylä 2003; D. Herron (ed.)), Rep. Univ. Jyv¨ askylä Dept. Math. Stat. 92 (2003).


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[24] P. Jones and S. Smirnov: Removability theorems for Sobolev functions and QC maps, Ark. Mat. 38 (2000), no. 2, 263–279. [25] P. Koskela and T. Nieminen: Quasiconformal removability and the quasihyperbolic metric, Indiana Univ. Math. J. 54 (2005), no. 1, 143–151. [26] R. Kulkarni and U. Pinkall: A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), 89–129. [27] H. Lindén: Quasihyperbolic Geodesics and Uniformity in Elementary Domains, Ph.D. Thesis, University of Helsinki, 2005. [28] G. Martin: Quasiconformal and bilipschitz mappings, uniform domains and the quasihyperbolic metric, Trans. Amer. Math. Soc. 292 (1985), 169–191. [29] G. Martin and B. Osgood: The quasihyperbolic metric and associated estimates on the hyperbolic metric, J. Anal. Math. 47 (1986), 37–53. [30] P. Seittenranta: Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc. 125 (1999), 511–533. [31] M. Vuorinen: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., Vol. 1319, Springer, Berlin, 1988. Peter H¨ ast¨ o E-mail: [email protected] Address: Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland


Uniform Spaces and Gromov Hyperbolicity David A Herron Abstract. This brief outline contains mostly definitions, background information, and statements of theorems. Along with the title topics, we also discuss the uniformization volume growth problem as well as certain capacity and slice condition characterizations of uniformity. Keywords. uniform spaces, Gromov hyperbolicity, quasihyperbolic metric, volume growth, Ahlfors regular spaces, Loewner spaces, slice conditions. 2000 MSC. Primary: 30C65; Secondary: 53C23, 30F45.


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2. Metric Space Background

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2.A. General Information

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2.B. Abstract Domains

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2.C. Maps and Gauges

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2.D. Length and Geodesics

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2.E. Connectivity Conditions

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2.F. Doubling and Dimensions

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2.G. Quasiconformal Deformations

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2.H. Quasihyperbolic Distance and Geodesics

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2.I. Modulus and Capacity

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2.J. Ahlfors Regular and Loewner Spaces

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2.K. Slice Conditions 3. Uniform Spaces

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3.A. Euclidean Setting

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3.B. Measure Metric Space Setting

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3.C. Basic Information

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4. Gromov Hyperbolicity 4.A. Thin Triangles Definition Version October 19, 2006. The author was supported by the NSF and the Charles Phelps Taft Research Center.

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4.B. Gromov Boundary

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4.C. Connection with Uniform Spaces

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5. Uniformization

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5.A. Uniformization Problem

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5.B. BHK Uniformization

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5.C. Bounded Geometry and its Consequences

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5.D. Lifts and Metric Doubling Measures

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5.E. Volume Growth Problem

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6. Characterizations of Uniform Spaces

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6.A. Metric Characterizations

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6.B. Gromov Boundary Characterizations

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6.C. Characterizations using QC Maps

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6.D. Capacity Conditions

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6.E. LLC and Slice Conditions

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References

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1. Introduction This survey is meant to supplement the talks I presented at the International Workshop on Quasiconformal Mappings and their Applications and at the International Conference on Geometric Function Theory, Special Functions and their Applications. Primarily, I provide here basic background material including definitions, terminology, and fundamental facts. I also list a few references, many of which themselves contain additional references to this material. I have made no attempt to render a complete list of references and apologize to all those whose work I have neglected to mention. The reader is absolutely encouraged to consult the many works referred to by the authors I do mention. The goal of these notes is to provide the reader with a foundation enabling them to understand the meaning and relevance of the recent work [BHK01], [BHR01], [BKR98] of Bonk, Heinonen, Koskela and Rohde along with [Her04] and [Her06]. I am delighted to thank Mario Bonk, Juha Heinonen and Pekka Koskela for numerous helpful discussions and hours of blackboard sessions regarding these topics. By now Euclidean uniform spaces (domains in Euclidean space in which points can be joined by short twisted double cone arcs) are well recognized as being the ‘nice’ spaces for quasiconformal function theory as well as many other areas of analysis (e.g., potential theory); see [Geh87], [Väi88] for Euclidean space and [Gre01], [CGN00], [CT95] for the Carnot-Carathéodory setting. In [BHK01] Bonk, Heinonen, and Koskela develop a uniformization theory which provides a

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two way correspondence between uniform spaces and Gromov hyperbolic spaces. In particular, they prove the following fundamental result; see [BHK01, Theorem 1.1]. There is a one-to-one (conformal) correspondence between quasiisometry classes of proper geodesic roughly starlike Gromov hyperbolic spaces and quasisimilarity classes of bounded locally compact uniform spaces. A simple, yet beautiful, example is the open unit disk in the plane. In terms of its Euclidean geometry, each pair of points can be joined by a twisted double cone which stays away from the boundary and is not much bigger than the distance between the given points (it is a uniform space). On the other hand, the disk also admits a non-euclidean geometry, in terms of its Poincaré hyperbolic metric, and as such the disk is a Gromov hyperbolic space. The Bonk, Heinonen, Koskela theory asserts that this phenomenon holds in a very general setting. The complete proof of their result is presented in Chapters 2-5 of [BHK01] and beyond the scope of our discussion. However, there are two basic results involved which are central to my workshop lectures: Fact 4.1 says that every locally compact uniform space has a Gromov hyperbolic quasihyperbolization; Fact 5.1 says that every (proper geodesic) Gromov hyperbolic space can be uniformized. I will describe what uniform spaces are, what their connection is with Gromov hyperbolicity, and explain some of the ideas behind the proofs. Time permitting, I will also look at the related question of when there exists a uniformization with the property that the associated measure (see (2.7)) has regular volume growth. My conference lecture will focus on §6.D and §6.E. For the remainder of this introduction, I advertise results from [Her04] and [Her06] hoping to wet the reader’s appetite for this flavor of metric measure space geometric function theory. See §2-§5 for precise definitions.

In [BHR01] and [BKR98] the authors investigate conformal deformations of the unit ball in Euclidean space. The primary object of study in these notes is the geometry of quasiconformal deformations of an abstract metric measure space (Ω, d, µ). Following BHKR, we consider a metric-density ρ on Ω and Ωρ = (Ω, dρ , µρ ) denotes the deformed space (see subsection 2.G). We are interested in the situation when this new space Ωρ is uniform (see Section 3) and describe this by calling such a ρ a uniformizing density. Every proper geodesic Gromov hyperbolic space can be uniformized, and, there is a natural canonical proper geodesic space associated with any locally compact abstract domain, namely, its quasihyperbolization; see Facts 4.1 and 5.1. However, in general the associated measure (see (2.7)) may fail to have Ahlfors regular volume growth. For example, applying the BHK uniformization to the quasihyperbolized Euclidean unit ball we obtain a new metric measure space which has exponential volume growth. The theory developed in [BHK01] is exploited in [Her06] to extend some results of [BHR01] to the setting of abstract metric measure spaces (Ω, d, µ). More importantly, we establish the result given below which provides an answer to the

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question: When does an abstract domain admit a quasiconformal deformation which is both uniformizing and such that the induced measure (2.7) satisfies the natural volume growth estimate? That is, when is there a conformal uniformizing density? In particular, the induced measure should be Ahlfors regular. Under certain reasonable minimal hypotheses, this occurs precisely when the conformal Assouad dimension of the space’s Gromov boundary is small enough. See §5.E for a discussion of the proof of the following. Theorem A. Let Ω be an abstract domain with bounded Q-geometry. Suppose Ω admits a bounded uniformizing conformal density. Then Ω has a Gromov hyperbolic roughly starlike quasihyperbolization and the conformal Assouad dimension of its Gromov boundary is strictly less than Q. The converse holds too, provided we assume that the Gromov boundary of Ω is uniformly perfect. The above result is quantitative: the asserted constants depend only on the data associated with Ω and the density. In what follows we consider metric measure spaces (Ω, d, µ) which satisfy the following basic minimal hypotheses: Ω is an abstract domain having bounded Q-geometry and a Gromov hyperbolic roughly starlike quasihyperbolization. Precise definitions are stated in subsections 2.B, 2.D, 2.H, 5.C; roughly, these hypotheses ensure that Ω has ‘enough’ of the local properties enjoyed by domains in Euclidean space. The data associated with these basic hypotheses consists of six parameters: Q (the ‘dimension’), M , m, λ (the bounded geometry constants), δ (the Gromov hyperbolicity constant) and κ (the rough starlike constant). There are a number of auxiliary results (namely, Theorems B-F) needed for the proof of Theorem A; all of these can be found in [Her04] or [Her06]. First we have the so-called Gehring-Hayman Inequality (cf. [GH62]); it is an essential tool for most of what follows. This was proved in [BKR98, Theorem 3.1] for deformations of the Euclidean unit ball and in [HR93] for quasiconformal images of uniform domains in Euclidean space; see also [BB03, Theorem 2.3], [BHK01, Chpt. 5] and [HN94]. Our proof of the following (see [Her04, Theorem A]) utilizes ideas from both [BKR98, Theorem 3.1] and [HR93, Theorem 1.1]. Theorem B. Let ρ be an Ahlfors Harnack density on a uniform Loewner metric measure space (Ω, d, µ). Then there exists a constant Λ such that for all ¯ quasihyperbolic geodesics [x, y]k with endpoints in Ω, ℓρ ([x, y]k ) ≤ Λ dρ (x, y). This result is quantitative: Λ depends only on the data associated with Ω. Throughout this article the symbol Λ will stand for this Gehring-Hayman Inequality constant. Here is a simple, but useful, consequence of the Gehring-Hayman Inequality: if there is an arc α joining some point w in Ω to some point ζ in ∂Ω with


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ℓρ (α) < ∞, then ℓρ (γ) < ∞ for every quasihyperbolic geodesic ray going to ζ. In fact, there is even a ‘radial limit theorem’ [Her04, Theorem B] which says that this is true for modQ -a.e. point of ∂Ω. Next we communicate the primary tool employed in our proof of Theorem A. It is based on a lifting procedure discussed in [BKR98, 2.7] and established for the Euclidean unit ball as [BHR01, Proposition 1.25]. See (5.9) and (2.5) for the definitions of ρν (the lift of ν) and δν,1/P (the quasimetric determined by ν). See §5.D for a discussion of the proof of the following. Theorem C. Assume the basic minimal hypotheses, that the Gromov boundary of Ω is uniformly perfect, and that P < Q. Suppose ν is a P -dimensional metric doubling measure on ∂G Ω. Then the lift ρ = ρν of ν is a doubling conformal density on Ω and the natural map (∂ρ Ω, dρ ) → (∂G Ω, δν,1/P ) is bilipschitz. The Bonk-Heinonen-Koskela uniformization theory is a crucial tool employed in all our arguments and permits us to replace the space Ω with a bounded uniform space Ωε where the geometry is more transparent; see Fact 5.1. A key ingredient in our proof of Theorem A is the following generalization of [BHR01, Proposition 2.11]. In particular, it asserts that a conformal density on a bounded uniform space is uniformizing if and only if the associated measure (2.7) is a doubling measure on the original space. (See §5.E for the precise definition of a doubling conformal density.) Theorem D. Assume the basic minimal hypotheses. Let Ωε be any BHKuniformization of Ω. Suppose ρ is a conformal density on Ω. Then the following are quantitatively equivalent: (a) (b) (c) (d) (e)

ρ is doubling on Ω. Ωρ is bounded and uniform. Ωρ is bounded and Q-Loewner. Ωρ is bounded, Q-Loewner and Ahlfors Q-regular. the identity map Ωρ → Ωε is quasisymmetric.

Again, this result is quantitative: the asserted constants depend only on the data associated with Ω and ρ, and the related data. Also, we point out that the proof of (b) shows that the quasihyperbolic geodesics in Ω will be uniform arcs in Ωρ . A crucial component of the proof of Theorem C is the following result which permits us to estimate dρ (x) = distρ (x, ∂ρ Ω) in terms of ρ(x)d(x). More precisely, it tells us that Ahlfors Harnack metric-densities are Koebe under the right conditions. The lower bound is immediate via the Harnack inequality. To obtain any upper bound, we at least need ∂ρ Ω 6= ∅. In fact, we require a condition which ensures that Ω has a uniformly thick boundary as seen from each point. With this in mind, we introduce the following notion: we say that (Ω, d, µ) satisfies a Whitney ball modulus property if there exists a constant m > 0 such that ¯ λ d(x)), ∂Ω; Ω) ≥ m modQ (B(x;

for all x ∈ Ω.

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Theorem E. Let ρ be a Ahlfors Harnack metric-density on a uniform Loewner abstract domain (Ω, d, µ). Suppose that Ω enjoys a Whitney ball modulus property. Then there is a constant K such that for all x ∈ Ω, K −1 ρ(x)d(x) ≤ dρ (x) ≤ Kρ(x)d(x);

the constant K depends only on the data associated with Ω. An important consequence of Theorem E is that the quasihyperbolizations of Ω and Ωρ are bilipschitz equivalent, and it follows that (Ωρ , kρ ) is a Gromov hyperbolic space. We mention that any uniform Loewner space with connected boundary satisfies a Whitney ball modulus property, provided it and its boundary are simultaneously bounded or unbounded. Similarly any bounded uniform Loewner space with a finite number of non-degenerate boundary components will enjoy this modulus property. Here is a sufficient condition for this property to hold which allows for a totally disconnected boundary. Theorem F. Let (Ω, d, µ) be a locally Loewner, uniform metric measure space. Assume Ω and ∂Ω are either both bounded or both unbounded. Suppose that for some p > 0, ∂Ω satisfies the Hausdorff p-content condition p ¯ r)) ≥ c rp (∂Ω ∩ B(ζ; H∞

for all 0 < r ≤ diam(∂Ω) and all ζ ∈ ∂Ω.

Then Ω enjoys a Whitney ball modulus property with a constant m which depends only on c and the data associated with Ω. In contrast to the Euclidean case, the converse to the above is false; see [Her04, Example 3.2] which furnishes a space with an isolated boundary point which nonetheless satisfies a Whitney ball modulus property. Our notation is relatively standard and, for the most part, conforms with that of [BHK01]. We write C = C(a, . . .) to indicate a constant C which depends only on the parameters a, . . .; the notation A . B means there exists a finite constant c with A ≤ cB, and A ≃ B means that both A . B and B . A hold. Typically a, b, c, C, K, . . . will be constants that depend on various parameters, and we try to make this as clear as possible often giving explicit values, however, at times C will denote some constant whose value depends only on the data present but may differ even on the same line of inequalities.

2. Metric Space Background Naturally there are scores of references for metric space geometry. Here is a brief list of some texts which I have found especially helpful: [BH99], [BBI01], [Hei01], [Sem01], [Sem99], [DS97], and of course the references mentioned in these works.


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2.A. General Information. In what follows (X, d) will always denote a generic metric space possessing no additional presumed properties. For the record, this means that d is a distance function; that is, d : X × X → R is positive semidefinite, symmetric, and satisfies the triangle inequality. We often write the distance between x and y as d(x, y) = |x − y|. The open ball (sphere) of radius r centered at the point x is B(x; r) := {y : |x−y| < r} (S(x; r) := {y : |x−y| = r}). When B = B(x; r) and λ > 0, λB := B(x; λ r). We say that X is a proper metric space if it has the Heine-Borel property that every closed ball is compact (or equivalently, the compact sets are exactly the closed and bounded sets). In general, we work in the setting of a metric measure space (X, d, µ) with X a non-complete locally complete (often locally compact) rectifiably connected metric space and µ a Borel regular measure satisfying µ[B(x; r)] > 0 for each ball. Recall that every metric space can be isometrically embedded into a complete ¯ denote the metric completion of a metric space X and metric space. We let X ¯ we call ∂X = X \ X the metric boundary of X. Then d(x) = dist(x, ∂X) is the distance from a point x ∈ X to the boundary ∂X of X; note that when ∂X is ¯ we have d(x) > 0 for all x ∈ X. For example, this holds when X closed in X, is locally compact. Of course, if X is complete to begin with, then ∂X = ∅ and d(x) = ∞ for all x ∈ X. We call X locally complete provided d(x) > 0 for all x ∈ X. In a locally complete metric space we make extensive use of the notation B(x) := B(x; d(x)). In this setting, we call λB(x) = B(x; λ d(x)) a Whitney ball in X with associated Whitney ball constant λ ∈ (0, 1).

It is convenient, at times, to consider quasimetric spaces (X, q). We call q a quasimetric on X if q : X × X → R is symmetric and positive definite but only satisfies q(x, y) ≤ K (q(x, z) + q(y, z)) for all x, y, z ∈ X in place of the triangle inequality. See [Hei01, 14.1], [Sem01] and [DS97].

Starting with a quasimetric q, there is a standard way to define a pseudometric d with d ≤ q (cf. [BH99, 1.24, p.14]), but it may happen that d(x, y) = 0 for some x 6= y. However, by first ‘snowflaking’ q and then applying this procedure we can arrive at an honest distance function; see [Hei01, Proposition 14.5] or [BH99, Proposition 3.21, p.435]. 2.1. Fact. Let q be a quasimetric on X. There is an ε0 > 0 depending only on the quasimetric constant K for q such that for all ε ∈ (0, ε0 ), the quasimetric qε (x, y) = q(x, y)ε is bilipschitz equivalent to an honest distance function d on X; in fact there is a constant L = L(ε, K) such that L−1 qε (x, y) ≤ d(x, y) ≤ qε (x, y)

for all x, y ∈ X.

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We remark that all the quasimetrics qε as defined above are QS equivalent to each other. Another useful notion, apparently introduced by Väisälä, is that of a metametric m : X × X → R which is symmetric, non-negative, satisfies the triangle inequality, but only m(x, y) = 0 =⇒ x = y and so possibly m(x, x) > 0. See [Väi05a, 4.2] for a treatment of metametric spaces. A metric space X is called uniformly perfect provided it has at least two points and there is a constant ϑ ∈ (0, 1) such that for all balls B ⊂ X, B \ ϑB 6= ∅ provided X \ B 6= ∅. This concept, which involves three points, is especially useful when dealing with quasisymmetric maps and also with doubling measures (see §2.F). The property of being uniformly perfect is preserved by quasisymmetric homeomorphisms, with the new constant depending only on the original constant and the quasisymmetry data; in particular, one can ask whether or not a conformal gauge is uniformly perfect (see §2.C). It is a routine exercise to see that uniformly perfect locally compact spaces contain quasisymmetrically embedded middle-third Cantor dusts. Using this fact, together with a scaling argument and properties of quasisymmetric homeomorphisms (e.g. [Hei01, 11.10,11.11]), one can verify a version of the following. For a simple more direct approach, which also provides the indicated explicit constants, see [Her06, Lemma 4.2]. 2.2. Fact. Suppose X is a uniformly perfect compact metric space. Then X satisfies the p-dimensional Hausdorff measure density condition Hp [B(x; r)] ≥

rp 6

for all 0 < r ≤ diam(X) and all x ∈ X,

where p = 1/ log2 (4/ϑ) and ϑ is the uniform perfectedness constant. The above result can be used in conjunction with Theorem F to see that the Whitney ball modulus property holds. 2.B. Abstract Domains. We call a metric measure space (Ω, d, µ) an abstract domain if Ω is a non-complete locally complete rectifiably connected metric space (and µ a Borel regular measure with dense support). An important example of such a space is, of course, a proper subdomain of Euclidean space with either Euclidean distance or the induced Euclidean length distance. Unless explicitly indicated otherwise, the adjective locally means that the modified property or condition holds in all Whitney-type balls λB(x) where 0 < λ < 1 is some fixed constant which we call the Whitney ball constant; when there are several such local conditions in play, we always take λ to be the minimum of all the associated Whitney ball constants.


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2.C. Maps and Gauges. An embedding f : X → Y from a metric space X into a metric space Y is quasisymmetric, abbreviated QS, if there is a homeomorphism η : [0, ∞) → [0, ∞) (called a distortion function) such that for all triples x, y, z ∈ X, |x − y| ≤ t|x − z| =⇒ |f x − f y| ≤ η(t)|f x − f z|.

These mappings were studied by Tukia and Väisälä in [TV80]; see also [Hei01]. The bilipschitz maps form an important subclass of the quasisymmetric maps; f : X → Y is bilipschitz if there is a constant L such that for all x, y ∈ X, L−1 |x − y| ≤ |f x − f y| ≤ L|x − y|.

More generally, a map f : X → Y is an (L, C)-quasiisometry if L ≥ 1, C ≥ 0 and for all x, y ∈ X, L−1 |x − y| − C ≤ |f x − f y| ≤ L|x − y| + C.

There seems to be no universal agreement regarding this terminology; some authors use the adjective quasiisometry to mean what we have called bilipschitz, and then a rough quasiisometry satisfies our definition of quasiisometry. So the reader should beware! Of course a (1, 0)-quasiisometry is simply called an isometry (onto its range). Note that the above definitions also make sense for mappings of quasimetric spaces. Given a metric (or a quasimetric) on X, we can form the conformal gauge G on X consisting of all metrics on X which are QS equivalent to the original (quasi)metric. That is, G is the family of all metrics ∂ on X such that the identity map (X, d) → (X, ∂) is QS. See [Hei01, Chapter 15] for more discussion of this topic. An embedding f : X → Y from a metric space X into a metric space Y is called quasim¨ obius, abbreviated QM, if there is a homeomorphism ϑ : [0, ∞) → [0, ∞) (called a distortion function) such that for all quadruples x, y, z, w of distinct points in X, |x, y, z, w| ≤ t =⇒ |f x, f y, f z, f w| ≤ ϑ(t)

where the absolute cross ratio is

|x − y||z − w| . |x − z||y − w| These mappings were introduced and investigated by Väisälä in [Väi85]; see also [Väi05a]. Every QS homeomorphism is QM; the converse holds in certain special cases. Clearly Möbius transformations are QM maps in Euclidean space; however, a Möbius transformation from the unit ball onto a half-space is not QS. The QM maps are more flexible than the QS. The QS and QM maps are defined by global conditions whereas QC (quasiconformal) maps only satisfy a local condition. I highly recommend Tyson’s recent survey article [Tys03]. Väisälä’s notes [Väi71] are the classical reference for QC maps in the Euclidean setting. These maps have been studied in the Heisenberg |x, y, z, w| =

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group setting and there is still much research underway there. Heinonen and Koskela strongly advanced the theory in the general metric space setting; see [HK95] and [HK98]. See Koskela’s notes [Kos07] for a ‘modern’ approach to QC maps in the Euclidean setting. There are three so-called definitions for QC maps: the metric definition, the geometric definition, and the analytic definition. We present the first two. A homeomorphism f : X → Y is (metrically) quasiconformal provided there is a constant H < ∞ such that for all x ∈ X, L(x, f, r) , l(x, f, r) rց0 L(x, f, r) = sup{|f (y) − f (x)| : |x − y| ≤ r} ,

lim sup H(x, f, r) ≤ H

where H(x, f, r) =

l(x, f, r) = inf{|f (y) − f (x)| : |x − y| ≥ r} .

A homeomorphism f : X → Y is (geometrically) quasiconformal provided there is a constant K < ∞ such that for all curve families Γ in X, K −1 mod(Γ) ≤ mod(f Γ) ≤ K mod(Γ).

Notice that unlike the metric definition, which makes sense for any pair of metric spaces, the geometric definition requires measure metric spaces. These are generally assumed to be Ahlfors Q-regular spaces (see §2.J) in which case mod(·) denotes the Q-modulus. 2.D. Length and Geodesics. The length of a continuous path γ : [0, 1] → X is defined in the usual way by n X |γ(ti ) − γ(ti−1 )| where 0 = t0 < t1 < · · · < tn = 1. ℓ(γ) := sup i=1

We call γ rectifiable when ℓ(γ) < ∞. We let Γ(x, y) = Γ(x, y; X) denote the collection of all rectifiable paths joining x and y in X; in general we should also indicate the metric in this notation, but it will always be understood from context. Väisälä’s notes [Väi71, §1-§5] provide an excellent reference for studying properties of curves, and the results are valid in the general metric space setting. Each rectifiable path γ : [0, 1] → X has an associated arclength function s : [0, 1] → [0, ℓ(γ)], given by s(t) = ℓ(γ[0, t]), which is of bounded variation. Given a Borel measurable function ρ : X → [0, ∞], we define Z Z 1 ρ ds := ρ(γ(t)) ds(t). γ

0

An arc in a metric space X is the homeomorphic image of an interval I ⊂ R. Given two points x and y on an arc α, we write α[x, y] to denote the subarc of α joining x and y. A geodesic in X is the image ϕ(I) of some isometric embedding ϕ : I → X where I ⊂ R is an interval; we use the adjectives segment, ray, or line (respectively) to indicate that I is bounded, semi-infinite, or all of R. When ϕ is


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L-bilipschitz we call ϕ(I) an L-quasigeodesic. More generally, if ϕ is (L, C)quasiisometric, then we call ϕ(I) an (L, C)-quasigeodesic. Thus γ is an Lquasigeodesic precisely when ∀x, y ∈ γ :

ℓ(γ[x, y]) ≤ L|x − y|;

classically, such curves in the plane R2 were called chord arc curves. A metric space is geodesic if each pair of points can be joined by a geodesic segment. We use the notation [x, y] to mean a (not necessarily unique) geodesic segment joining points x, y; such geodesics always exist if our space is geodesic, but may not be unique. (If there is some other distance function, such as k, then we write [x, y]k to denote a k-geodesic joining x, y). We consider a given geodesic [x, y] as being ordered from x to y (so we can use phrases such as the ‘first’ point encountered). An unbounded metric space is roughly κ-starlike with respect to a base point w if each point lies within distance κ of some geodesic ray emanating from w. The geodesic boundary ∂g X of an unbounded geodesic metric space X is the set of equivalence classes of geodesic rays in X where two such rays are considered equivalent when they are at a finite Hausdorff distance from each other. Equivalently, if α, β : [0, ∞) → X are geodesic rays in X, then α ≃ β if supt |α(t) − β(t)| < ∞. The geodesic boundary of Rn is the sphere Sn−1 . The geodesic boundary of hyperbolic n-space (Bn , h) is also the sphere Sn−1 . Every metric space (X, d) admits a natural (or intrinsic) metric, the so-called length distance given by l(x, y) := inf{ℓ(γ) : γ a rectifiable curve joining x, y in Ω}. A metric space (X, d) is a length space provided d(x, y) = l(x, y) for all points x, y ∈ X; it is also common to call such a d an intrinsic distance function. Notice that an l-geodesic [x, y]l is a shortest curve joining x and y. The Hopf-Rinow Theorem (see [Gro99, p.9], [BBI01, p.51], [BH99, p.35]) says that every locally compact length space is proper (and therefore geodesic). In a general length space, when geodesics may not exist, one works with so-called short arcs; see [Väi05a]. id

Since |x−y| ≤ ℓ(x, y) for all x, y, the identity map (X, l) → (X, d) is Lipschitz continuous. It is important to know when this map will be a homeomorphism (cf. [BHK01, Lemma A.4, p.92]). Notice that the identity map (X, d) → (X, l) is uniformly locally Lipschitz when X is locally quasiconvex; see §2.E. More generally, one can show that the identity map (X, l) → (X, d) is a homeomorphism precisely when X satisfies a weak notion of local quasiconvexity; see [BH07]. 2.E. Connectivity Conditions. A metric space (X, d) is a-quasiconvex provided each pair of points can be joined by a path whose length is at most a times the distance between its endpoints. A locally complete space X is locally quasiconvex if there exists a constant a ≥ 1 such that for all z ∈ X, points x, y ∈ λB(z) can be joined by a rectifiable arc α in X with ℓ(α) ≤ a|x − y|; we

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abbreviate this by the phrase ‘X is locally a-quasiconvex’. (Here it is understood that there is some Whitney ball constant λ ∈ (0, 1) which may also depend on other parameters). A space (X, d) is c-linearly locally connected, or c-LLC, if c ≥ 1 and the following conditions hold for all x ∈ X and all r > 0: (LLC1 )

points in B(x; r) can be joined in B(x; c r)

and (LLC2 )

¯ r) can be joined in X \ B(x; ¯ r/c). points in X \ B(x;

Here the phrase ‘can be joined’ means ‘can be joined by a continuum’. We also use the term LLC with respect to arcs in which case ‘can be joined’ means ‘can be joined by a rectifiable arc’. Note that quasiconvexity implies LLC1 (even with respect to arcs). The generic example of a space which does not satisfy the LLC2 condition is the interior of an infinite Euclidean cylinder such as Bn−1 × R ⊂ Rn . However, for 2 ≤ k < n the regions Bn−k × Rk ⊂ Rn are easily seen to be 1-LLC2 . The complement of a semi-infinite slab (e.g., Rn \ {(x1 , . . . , xn ) : x1 ≥ 0, |xn | ≤ 1}) fails to be LLC1 . Ahlfors regular Loewner spaces are LLC; see [HK98, Theorem 3.13]. Uniform domains also enjoy this property, but not necessarily uniform spaces. The LLC condition was invented by Gehring who first used it to characterize quasidisks; see [Geh82] and the references mentioned therein. 2.F. Doubling and Dimensions. The p-dimensional Hausdorff measure of a set A ⊂ X is given by Hp (A) := limr→0 Hrp (A) where X Hrp (A) := inf{ diam(Bi )p : A ⊂ ∪Bi , Bi balls with diam(Bi ) ≤ r}.

p The Hausdorff p-content of A is just H∞ (A). The Hausdorff dimension of A is determined by dimH (A) := inf {p > 0 : Hp (A) = 0} .

We also require the Assouad dimension of X which is given by

dimA (X) := inf{p : #S ≤ C(R/r)p for all S ⊂ X with r ≤ |x − y| ≤ R for all x, y ∈ S}

where #S denotes the cardinality of the set S. See [Hei01, 10.15] or [Luu98, 3.2]. The spaces with finite Assouad dimension are precisely the doubling spaces (which we discuss below in more detail). Finally, the conformal Assouad dimension of a metric space X is c-dimA (X) := inf{dimA (X, d) : d ∈ G},

where G is the conformal gauge on X determined by the original metric; see [Hei01, 15.8, p.125].


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A metric space (X, d) satisfies a (metric) doubling condition if there is a constant N such that each ball in X of radius R can be covered by at most N balls of radius R/2; these are precisely the spaces of finite Assouad dimension. A Borel measure ν is a doubling measure on X if there is a constant D = Dν such that ν[B(x; 2r)] ≤ D ν[B(x; r)] for all x ∈ X and all r > 0.

A Borel measure ν on X is p-homogeneous if there is a constant C = Cν such that p R ν[B(x; R)] for all x ∈ X and all 0 < r ≤ R. ≤C ν[B(x; r)] r Obviously every homogeneous measure is doubling; the converse holds too with C = D and p = log2 (D). Every Ahlfors Q-regular measure is Q-homogeneous. The existence of a doubling measure is easily seen to imply a metric doubling condition; the converse holds if our metric space is complete. Here is a precise statement of this result, which is due to Vol’berg and Konyagin for compact spaces, and Luukkainen and Saksman for complete spaces (see [Hei01, Theorem 13.5]). 2.3. Fact. A complete doubling space X carries a p-homogeneous measure for each p > dimA (X). An especially important property of doubling measures is their exponential decay on uniformly perfect spaces, which we record as follows; see [Hei01, (13.2)] or [Sem99, Lemma B.4.7, p.420]. 2.4. Fact. Let ν be a doubling measure on a uniformly perfect metric space. There are constants C ≥ 1 and α > 0, depending only on the doubling constant for ν and the uniformly perfect constant, such that for all balls B(z; r) ⊂ B(x; R), r α ν[B(z; r)] ≤C . ν[B(x; R)] R

Now we discuss an interesting way to deform the geometry of a doubling space. Let ν be a doubling measure on a metric space (X, d). For each α > 0 we define δ = δν,α by ¯ |x − y|) ∪ B(y; ¯ |x − y|); (2.5) δ(x, y) := ν[B(xy)]α , where B(xy) := B(x;

see [DS97, §16.2], [Sem99, (B.3.6)], [Hei01, 14.11]. This always defines a quasimetric on X, and, when X is uniformly perfect, the identity map (X, d) → (X, δ) will be quasisymmetric and (X, δ, ν) will be Ahlfors (1/α)-regular. Moreover, there is an α0 > 0 (depending only on the doubling constant for ν) such that for all 0 < α < α0 , δν,α is bilipschitz equivalent to an honest distance function on X (see Fact 2.1). In particular, if ν is p-homogeneous, then δν,1/p is already bilipschitz equivalent to an honest distance function (e.g., if (X, d, ν) is Ahlfors p-regular, then δν,1/p is bilipschitz equivalent to d). In conjunction with the above chain of ideas, we declare ν to be a p-dimensional metric doubling measure on X if ν is a doubling measure on X with the property that δν,1/p is bilipschitz equivalent to a distance on X. For example, a

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p-homogeneous measure will be a p-dimensional metric doubling measure. We summarize the above comments; see [Hei01, 14.11,14.14], [Sem99, B.3.7, B.4.6, p.421], [DS97, 16.5,16.7,16.8]. 2.6. Fact. Let ν be a p-dimensional metric doubling measure on a uniformly perfect metric space (X, d). Define δ = δν,1/p as in (2.5). Then δ is a quasimetric on X which is bilipschitz equivalent to a distance function on X, the identity map (X, d) → (X, δ) is quasisymmetric, and (X, δ, ν) is an Ahlfors p-regular space. All of the new parameters depend only on the original data for X and ν. There is one final comment we wish to point out regarding the quasimetrics δν,α . As above, suppose ν is a doubling measure on a metric space (X, d), and suppose X has another metric, say, ∂ which is QS equivalent to d. Then by using the doubling property of ν in conjunction with quasisymmetry we see that ν[Bd (xy)] ≃ ν[B∂ (xy)] (where these sets are defined as above using balls centered at x and y in the appropriate metrics); here the constant depends only on the doubling constant and the quasisymmetry data. It therefore follows that the quasimetric δd (defined as in (2.5) via Bd (xy)) is bilipschitz equivalent to δ∂ (defined via B∂ (xy)). We note the important fact that quasisymmetric homeomorphisms preserve these doubling conditions; cf. [Hei01, Theorem 10.18] or [DS97, Lemma 16.4]. In particular, the notions of doubling measure, the quasimetrics δν,α , and metric doubling measures do not depend on the given distance function per se; they all make sense for a conformal gauge. 2.G. Quasiconformal Deformations. Given an abstract domain (Ω, d, µ) and a positive Borel measurable function ρ on Ω, we wish to define a new metric measure space Ωρ = (Ω, dρ , µρ ) which is a quasiconformal deformation of Ω. (Above in §2.F we described another method for deforming the geometry of Ω which was based on having a doubling measure. See Fact 2.6.) We start by defining the ρ-length of a rectifiable curve γ via Z ℓρ (γ) := ρ ds γ

and then the ρ-distance between two points x, y is dρ (x, y) := inf{ℓρ (γ) : γ a rectifiable curve joining x, y in Ω}; see §2.D. The careful reader no doubt recognizes that, in general, dρ (x, y) could be zero or even infinite; in order to ensure that dρ be an honest distance function, we must require that 0 < dρ (x, y) < ∞ for all points x, y ∈ Ω. We designate this by calling such ρ a metric-density on Ω. One way to guarantee this is to ask that ρ be locally bounded away from zero and infinity. In practice, our densities will always satisfy a Harnack inequality—see below—so this is never a problem for us.


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The ρ-balls (etc.) are written as Bρ (x; r); these are the metric balls in Ωρ , so Bρ (x; r) = {y ∈ Ω : dρ (x, y) < r}. We define a new measure µρ by Z (2.7) µρ (E) := ρQ dµ. E

Here Q is usually the Hausdorff dimension of (Ω, d). When Ωρ is non-complete (which will often be the case for us), we can form ¯ ρ \ Ωρ and define dρ (x) = distρ (x, ∂ρ Ω). In this setting we also employ ∂ρ Ω = Ω the notation Bρ (x) = Bρ (x; dρ (x)); thus λBρ (x) is a Whitney ball in Ωρ . We are especially interested in the metric-densities ρ for which Ωρ is a uniform space, and we call such a ρ a uniformizing density (which implicitly includes the hypothesis that Ωρ is non-complete). The Bonk-Heinonen-Koskela theory produces uniformizing densities on proper geodesic Gromov hyperbolic spaces; see Fact 4.1. Some other classes of metric-densities which we wish to single out for attention include Harnack, Ahlfors, and Koebe densities; their definitions follow below. We let Hρ , Aρ , Kρ denote the parameters associated with these densities. Before delving into the technical definitions, we wish to make a few comments. The reader no doubt has encountered deformations of Euclidean domains Ω ⊂ Rn by continuous densities ρ; in this setting Ωρ is a conformal deformation of Ω, meaning that the identify map Ω → Ωρ is conformal (i.e., metrically 1quasiconformal). However, in our more general setting, even for the case ρ = 1 say, the identity map Ω → Ωρ = Ωl may fail to be quasiconformal (e.g., if Ω does not satisfy some sort of local quasiconvexity condition). A similar phenomenon holds for Borel metric-densities, even for domains Ω ⊂ Rn . Nonetheless, when Ω is locally quasiconvex and ρ is a Harnack metric-density, Lemma 2.8 below reveals that the identity map Ω → Ωρ is QC (and according to Proposition 2.9 even QS under the right circumstances). This is a good thing: we want Ωρ to be a quasiconformal deformation of Ω. With this in mind, we pronounce the following definitions. First, we declare ρ to be a bounded density if the deformed space Ωρ is bounded, i.e., diamρ (Ω) < ∞.

Next, we call ρ a Harnack density provided it satisfies a uniform local Harnack type inequality: for all points x in Ω, (H)

ρ(y) 1 ≤ ≤H H ρ(x)

for all y ∈ λB(x).

Here H = Hρ ≥ 1 and 0 < λ < 1 (generally λ will be small). Note that in contrast to the situation in [BKR98, p.637], the validity of (H) for some 0 < λ < 1 need not mean a similar set of inequalities will hold for λ = 1/2. The condition (H) provides local control and permits the use of standard chaining type arguments; e.g., see Lemma 2.13. We call ρ an Ahlfors density if the associated metric measure space Ωρ = (Ω, dρ , µρ ) is Ahlfors upper Q-regular (cf. §2.J); i.e., if µρ satisfies a global upper

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Ahlfors Q-regular volume growth estimate: there is a constant A = Aρ such that for all points x in Ω, (A)

µρ [Bρ (x; r)] ≤ A rQ

for all r > 0.

The positive real number Q is generally the Hausdorff dimension of our space; it must agree with the number Q appearing in the definition of a Loewner space (a notion also discussed in §2.J). The volume growth condition (A) ensures that Ωρ satisfies an upper mass condition and so provides modulus estimates via Facts 2.15, 2.16, 2.17. Below (in §2.H) we discuss the density 1/d which determines the quasihyperbolic distance; of course this is a continuous Harnack density, but in general 1/d does not satisfy the volume growth requirement (A). We call ρ a Koebe density if Ωρ is non-complete and there is a constant K = Kρ such that dρ (x) = distρ (x, ∂ρ Ω) enjoys the property 1 dρ (x) ≤ ≤K for all x ∈ Ω. K ρ(x)d(x) (Note that when ρ is a Harnack density, dρ (x) ≥ (λ/H)ρ(x)d(x) always holds, and so it is the upper estimate which is needed.) For example, if ρ = |f ′ | where f is a holomorphic homeomorphism defined in a subdomain Ω of the complex plane, then a classical theorem in univalent function theory asserts that ρ is a Koebe density with constant K = 4. As another example we note that Theorem E asserts that any Harnack Ahlfors density on a uniform Loewner space (with sufficiently ‘thick’ boundary) is a Koebe density; see [Her04, Theorem E]. We point out that when ρ is a Koebe density on (Ω, d), the identity map (Ω, k) → (Ωρ , kρ ) is easily seen to be Kρ -bilipschitz; here (Ωρ , kρ ) denotes the quasihyperbolization of Ωρ . (K)

We employ the terminology conformal density for a metric-density which is Harnack, Ahlfors, and Koebe. A basic example of a conformal density is ρ = |f ′ | for any holomorphic homeomorphism |f ′ | defined in a subdomain of the complex plane; we refer to [BKR98, Section 2] for other examples of conformal densities on the Euclidean unit ball. The reader should be aware that the phrase ‘ρ is a conformal density’ does not necessarily mean that the identity map Ω → Ωρ is quasiconformal (unless Ω is locally quasiconvex). Here are some especially useful estimates which also provide information concerning the identity map Ω → Ωρ for certain densities. Roughly speaking, this map is locally bilipschitz (therefore quasiconformal) for Harnack densities and uniformly locally quasisymmetric for Harnack Koebe densities, provided Ω is locally quasiconvex. Proposition 2.9 gives a significant strengthening of this result. 2.8. Lemma. Let ρ : Ω → (0, ∞) be a Harnack density on a locally aquasiconvex abstract domain (Ω, d). Put η = λ/2a. Then for all z ∈ Ω, 1 dρ (x, y) ≤ ≤ aH for all points x 6= y in ηB(z); H ρ(z)|x − y|


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in particular, 1 diamρ [ηB(z)] ≤ ≤ 2aH. H ηρ(z)d(z) If ρ is also a Koebe density, then for all 0 ≤ ϑ ≤ λ/2C and all x ∈ Ω, C −1 ϑB(x) ⊂ ϑBρ (x) ⊂ CϑB(x),

where C = aHK. Here H = Hρ , K = Kρ and λ is the Whitney ball constant. One immediate consequence of Lemma 2.8 is that the identity map Ω → Ωρ is metrically quasiconformal with linear dilatation aH 2 . In addition, because of the definition of the associated measure (see (2.7)), a straightforward calculation reveals that this identity map is geometrically quasiconformal with inner dilatation H Q and outer dilatation (aH)Q . (Here we assume a Harnack density on a locally quasiconvex Ω.) It is therefore natural to inquire about possible quasisymmetry properties of this identity map. Heinonen and Koskela proved that a quasiconformal map of bounded Ahlfors regular spaces, with domain a Loewner space and a linearly locally connected target space, is in fact quasisymmetric [HK98, Theorem 4.9]. The corollary to the following analog of their result is used in the proof of Theorem D; note that here our domain space is not assumed to be Ahlfors regular. 2.9. Proposition. Let Ω be a bounded locally quasiconvex Q-Loewner space. Suppose ρ is a conformal density on Ω with Ωρ a bounded linearly locally connected space. Then the identity map Ω → Ωρ satisfies the weak-quasisymmetry condition ∀x, y, z ∈ Ω :

|x − y| ≤ |x − z| =⇒ dρ (x, y) ≤ Ldρ (x, z)

for some constant L which depends only on the data associated with Ω, ρ, Ωρ , and the ratios r, q given in the proof. 2.10. Corollary. Let Ω be a bounded quasiconvex Q-Loewner space. Suppose ρ is a conformal density on Ω and Ωρ is a bounded Q-Loewner space. Then the identity map Ω → Ωρ is quasisymmetric with a distortion function which depends only on the data associated with Ω, ρ, Ωρ , and the ratios r, q given in the proof of Proposition 2.9. As an exercise to help understand the various properties of these metricdensities, the interested reader can provide a proof for the following [Her06, Lemma 2.6]. 2.11. Lemma. Suppose (Ω, d, µ) is a locally a-quasiconvex abstract domain. Let τ be a positive Borel function on Ω which is locally bounded away from 0 and ∞. Put ∆ = Ωτ . If σ is a metric-density on ∆, then its pull-back ρ = σ τ is a metric-density on Ω, Ωρ = ∆σ , and (a) ρ and σ either are, or are not, both Ahlfors regular (with Aρ = Aσ ), (b1) if σ, τ are both Koebe, then so is ρ with Kρ = Kσ Kτ ,

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(c1) if σ, τ are both Harnack, then so is ρ with Hρ = Hσ Hτ . On the other hand, if ρ is a metric-density on Ω, then its push-forward σ = ρ τ −1 is a metric-density on ∆, ∆σ = Ωρ , (a) holds, and (b2) if ρ, τ are Koebe, then so is σ with Kσ = Kρ Kτ , (c2) if ρ, τ are Harnack and τ is Koebe, then σ is Harnack with Hσ = Hρ Hτ . 2.H. Quasihyperbolic Distance and Geodesics. The quasihyperbolic distance in an abstract domain (Ω, d) is defined by Z ds k(x, y) = kΩ (x, y) := inf ℓk (γ) = inf γ d(z)

where the infimum is taken over all rectifiable curves γ which join x, y in Ω. The quasihyperbolization of an abstract domain (Ω, d) is the metric space (Ω, k) obtained by using quasihyperbolic distance. It is not hard to see that (Ω, k) is complete, provided the identity map (Ω, ℓ) → (Ω, d) is a homeomorphism; see [BHK01, Proposition 2.8]. Thus by the Hopf-Rinow theorem ([Gro99, p.9], [BBI01, p.51], [BH99, p.35]), every locally compact abstract domain has a proper (hence geodesic) quasihyperbolization. We call the geodesics in (Ω, k) quasihyperbolic geodesics; see §2.D. Note that when ρ is a Koebe density on Ω, the identity map (Ω, k) → (Ωρ , kρ ) is bilipschitz and we find that quasihyperbolic geodesics in Ω are quasihyperbolic quasigeodesics in Ωρ ; that is, a geodesic in (Ω, k) will be a quasigeodesic in (Ωρ , kρ ) (the quasihyperbolization of Ωρ ). We remind the reader of the following basic estimates for quasihyperbolic distance, first established by Gehring and Palka [GP76, Lemma 2.1]: d(x) ℓ(x, y) |x − y| . k(x, y) ≥ log 1 + ≥ j(x, y) = log 1 + ≥ log d(x) ∧ d(y) d(x) ∧ d(y) d(y) See also [BHK01, (2.3),(2.4)]. The first inequality above is a special case of the more general (and easily proved) inequality, ℓ(γ) ℓk (γ) ≥ log 1 + d(x) ∧ d(y)

which holds for any rectifiable curve γ with endpoints x, y.

An immediate consequence of the above inequalities is that the identity map (Ω, k) → (Ω, d) is continuous; indeed, Bk (x; R) ⊂ (eR − 1)B(x)

for all x ∈ Ω and all R > 0,

where Bk (x; R) denotes the R-ball centered at x in (Ω, k). It is important to know when this map will be a homeomorphism (which, according to [BHK01, Lemma A.4, p.92], will be the case if and only if the identity map (Ω, ℓ) → (Ω, d) is a homeomorphism). The following provides quantitative information concerning this question; it is easy to verify via simple estimates for the quasihyperbolic lengths of the ‘promised short arcs’.


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2.12. Lemma. Suppose that (Ω, d) is a locally a-quasiconvex abstract domain. Then for all x ∈ Ω and all R > 0, τ B(x) ⊂ Bk (x; R)

provided 0 ≤ τ ≤ min{λ, R/[a(1 + R)]}.

As an exercise, the reader can check that for a domain in Rn , |x − y| ≤ [d(x) + d(y)]/2 =⇒ k(x, y) ≤ 2. Thus (2/3)B(x) ⊂ Bk (x; 2). The Harnack inequality (H), as stated in §2.G, only requires that ρ be essentially constant on Whitney type balls. We can do the usual chaining type arguments to see that such a density ρ will satisfy a Harnack type inequality on much bigger sets, of course with a change in the Harnack constant. Here is a useful example of this phenomena. 2.13. Lemma. Let ρ be a Harnack density on an abstract domain (Ω, d). If x, y ∈ Ω satisfy k(x, y) ≤ K, then 1/H1 ≤ ρ(y)/ρ(x) ≤ H1 , where H1 = H1 (K, Hρ , λ). We conclude this subsection with a covering lemma for quasihyperbolic geodesics. 2.14. Lemma. Suppose that (Ω, d) is a locally a-quasiconvex abstract domain. Let γ be a quasihyperbolic geodesic segment or ray in Ω with endpoint x0 . Let x0 , x1 , x2 , . . . be successive points along γ with k(xi , xi−1 ) = K ≤ log(1 + τ ) where τ = Pmin{λ, 1/2a}. Then the balls Bi = τ B(xi ) cover γ and have bounded overlap: χBi ≤ Cχ∪Bi , where C = 1 + 4/K.

2.I. Modulus and Capacity. For p ≥ 1 we define the p-modulus of a family Γ of curves in a metric measure space (X, d, µ) by Z modp Γ := inf ρp dµ,

where the infimum is taken over all Borel functions ρ : X → [0, ∞] satisfying R ρ ds ≥ 1 for all locally rectifiable curves γ ∈ Γ. Then the p-modulus of a pair γ of disjoint compact sets E, F ⊂ X is modp (E, F ; X) := modp Γ(E, F ; X)

where Γ(E, F ; X) is the family of all curves joining the sets E, F in X. We also let Γr (E, F ; X) be the subfamily of Γ(E, F ; X) consisting of the rectifiable paths joining E, F . An important property is that under fairly general circumstances, modp (E, F ; X) agrees with the p-capacity of the pair E, F . There is extensive literature regarding these “capacity equals modulus” results; for a start, see [HK98, Proposition 2.17]. For the reader’s convenience, we cite the following modulus estimates. First we have the standard Long Curves Estimate; see [HK98, 3.15]. 2.15. Fact. Let x ∈ X and suppose that the upper mass condition µ[B(x; R)] ≤ M Rp holds for some R > 0. Let Γ be a family of curves in B(x; R) and suppose that each γ ∈ Γ has arclength ℓ(γ) ≥ L > 0. Then modp Γ ≤ L−p µ[B(x; R)] ≤ M (R/L)p .

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Next we record the Spherical Ring Estimate; see [HK98, 3.14, p.17]. 2.16. Fact. Let x ∈ X, 0 < 2r ≤ R, and suppose that the upper mass condition µ[B(x; t)] ≤ M tp holds for all 0 < t < r + R. Then ¯ r), X \ B(x; R); X) ≤ C (log(R/r))1−p , modp (B(x; where C = 2p+1 M/ log 2.

Finally, we require the following Basic Modulus Estimate; see [BKR98, Lemma 3.2]. 2.17. Fact. Let (X, d, µ) be a metric measure space. Assume that ρ is a metricdensity on X whose associated measure (2.7) satisfies the Ahlfors volume growth condition (A) at some point x ∈ E ⊂ X. Suppose that L > λ ≥ diamρ E, and that Γ is some family of curves γ in X each having one endpoint in E and satisfying ℓρ (γ) ≥ L. Then modQ Γ ≤ C (log (1 + L/λ))1−Q ,

where C = 2Q+1 A/ log 2.

2.18. Corollary. Let (X, d, µ) be a metric measure space. Assume that for some point x ∈ E ⊂ X, the upper mass condition µ[B(x; r)] ≤ M rQ holds for all r > 0. Suppose that Γ is a family of curves γ in X each having one endpoint in E and satisfying ℓ(γ) ≥ L > diam E. Then modQ Γ ≤ C (log (1 + L/ diam E))1−Q ,

where C = 2Q+1 M/ log 2.

In Euclidean space Rn , the n-modulus is also called the conformal modulus and simply denoted by mod(·). Below we state some well-known geometric estimates for the conformal modulus mod(E, F ; Ω). Here and elsewhere in these notes, ∆(E, F ) := dist(E, F )/ min{diam(E), diam(F )} is the relative distance between the pair E, F of nondegenerate disjoint continua. 2.19. Facts. Let E, F be disjoint compact sets in Rn . ¯ t), then (a) If E, F are separated by the spherical ring B(x; s) \ B(x; mod(E, F ; Rn ) ≤ ωn−1 (log(s/t))1−n .

(b) If E ∩ S(x; r) 6= ∅ 6= F ∩ S(x; r) for all t < r < s, then mod(E, F ) ≥ σn log(s/t).

(c) If both E and F are connected, then

σn log(1 + 1/∆(E, F )) ≤ mod(E, F ; Rn ) ≤ Ωn (1 + 1/∆(E, F ))n .

(d) (Comparison Principle) If A, B, E, F ⊂ Ω with A, B also compacta, then mod(E, F ; Ω) ≥ 3−n min{mod(E, A; Ω), mod(F, B; Ω), I},

where I = inf{mod(α, β; Ω) | α ∈ Γr (E, A; Ω), β ∈ Γr (F, B; Ω)}.


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(e) (Teichm¨ uller Estimate) If E, F are both connected, then for all x, y ∈ E and z, w ∈ F |x − z||y − w| n mod(E, F ; R ) ≥ τ |x − y||z − w| where τ (r) is the capacity of the Teichm¨ uller ring

Rn \ {−1 ≤ x1 ≤ 0 or x1 ≥ r};

i.e, τ (r) = mod([−e1 , 0], [re1 , ∞]; Rn ). (f) There exists λ = λ(n) ∈ [6, 5e(n−1)/2 ) such that when E, F are both connected and ∆(E, F ) ≥ 1, 21−n ωn−1 [log(λ∆(E, F ))]1−n ≤ mod(E, F ; Rn ) ≤ ωn−1 [log(∆(E, F ))]1−n .

(g) (Carleman Inequality) For E ⊂ Ω,

mod(E, ∂Ω; Ω) ≥ nn−1 ωn−1 (log(|Ω|/|E|))1−n .

Here σn and ωn−1 , Ωn are the spherical cap constant and the measures of the (n − 1)-sphere, n-ball respectively. Most of these estimates can be found in [Väi71] or [Vuo88]. Lemma 2.5 in [BH06] gives a precise formula for mod(E, F ; Rn ) in the case when E, F are disjoint closed balls. 2.J. Ahlfors Regular and Loewner Spaces. A metric measure space (X, d, µ) is Ahlfors Q-regular provided there exists a finite constant M = Mµ such that for all x ∈ X and all 0 < r ≤ diam Ω, M −1 rQ ≤ µ[B(x; r)] ≤ M rQ .

The positive real number Q will then be the Hausdorff dimension of (X, d), and the Q-dimensional Hausdorff measure HQ on X will also satisfy the above inequalities (possibly with a change in the constant M ). A metric space (X, d) is Ahlfors Q-regular if (X, d, HQ ) is Ahlfors Q-regular. We use the adjectives upper or lower to indicate that only one of these inequalities is in force, and—in the abstract domain setting—add the adjective locally to mean that the required inequality holds (or, inequalities hold) for Whitney balls (i.e., for radii 0 < r ≤ λd(x)). There is an interesting result which gives upper estimates for the Assouad dimension of subsets of Ahlfors regular spaces. See [BHR01, 3.12], [DS97, 5.8], [Luu98, 5.2]. 2.20. Fact. Suppose X is an Ahlfors Q-regular space and let M ⊂ X. Then dimA M < Q if and only if M is porous in X; the constants depend only on each other and the HQ -regularity constant. The notion of a Loewner space was introduced by Heinonen and Koskela in their study [HK98] of quasiconformal mappings of metric spaces; Heinonen’s recent monograph [Hei01] renders an enlightening account of these ideas. A

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path-connected metric measure space (X, d, µ) is a Q-Loewner space, Q > 1, provided the Loewner control function ϕ(t) := inf{modQ (E, F ; X) : ∆(E, F ) ≤ t}

is strictly positive for all t > 0; here E, F are non-degenerate disjoint continua in Ω and ∆(E, F ) := dist(E, F )/ min{diam(E), diam(F )} is the relative size of the pair E, F . Note that we always have Q ≥ dimH (Ω) ≥ 1. When (Ω, d, µ) is an n-Loewner space with Ω ⊂ Rn a domain and d, µ are Euclidean distance and Lebesgue n-measure respectively, we simply call Ω a Loewner domain. This is a generalization of Väisälä’s notion of a broad domain (which he introduced in his analysis [Väi89, 2.15] of space domains QC equivalent to a ball, and also used in his study [NV91, 3.8] of John disks), which in turn is an analog of the quasiextremal distance domains first studied by Gehring and Martio [GM85]. We call Ω ( Rn a ψ-QED domain if ψ : [0, ∞) → [0, ∞) is a homeomorphism and for all disjoint continua E, F in Ω, mod(E, F ; Ω) ≥ ψ(mod(E, F ; Rn )).

Clearly, ψ(t) ≤ t is a necessary restriction on such ψ. Also, every ψ-QED domain is Loewner. The typical nonlinear functions ψ that arise in the literature have the form ψp,M (t) = M −1 min{tp , t1/p } with p, M ≥ 1, a condition we call M -QEDp , or simply M -QED when p = 1. The most important, and original, inequalities of this form are the M -QED conditions corresponding to ψ(t) = t/M for some constant M ≥ 1. This idea was introduced by Gehring and Martio who called such regions quasiextremal distance domains. The terminology arises from the fact that the quantity mod(E, F ; Ω)1/(1−n) is the extremal distance between E and F in Ω. When we speak of a QED domain or a QED condition, we always mean an M -QED domain or an M -QED condition for some M ≥ 1. As in [HK96] we can consider the location of the continua E, F as well as looking at special types of continua. In particular we can relax the ψ-QED inequality by requiring it to hold only for all disjoint closed balls (or just closed Whitney balls) to get the class ψ-QEDb (or ψ-QEDwb , respectively). Precise definitions can be found in [BH06]. Every a-uniform domain in Rn is M -QED for some M = M (a, n); this follows easily from Jones’ extension result for Sobolev spaces [Jon81, Theorem 1]. Also, it is trivially true that QED =⇒ ψ − QED =⇒ ψ − QEDb =⇒ ψ − QEDwb .

The converse of the middle implication fails; see[HK96, Example 4.1] and [BH06, Example 4.2]. According to [BH06, Theorem 3.3], the last implication is reversible modulo a quantitative change in ψ. In addition, we always have ψ − QED ⇐⇒ Loewner =⇒ QEDwb =⇒ ψ − QEDwb ⇐⇒ Loewnerwb


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where the last condition means that the Loewner condition is assumed only for Whitney balls. Examples 4.2 and 4.3 in [BH06] illustrate that in general the converses of the middle two implications fail to hold. The first equivalence is established in [BH06, Theorem 1.3]. It remains open as to whether or not Loewner domains (i.e. ψ-QED domains) are always QED. 2.K. Slice Conditions. There are various so-called slice conditions each designed to handle their own specific problem. The ideas here are due to Buckley et al. and his exposition [Buc03] is the place to begin reading about this topic. He and his many co-authors have utilized an assortment of slice conditions to investigate all kinds of different problems. A non-empty bounded open set S ⊂ X is called a C-slice separating x, y provided ∀ α ∈ Γ(x, y) :

ℓ(α ∩ S) ≥ diam(S)/C

and C −1 B(x) ∩ S = ∅ = S ∩ C −1 B(y) .

A set of C-slices for x, y ∈ X is a collection S of pairwise disjoint C-slices separating x, y in X. One can show (see [BS03, (2.1)]) that the cardinality of any such set S of C-slices separating x, y is always bounded by #S ≤ C 2 k(x, y). We are interested in knowing when we can reverse this inequality. Since there may be no C-slices separating x, y, we consider the quantity dws (x, y) = dws (x, y; C) = dX ws (x, y; C) := 1 + sup #S where the supremum is taken over all S which are sets of C-slices in X separating x, y, and #S denotes the cardinality of S. We call (X, d) a weak C-slice space provided for all x, y ∈ X, k(x, y) ≤ C dws (x, y; C),

Thus in these spaces dws (x, y) ≃ k(x, y), at least when k(x, y) ≥ 2. The weak slice condition was introduced in [BO99, Section 5]; see also [BS03], [Buc03], [Buc04]. When the weak C-slice space (X, d) is a domain Ω ( Rn , we call Ω a weak C-slice domain. The following rather technical lemma is quite useful for obtaining an upper bound for the cardinality of a set of slices; in weak slice spaces it provides an upper bound for quasihyperbolic distances. It is the case α = 0 of [BS03, Lemma 2.17]. 2.21. Lemma. Let Γ be a 1-rectifiable subset of a rectifiably connected metric space (X, d). Suppose ϕ : Γ → [ε, ∞) (with ε > 0) and S is a collection of disjoint non-empty bounded subsets of X. Suppose also that there exist positive constants b, c such that (a) ∀S ∈ S : ℓ(S ∩ Γ) ≥ c diam(S) , (b) ∀S ∈ S , ∀z ∈ S ∩ Γ : ϕ(z) ≤ diam(S) ,

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ℓ(ϕ−1 (0, t]) ≤ b t .

Then the cardinality of S is at most #S ≤ 2(b/c) log2 (4ℓ(Γ)/cε).

3. Uniform Spaces Roughly speaking, a space is uniform provided points in it can be joined by socalled bounded turning twisted double cone arcs, i.e. paths which are not too long and which stay away from the regions boundary. Uniform domains in Euclidean space were first studied by John [Joh61] and Martio and Sarvas [MS79] who proved injectivity and approximation results for them. They are well recognized as being the ‘nice’ domains for quasiconformal function theory as well as many other areas of geometric analysis (e.g., potential theory); see [Geh87] and [Väi88]. Every (bounded) Lipschitz domain is uniform, but generic uniform domains may very well have fractal boundary. Recently, uniform subdomains of the Heisenberg groups, as well as more general Carnot groups, have become a focus of study; see [CT95], [CGN00], [Gre01]. 3.A. Euclidean Setting. When our uniform space (see the definition given below in §3.B) (Ω, d) is a domain Ω ⊂ Rn with Euclidean distance, we simply call Ω a uniform domain. Every plane uniform domain is a quasicircle domain (each of its boundary components is either a point or a quasicircle), and a finitely connected plane domain is uniform if and only if it is a quasicircle domain. However, the plane punctured at the integers is not uniform. Such nice topological information is not true for uniform domains in higher dimensions. For example, a ball with a radius removed is uniform; this is not true when n = 2. For domains in Rn we can consider uniformity both with respect to the Euclidean distance and with respect to the induced length metric also. The latter class of domains are usually called inner uniform; cf. [Väi98]. For example, a slit disk in the plane is not uniform (with respect to Euclidean distance) but it is an inner uniform domain. On the other hand, an infinite strip, or the inside of an infinite cylinder in space, is not uniform nor inner uniform. The region between two parallel planes is not uniform nor inner uniform. Every quasiball is uniform. 3.B. Measure Metric Space Setting. Following [BHK01], a uniform space is an abstract domain (so, a non-complete, locally complete, rectifiably connected metric space) (Ω, d) with the property that there is some constant a ≥ 1 such that each pair of points can be joined by an a-uniform arc. A rectifiable arc γ joining x, y in Ω is an a-uniform arc provided and

ℓ(γ) ≤ a|x − y|

min{ℓ(γ[x, z]), ℓ(γ[y, z])} ≤ a d(z) for all z ∈ γ. Here ℓ(γ) is the arclength of γ and γ[x, z] denotes the subarc of γ between x, z. The second inequality above ensures that Ω contains the twisted double cone ∪{B(z; ℓ(z)/a) : z ∈ γ} where ℓ(z) denotes the left-hand-side of this inequality;


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the first inequality asserts that this twisted double cone is not too ‘crooked’. Consequently, we call γ a double a-cone arc if it satisfies the second inequality above (the phrases cigar arc and corkscrew are also used). 3.C. Basic Information. An important, and characteristic, property of uniform spaces is that quasihyperbolic geodesics are uniform arcs. (See [GO79, Theorems 1,2] for domains in Euclidean space and [BHK01, Theorem 2.10] for general metric spaces.) Slight alterations to the proof of [BHK01, Theorem 2.10] yield the following generalization of this property. 3.1. Fact. In an a-uniform space, quasihyperbolic c-quasigeodesics are b-uniform arcs where b = b(a, c). In general, quasihyperbolic geodesics may not exist; see [Väi99, 3.5] for an example due to P. Alestalo. However, one can still show that quasihyperbolically short arcs are uniform arcs. One can prove that boundary points in a locally compact uniform space can be joined by quasihyperbolic geodesics, and these geodesics are still uniform arcs. Another crucial piece of information is a characterization of uniformity due to Gehring and Osgood [GO79, Theorems 1,2]; Bonk, Heinonen, and Koskela [BHK01, Lemma 2.13] verified the necessity of this condition for the metric space setting, while the Gehring-Osgood argument can be modified to establish the sufficiency. Recalling the basic estimates for quasihyperbolic distance, we see that uniform spaces are precisely those abstract domains in which the quasihyperbolic distance is bilipschitz equivalent to the j distance. See also Theorem 6.1. 3.2. Fact. An abstract domain is a-uniform if and only if k(x, y) ≤ b j(x, y) for all points x, y. The constants a and b depend only on each other. We conclude this subsection with a useful fact regarding bounded uniform spaces; see [Her06, Lemmas 2.12,2.13]. 3.3. Lemma. Let ρ be a Harnack Koebe density on a bounded a-uniform space (Ω, d). Suppose that Ωρ is bounded and that quasihyperbolic geodesics in Ω are double a-cone arcs in Ωρ . Then for any positive constant C, diamρ [CB(z)] ≃ dρ (z)

for all z ∈ Ω,

where the constant depends only on C, Hρ , Kρ , a, λ and the quantity q given in the proof.

4. Gromov Hyperbolicity Good sources for information concerning Gromov hyperbolicity include [BHK01], [BBI01], [BS00], [BH99], [Bon96] and especially the references mentioned in these works. Väisälä has an especially nice treatment [Väi05a] of Gromov hyperbolicity for spaces which are not assumed to be geodesic nor proper. Note however that Bonk and Schramm have demonstrated that every Gromov δ-hyperbolic metric

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space can be isometrically embedded into some complete geodesic δ-hyperbolic space; see [BS00, Theorem 4.1]. Hästö [Häs06] has an intriguing result giving a striking contrast between the hyperbolicity of (Ω, j) versus that of (Ω, ˜j) for Ω ( Rn : the latter space is always Gromov hyperbolic whereas the former is Gromov hyperbolic precisely when Ω has exactly one boundary point. This is quite surprising as these spaces are bilipschitz equivalent (indeed, ˜j ≤ j ≤ 2˜j). It is known that for intrinsic spaces, so also for geodesic spaces, Gromov hyperbolicity is preserved under (L, C)-quasiisometries. In particular, Hästö’s result illustrates the failure of this property in the non-intrinsic setting. 4.A. Thin Triangles Definition. A geodesic metric space is Gromov hyperbolic if its geodesic triangles are δ-thin for some δ > 0, which means that each point on the edge of any geodesic triangle is within distance δ of some point on one of the other two edges. That is, if [x, y] ∪ [y, z] ∪ [z, x] is a geodesic triangle, then for all u ∈ [x, y], dist(u, [x, z] ∪ [y, z]) ≤ δ. (Recall that [x, y] denotes some arbitrary, but fixed, geodesic joining x, y.) There is a more general definition which applies to non-geodesic spaces. It is based on the Gromov product 1 for points x, y, w in the space . (x|y)w := (|x − w| + |y − w| − |x − y|) 2 The Gromov product is useful even in geodesic spaces; it can be extended to the Gromov boundary and then used to define a canonical conformal gauge there. Roughly speaking, all simply connected manifolds with negative curvature are Gromov hyperbolic; e.g., every CAT(κ) space with κ < 0. For a specific example, consider any bounded strictly pseudoconvex domain Ω (with sufficiently smooth boundary) in complex n-space together with any of the classical hyperbolic distances h; a result of Balogh and Bonk [BB00] asserts that (Ω, h) is a Gromov hyperbolic space (with ∂G Ω = ∂Ω, the Euclidean boundary, and canonical conformal gauge determined by the Carnot-Carathéodory distance on ∂Ω). 4.B. Gromov Boundary. The Gromov boundary ∂G H of a proper geodesic Gromov hyperbolic metric space (H, h) is defined as the set of equivalence classes of geodesic rays, with two such rays being equivalent if they have finite Hausdorff distance. That is, ∂G H is the geodesic boundary of H; see §2.D. An alternative description can be given in terms of (equivalent) sequences which converge at infinity; in particular, this allows us to extend the Gromov product to the boundary (cf. [Väi05a, 5.7] or [BH99, pp. 431-436]). This in turn yields a canonically defined conformal gauge on the Gromov boundary generated by the quasimetrics qw,ε (ξ, η) = exp[−ε(ξ|η)w ]

for points ξ, η ∈ ∂G H.

For 0 < ε < ε(δ) = log 2/(4δ), each quasimetric qε is bilipschitz equivalent to an honest metric on the boundary, and all these metric spaces are QS equivalent to each other; in particular they all generate the same topology on ∂G H, and ∂G H is compact. See Fact 2.1, [BH99, Proposition 3.21, pp.435-436], [BHK01, p.18].


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4.C. Connection with Uniform Spaces. Bonk, Heinonen and Koskela established the following fundamental connection between uniform spaces and Gromov hyperbolicity; see [BHK01, Proposition 2.8, Theorem 3.6] 4.1. Fact. The quasihyperbolization (Ω, k) of a locally compact a-uniform space (Ω, d) is proper, geodesic and δ-hyperbolic where δ = δ(a) = 10000a8 . When (Ω, d) is bounded, (Ω, k) is roughly κ-starlike with κ = 5000a8 . ˙ of In fact they also prove that the Gromov boundary ∂G Ω ‘is’ the boundary ∂Ω Ω in the one-point extension Ω˙ of Ω; see [BHK01, Proposition 3.12]. Moreover, in the bounded case, the canonical gauge on ∂G Ω is naturally quasisymmetrically equivalent to the conformal gauge determined by d on ∂Ω. See [Väi05b] for similar results in the Banach space setting.

5. Uniformization The celebrated Riemann Mapping Theorem asserts that every simply connected proper subdomain of the plane can be mapped conformally onto the unit disk, and hence supports a bounded conformal uniformizing metric-density, namely, ρ = |f ′ | where f is the Riemann map. Koebe proved a similar result for finitely connected plane domains: any one of these can always be conformally mapped onto a circle domain (meaning that each boundary component is either a point or a circle). In space, every conformal map is (the restriction of) a Möbius transformation, and thus the only space regions conformally equivalent to a ball are balls and half-spaces. The problem of determining which space domains are QC equivalent to a ball has been investigated for more than four decades by now (see [GV65]), and the most significant result (that I know of) is Väisälä’s characterization in [Väi89] describing the cylindrical domains (Ω = D × R ⊂ R3 ) which are QC equivalent to B3 . 5.A. Uniformization Problem. Here we consider the metric space analog of the Riemann Mapping Problem. We seek to characterize the abstract domains which can be quasiconformally deformed into a uniform space. We ask the question: What are necessary and sufficient conditions for an abstract domain to support a conformal uniformizing metric-density? Theorem A provides an initial answer. 5.B. BHK Uniformization. Bonk, Heinonen and Koskela developed a uniformization theory, which they call dampening, valid for proper geodesic Gromov hyperbolic spaces. Their theory produces the following; see [BHK01, Proposition 4.5, Chapter 5]. (They established far more than we mention here:-) 5.1. Fact. Let (H, h) be an unbounded proper geodesic Gromov δ-hyperbolic space. Fix a base point w ∈ H. For ε > 0 define ρε (x) = exp[−εh(x, w)] and let Hε = (H, dε ) where dε = hρε . Then:

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(a) The geodesics in H are double a-cone arcs in Hε with a = a(ε, δ) = e1+8εδ . (b) There is a constant ε0 = ε0 (δ) such that for all ε ∈ (0, ε0 ], ∀x, y ∈ H , ∀ geodesics [x, y] :

here ℓε = ℓρε .

ℓε [x, y] ≤ 20 dε (x, y);

In fact, Hε is always bounded, and thus when ε ≤ ε0 we see that (H, h) has been deformed, or dampened, (via the natural metric-density ρε ) to a bounded 20-uniform space Hε . We briefly describe their theory in the special case which concerns us. We consider a locally compact abstract domain (Ω, d) with the property that the identity map (Ω, ℓ) → (Ω, d) is a homeomorphism (so the identity (Ω, k) → (Ω, d) is also a homeomorphism) and such that its quasihyperbolization (Ω, k) is a Gromov hyperbolic space. According to Fact 5.1, the space (Ω, k) admits a uniformizing density of the form ρε (x) := exp[−εk(x, w)]; here w ∈ Ω is a fixed base point and ε > 0 a sufficiently small parameter. More precisely, when (Ω, k) is δ-hyperbolic and 0 < ε < ε(δ), the quasihyperbolic geodesics in Ω are 20-uniform arcs in (Ω, dε ). (A careful check of BHK shows that ε(δ) = [42(5 + 192δ + 1920δ 2 )]−1 ≤ (300 max{1, δ})−2 . :-) Here dε stands for the distance function obtained by conformally deforming k via the metricdensity ρε . Since k was obtained from the original distance function d via the quasihyperbolic density 1/d, Ωε = (Ω, dε ) is a conformal deformation of (Ω, d) via the metric-density πε (x) := ρε (x)/d(x) = d(x)−1 exp(−εk(x, w)) ; again, πε will be a uniformizing density when 0 < ε < ε(δ) = (300 max{1, δ})−2 . In order to determine when πε will be a Harnack or Koebe density, we need the following information concerning ρε (see [BHK01, (4.4),(4.6),(4.17)]): ρε (x) ≤ eεk(x,y) , ρε (y) ρε (x) ρε (x) (5.3) ≤ dε (x) ≤ (2eεκ − 1) . eε ε The first set of inequalities (5.2) hold for all points x, y ∈ Ω and all ε > 0. They guarantee that the identity map (Ω, k) → (Ω, dε ) is locally bilipschitz, so (Ω, dε ) is locally compact and rectifiably connected. On the other hand, Ωε = (Ω, dε ) is ¯ ε , put ∂ε Ω = Ω ¯ ε \ Ω and let dε (x) = distε (x, ∂ε Ω). non-complete, so we can form Ω (See [BHK01, pp.27-28]). We find that the leftmost inequality in (5.3) holds for all x ∈ Ω and any ε > 0. However, in order to obtain the rightmost inequality in (5.3), we must further require that (Ω, k) be roughly κ-starlike. (5.2)

e−εk(x,y) ≤

Now using (5.2), and obvious inequalities for 1/d, we deduce that e−2R ≤ e−(ε+1)R ≤

πε (y) ≤ e(ε+1)R ≤ e2R πε (x)


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for all points x, y with k(x, y) ≤ R. According to Lemma 2.12, when (Ω, d) is locally a-quasiconvex we have τ B(x) ⊂ Bk (x; R)

provided 0 < τ ≤ min{λ, R/[a(1 + R)]}.

Taking R = a (say) and τ = min{λ, 1/2a} we find that for all x ∈ Ω and all y ∈ τ B(x), πε (y) ≤ e(ε+1)a ≤ e2a ; e−2a ≤ e−(ε+1)a ≤ πε (x) that is, πε is a Harnack density with constants H = e2a and τ which are independent of ε. Finally, since Ωε is non-complete, we can ask whether or not πε is a Koebe density. Since πε (x) = ρε (x)/d(x), we see from (5.3) that πε will be a Koebe density, with constant K = (2eεκ − 1)/ε (assuming εe ≤ 1), provided (Ω, k) is roughly κ-starlike (and (Ω, d) uniformly locally quasiconvex). These conditions describing when πε will be a Harnack or Koebe density do not require that (Ω, k) be Gromov hyperbolic. We record the above information for later reference; see also Lemma 2.8 and Corollary 5.8. Note too that (Ω, dε ) is bounded with diamε Ωε ≤ 2/ε. 5.4. Lemma. Let (Ω, d) be a locally a-quasiconvex abstract domain and fix a base point w ∈ Ω. Then for any ε > 0, πε (x) = ρε (x)/d(x) = d(x)−1 exp(−εk(x, w))

is a Harnack density with constant H = e2a . If in addition (Ω, k) is roughly κstarlike, then πε is also a Koebe density, with constant K = max{εe, (2eεκ −1)/ε}. The above, in conjunction with Lemma 2.11 and Proposition 5.6(b), provides a one-to-one correspondence between conformal metric-densities on Ω and the same on Ωε . Here is a precise statement of this. 5.5. Corollary. Suppose (Ω, d, µ) is an abstract domain having bounded geometry and a Gromov hyperbolic roughly starlike quasihyperbolization (Ω, k). Let Ωε = (Ω, dε , µε ) be the deformation of Ω via the density πε defined just above. If σ is a conformal density on Ωε , then its pull-back ρ = σ πε is a conformal density on Ω, and conversely if ρ is a conformal density on Ω, then its pushforward σ = ρ πε−1 is a conformal density on Ωε . In both cases Ωρ = (Ωε )σ , and the metric-density parameters depend only on each other and the data associated with Ω. 5.C. Bounded Geometry and its Consequences. Recall our definition that an abstract domain (Ω, d, µ) is a locally complete, rectifiably connected, noncomplete metric measure space; these are our standing metric hypotheses. We say that (Ω, d, µ) has bounded Q-geometry, Q > 1, provided it is both locally upper Ahlfors Q-regular (see §2.J) and weakly locally Q-Loewner ; this latter condition means that there exists a positive constant m such that for all x ∈ Ω, and all non-degenerate disjoint continua E, F in λB(x), ∆(E, F ) ≤ 16 =⇒

mod Q (E, F ; Ω) ≥ m.

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Any space Ω with the property that Whitney balls λB(x) (with a fixed parameter) are uniformly bilipschitz equivalent to Euclidean balls (in a fixed dimension) is easily seen to have bounded geometry. Other examples include Riemannian manifolds with Ricci bounded geometry as well as the exotic examples of Bourdon-Pajot and Laakso for any Q > 1; see [BHK01, Exs.9.7, p.86] and the references mentioned there. (Note that by Proposition 5.6, (Ω, d, µ) has bounded geometry if and only if its quasihyperbolization satisfies the condition studied in [BHK01, Chpt.9].) The first part of bounded Q-geometry is a necessary condition for Ω to support any conformal density (at least when Ω is locally quasiconvex). The second part of bounded Q-geometry, the weak local Loewner criterion, can also be described in terms of Poincaré inequalities as explained in [BHK01, Proposition 9.4] and [HK98, §5]; it ensures that there are plenty of curves available (e.g., it gives local quasiconvexity). However, to substantiate this existence of many curves requires the use of certain modulus estimates (see Facts 2.15,2.16), and these estimates in turn require an upper mass condition. The Loewner part of bounded Q-geometry also performs an essential role in two other places. First, it is a key player in the proof of Proposition 2.9, which is the crucial ingredient in the proof of (d) implies (e) in Theorem D. Second, for uniform Loewner spaces with appropriately ‘thick’ boundaries, the Koebe condition for a metric-density follows from the Harnack and Ahlfors condition; this fact is utilized in the proof of Theorem C. Bounded geometry provides a number of essential properties for our underlying space. 5.6. Proposition. Let (Ω, d, µ) be an abstract domain having bounded Qgeometry (with constants M, m, λ). Then: (a) µ is locally Ahlfors Q-regular. (b) Ω is locally quasiconvex with constants which depend only on Q, M, m, λ. (c) Ω is locally Q-Loewner with a control function ψ and parameters κ, ε0 which depend only on the data Q, M, m, λ associated with Ω. Here is a useful consequence of the above. 5.7. Corollary. Let ρ be a Harnack Koebe density on an abstract domain (Ω, d, µ) having bounded Q-geometry. Then Ωρ is locally Ahlfors Q-regular and locally Q-Loewner with parameters and a control function which depend only on the data associated with ρ and Ω. Note that if Ωρ above is also uniform, then it would be globally Loewner by [BHK01, Theorem 6.4]. We record the following consequence of this observation. 5.8. Corollary. Let (Ω, d, µ) be an abstract domain having bounded Q-geometry (with constants M, m) and a Gromov δ-hyperbolic roughly κ-starlike quasihyperbolization (Ω, k). Then any BHK-uniformization Ωε = (Ω, dε , µε ) of Ω is a


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bounded locally Ahlfors Q-regular locally Q-Loewner space (hence of bounded Qgeometry), and even Q-Loewner when ε < ε(δ). Here the various parameters and control functions depend only on δ, κ, Q, M, m, λ, ε. 5.D. Lifts and Metric Doubling Measures. Here we discuss the ideas behind Theorem C and briefly outline the proof. Recall the notion of a metric doubling measure discussed near the end of §2.F. We define the lift ρν of ν via the formula

ν(Σx )1/P ; ρν (x) := d(x)

(5.9)

here ν is a P -dimensional metric doubling measure on ∂G Ω and, for some fixed base point w ∈ Ω, the shadow of a point x ∈ Ω is Σx := {ζ ∈ ∂G Ω : k(x, [w, ζ)) ≤ R}. It is not hard to see that there are constants R = R(δ, κ, ε) and C = C(δ, κ, ε) such that (5.10)

Sx := ∂ε Ω ∩ 2Bε (x) ⊂ Σx ⊂ ∂ε Ω ∩ CBε (x);

of course we are using the natural identification of ∂G Ω with ∂ε Ω. Employing (5.10), the doubling property of ν, and the fact that πε is Koebe, it is straightforward to verify that the push-forward of ρν (as defined in (5.9)) via the uniformizing density πε gives a density on Ωε which is bilipschitz equivalent to the density defined via the formula ρ(x) :=

ν(Sx )1/P dε (x)

where Sx := ∂ε Ω ∩ 2Bε (x).

Theorem C now follows from Corollary 5.5 once we verify that ρ is a Borel Harnack Ahlfors Koebe doubling metric-density on Ωε . (⌣) ¨ The first two of these are easy. The Koebe property follows from Theorem E once we know the Ahlfors volume growth property. To see that Theorem E can be applied, we first use Fact 2.2 along with Theorem F to see that the Whitney ball modulus property holds. To establish the Ahlfors property we need the doubling property. This in turn requires the following ‘quasihyperbolic doubling’ result. 5.11. Proposition. Let (Ω, d, µ) be a locally a-quasiconvex locally Ahlfors Qregular abstract domain. Then its quasihyperbolization (Ω, k) is locally doubling in the sense that if Σ ⊂ Ω is a set of points satisfying 0 < t ≤ k(x, y) ≤ T < ∞

for all x, y ∈ Σ, x 6= y,

then the cardinality of Σ is bounded by #Σ ≤ 2M 2 8Q (2M 2 24Q )8aT /λt . Here M is the local regularity constant and λ the Whitney ball constant.

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Here is an interesting application of Theorems C and D which provides a characterization of doubling conformal densities in terms of metric doubling measures. We say that a metric-density ρ on Ω is induced by a metric doubling measure ν on ∂G Ω if there exists a constant C ≥ 1 such that C −1 ρν (x) ≤ ρ(x) ≤ Cρν (x)

for µ-a.e. x ∈ Ω.

5.12. Theorem. Assume the basic minimal hypotheses, that the Gromov boundary of Ω is uniformly perfect, and P < Q. A metric-density ρ on Ω is a doubling conformal density, with (∂ρ Ω, dρ ) Ahlfors P -regular, if and only if ρ is induced by some P -dimensional metric doubling measure on ∂G Ω. 5.E. Volume Growth Problem. Here we briefly outline the proof of Theorem A. Recall that this result provides our answer to the problem of deciding when there exists a uniformizing conformal density (so, in particular, the associated measure (2.7) should have Ahlfors regular volume growth). The necessity in this result follows from Fact 4.1 along with Fact 2.20. The real work involved is in establishing the sufficiency. The first major step is to prove Theorem D. Following [BHR01], we say that a metric-density ρ, on a uniform space (Ω, d, µ), is doubling provided µρ is a doubling measure on (Ω, d); i.e., there exists a constant D = Dρ such that for all x ∈ Ω,

(D)

µρ [B(x; 2r)] ≤ D µρ [B(x; r)]

for all r > 0.

When Ω is not uniform, the above doubling condition may fail to hold even if ρ is ‘nice’; e.g., consider ρ = |f ′ | on Ω = {x + iy : |y| < 1} in the complex plane, where f is a conformal homeomorphism of Ω onto the unit disk. To compensate for this we employ the following definition: a conformal density ρ, on an abstract domain (Ω, d, µ) with Gromov hyperbolic quasihyperbolization, is doubling if its push-forward is doubling on some BHK-uniformization Ωε of Ω. (Since all such spaces Ωε are QS equivalent, and doubling measures can be defined for conformal gauges, there is no ambiguity here. See §2.C for a discussion of conformal gauges, and also the very last paragraph of §2.F.) Notice that the doubling condition (D) uses d-balls but µρ -measure and thus interweaves the measure properties of the deformed space with the metric properties of the original (or uniformized) space. Our proof of Theorem D requires the Gehring-Hayman Inequality (Theorem B), Corollary 2.10, and the following two results. First we give a necessary condition for a metric-density to be doubling. 5.13. Proposition. Let ρ be a Harnack density on an a-uniform locally Ahlfors Q-regular space (Ω, d, µ) (with constants H, M, λ). Suppose that ρ is also doubling on Ω (with constant D). Then quasihyperbolic geodesics in Ω are double c-cone arcs in Ωρ where c = c(D, H, M, Q, a, λ). If in addition Ω is bounded, then so is Ωρ and ∂Ω ⊂ ∂ρ Ω, with equality holding when Ω is also locally Q-Loewner. Next we state a sufficient condition for a metric-density to be doubling.


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5.14. Proposition. Let ρ be a conformal density on an a-uniform locally lower Ahlfors Q-regular space (Ω, d, µ) (with constants H, A, K, M, λ). Suppose that quasihyperbolic geodesics in Ω are double c-cone arcs in Ωρ . Then Ωρ is Ahlfors Q-regular with a parameter which depends only on c and the data H, A, K, M, Q, a, λ associated with ρ and Ω. If in addition Ω and Ωρ are both bounded, then ρ is doubling on Ω with a parameter which depends only on the aforementioned data and the quantity q given in the proof of Lemma 3.3. The second major step in the proof of Theorem A is to establish Theorem C. Its proof is outlined above in §5.D. That done, we use Theorem C to obtain a doubling conformal density, which by Theorem D is also bounded and uniformizing.

6. Characterizations of Uniform Spaces Here we mention a number of characterizations for uniform spaces and uniform domains. In [Väi88] Väisälä provides a complete description of the various possible twisted double cone conditions (which he calls length cigars, diameter cigars, distance cigars, and M¨ obius cigars). The work [Mar80] of Martio should also be mentioned. 6.A. Metric Characterizations. We have already mentioned that uniform spaces are precisely the abstract domains in which the quasihyperbolic metric is bilipschitz equivalent to the so-called j metric; see Fact 3.2. It turns out that the following seemingly weaker quasihyperbolic metric condition also characterizes uniform spaces. For uniform subdomains of Banach spaces this result is due to Väisälä [Väi91, 6.16, 6.17]. Here we write r(x, y) :=

|x − y| d(x) ∧ d(y)

to denote the so-called relative distance between x, y. See [BH07] for the following version. 6.1. Theorem. A locally quasiconvex abstract domain is uniform if and only if there is a homeomorphism ϑ : [0, ∞) → [0, ∞) satisfying lim supt→∞ ϑ(t)/t < 1, and such that for all points x, y, k(x, y) ≤ ϑ (r(x, y)). The uniformity constant depends only on ϑ, and conversely in an a-uniform space, one can always take ϑ(t) = b log(1 + t) with b = b(a). 6.B. Gromov Boundary Characterizations. We call Ω ( Rn a Gromov domain if its quasihyperbolization (Ω, k) is Gromov hyperbolic. Bonk, Heinonen and Koskela corroborated the following [BHK01, Proposition 7.12]. 6.2. Fact. A Gromov domain in Euclidean space is uniform if and only if it is linearly locally connected. They utilized the above to establish the following [BHK01, Theorem 7.11].

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6.3. Fact. A (bounded) Gromov domain in Euclidean space is uniform if and only if the canonical gauge on the Gromov boundary is quasisymmetrically equivalent to the Euclidean gauge on the Euclidean boundary. The careful reader will recognize that the Bonk-Heinonen-Koskela results were established for regions on the sphere (i.e., using the spherical metric). Results of Balogh and Buckley [BB06] are useful in this regard. Väisälä has recently proven a Banach space analog of the above result; see [Väi05b]. Along with providing a dimension free version of this result, he also considers arbitrary domains (not just bounded) and replaces QS equivalence with QM equivalence. In addition, he provides an example of a Gromov hyperbolic domain which is LLC but not uniform. 6.C. Characterizations using QC Maps. It is evident that bilipschitz homeomorphisms map uniform spaces to uniform spaces. This also holds true for QS and QM maps of uniform domains in Euclidean space and in Banach spaces [Väi99, Theorem 10.22], but not in the general metric space setting, and not for QC maps. On the other hand, according to [BKR98, 2.4], the average derivative of a quasiconformal map f : Bn → Ω ⊂ Rn is a conformal metric density on Bn (a uniform n-Loewner n-regular space). Thus we can appeal to Theorem D and read off a number of conditions which characterize when Ω will be uniform. 6.D. Capacity Conditions. It is known that given 0 < λ ≤ 1/2, there exists a constant c = c(λ, n) > 0 such that ∀ x, y ∈ Ω :

k(x, y) ≥ 2

=⇒

¯ ¯ mod(λB(x), λB(y); D) ≥ c/k(x, y)n−1 ;

this is valid for any proper subdomain Ω of Rn . To prove it, one starts by using Lemma 2.14 to select an appropriate cover of any quasihyperbolic geodesic joining x, y, and then a standard application of the Poincaré inequality applied to adjacent balls leads to the asserted inequality. See the proof of [HK96, Theorem 6.1]. Let C > 0 and 0 < λ ≤ 1/2. A proper subdomain Ω of Rn is a (C, λ)-k-cap domain provided ∀ x, y ∈ D :

k(x, y) ≥ 2

=⇒

¯ ¯ mod(λB(x), λB(y); D) ≤ C/k(x, y)n−1 .

¯ ¯ Thus in a k-cap domain D, we have mod(λB(x), λB(y); D) ≃ k(x, y)1−n for points with k(x, y) ≥ 2, with constants of comparison dependent only on λ, n, and the k-cap parameter. This is the two-sided version of a condition introduced by Buckley in [Buc04] to study quasiconformal images of domains which satisfy a quasihyperbolic boundary condition. As explained on p.26 of that paper, a (C, λ)-k-cap condition implies a (C ′ , λ′ )-k-cap condition for some C ′ = C ′ (C, λ, λ′ , n). We mainly consider the case λ = 1/2, and refer to a (C, 1/2)-k-cap domain simply as a C-k-cap domain.


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Every uniform domain in Rn is a k-cap domain, and the class of k-cap domains is invariant under quasiconformal mappings (with a quantitative change of parameter C). For proofs of these statements see [Buc04]. Recently we established the following characterization for uniform domains in Euclidean space; see [BH06, Theorem 3.5]. 6.4. Theorem. A proper subdomain of Rn is uniform if and only if it is both QEDwb and a k-cap domain. This result is quantitative. 6.E. LLC and Slice Conditions. By utilizing certain slice conditions, Balogh and Buckley [BB03] established a number of geometric characterizations for Gromov hyperbolic spaces. Here we mention the following new characterization of uniform spaces; see [BH07] 6.5. Theorem. An abstract domain is uniform and LEC if and only if it is quasiconvex, LLC2 with respect to arcs, and satisfies a weak slice condition. These implications are quantitative. We have modified the usual LLC conditions (see §2.E) by requiring that the points in question be joinable by rectifiable arcs (rather than just by continua). Every uniform domain in Rn is LLC. In fact every uniform space is quasiconvex and thus LLC1 with respect to arcs. However, uniform spaces need not be LLC2 ; e.g., an ‘asterik’ type space (the disjoint union of a point and a bunch of line segments or rays, with its intrinsic length distance) may be uniform but not LLC2 . We say that a locally complete metric space is locally externally connected, abbreviated LEC, provided there is a constant c ≥ 1 such that the (LLC2 ) condition holds for all points x ∈ Ω and all r ∈ (0, d(x)/c).

References [BB00] [BB03] [BB06] [Bon96] [BHK01] [BHR01] [BKR98] [BS00] [BH99]

Z.M. Balogh and M. Bonk, Gromov hyperbolicty and the Kobayashi metric on strictly pseudoconvex domains, Commet. Math. Helv. 75 (2000), no. 3, 504–533. Z.M. Balogh and S.M. Buckley, Geometric characterizations of Gromov hyperbolicty, Invent. Math. 153 (2003), 261–301. , Sphericalization and flattening, Conform. Geom. Dyn, to appear (2006) . M. Bonk, Quasi-geodesic segments and Gromov hyperbolic spaces, Geometriae Dedicata 62 (1996), 281–298. M. Bonk, J. Heinonen, and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001), 1–99. M. Bonk, J. Heinonen, and S. Rohde, Doubling conformal densities, J. reine angew. Math. 541 (2001), 117–141. M. Bonk, P. Koskela, and S. Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. 77 (1998), no. 3, 635–664. M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266–306. M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, SpringerVerlag, Berlin, 1999.

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S. Buckley, Slice conditions and their applications, Future Trends In Geometric Function Theory (Univ. Jyv¨ askylä), vol. 92, Rep. Univ. Jyv¨ askylä Dept. Math. Stat., 2003, RNC Workshop held in Jyv¨ askylä, June 15-18, 2003, pp. 63–76. S.M. Buckley, Quasiconfomal images of H¨ older domains, Ann. Acad. Sci. Fenn. Ser. Math. 29 (2004), 21–42. S. Buckley and D.A. Herron, Uniform domains and capacity, Israel J. Math, to appear (2006). , Uniform and weak slice spaces, in preparation (2007). S.M. Buckley and J. O’Shea, Weighted Trudinger-type inequalities, Indiana Univ. Math. J. 48 (1999), 85–114. S.M. Buckley and A. Stanoyevitch, Distinguishing properties of weak slice conditions, Conform. Geom. Dyn. 7 (2003), 49–75. D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, American Mathematical Society, Providence, RI, 2001. L. Capogna, N. Garofalo, and D-M Nhieu, Examples of uniform and NTA domains in Carnot groups, Proceedings on Analysis and Geometry (Novosibirsk), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., 2000, (Novosibirsk Akad., 1999), pp. 103–121. L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Manuscripta Math. 86 (1995), no. 3, 267–281. G. David and S. Semmes, Fractured fractals and broken dreams:self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, vol. 7, Oxford University Press, Oxford, 1997. F.W. Gehring, Characteristic properties of quasidisks, Les Presses de l’Université de Montréal, Montréal, Quebec, 1982. , Uniform domains and the ubiquitous quasidisk, Jahresber. Deutsch. Math.Verein 89 (1987), 88–103. F.W. Gehring and W.K. Hayman, An inequality in the theory of conformal mapping, J. Math. Pures Appl. 41 (1962), no. 9, 353–361. F.W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math. 45 (1985), 181–206. F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Analyse Math. 36 (1979), 50–74. F.W. Gehring and B.P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. F.W. Gehring and J. Väis¨ alä, The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1–70. A.V. Greshnov, On uniform and NTA-domains on Carnot groups, Sibirsk. Mat. Zh. 42 (2001), no. 5, 1018–1035. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999. P. Hästö, Gromov hyperbolicity of the jG and ˜jG metrics, Proc. Amer. Math. Soc. 134 (2006), 1137–1142. J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001. J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61–79. J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta. Math. 181 (1998), 1–61. J. Heinonen and R. Nakki, Quasiconformal distortion on arcs, J. d’Analyse Math. 63 (1994), 19–53. J. Heinonen and S. Rohde, The Gehring-Hayman inequality for quasihyperbolic geodesics, Math. Proc. Camb. Phil. Soc. 114 (1993), 393–405.


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David A Herron E-mail: [email protected] Address: Department of Mathematics, University of Cincinnati, OH 45221, USA


p-Laplace operator, quasiregular mappings, and Picard-type theorems Ilkka Holopainen and Pekka Pankka Abstract. We describe the role of p-harmonic functions and forms in the theory of quasiregular mappings. Keywords. Quasiregular mapping, p-harmonic function, p-harmonic form, conformal capacity. 2000 MSC. Primary 58J60, 30C65; Secondary 53C20, 31C12, 35J60.


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2. A-harmonic functions

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3. Morphism property and its consequences

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3.1. Sketch of the proof of Reshetnyak’s theorem

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Further properties of f

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4. Modulus and capacity inequalities

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5. Liouville-type results for A-harmonic functions

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6. Liouville-type results for quasiregular mappings

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7. Picard-type theorems

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8. Quasiregular mappings, p-harmonic forms, and de Rham cohomology 145 References

148

1. Introduction In this survey we emphasize the importance of the p-Laplace operator as a tool to prove basic properties of quasiregular mappings, as well as Liouvilleand Picard-type results for quasiregular mappings between given Riemannian manifolds. Quasiregular mappings were introduced by Reshetnyak in the mid sixties in a series of papers; see e.g. [36], [37], and [38]. An interest in studying these mappings arises from a question about the existence of a geometric function theory in real dimensions n ≥ 3 generalizing the theory of holomorphic functions C → C.

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Definition 1.1. A continuous mapping f : U → Rn of a domain U ⊂ Rn is called quasiregular (or a mapping of bounded distortion) if 1,n (1) f ∈ Wloc (U ; Rn ), and (2) there exists a constant K ≥ 1 such that

|f ′ (x)|n ≤ K Jf (x) for a.e. x ∈ U.

The condition (1) means that the coordinate functions of f belong to the 1,n local Sobolev space Wloc (U ) consisting of locally n-integrable functions whose distributional (first) partial derivatives are also locally n-integrable. In Condition (2) f ′ (x) denotes the formal derivative of f at x, i.e. the n × n matrix Dj fi (x) defined by the partial derivatives of the coordinate functions fi of f . Furthermore, |f ′ (x)| = max|f ′ (x)h| |h|=1

′

is the operator norm of f (x) and Jf (x) = det f ′ (x) is the Jacobian determinant of f at x. They exist a.e. by (1). The smallest possible K in Condition (2) is the outer dilatation KO (f ) of f . If f is quasiregular, then Jf (x) ≤ K ′ ℓ(f ′ (x))n

a.e.

for some constant K ′ ≥ 1, where

ℓ(f ′ (x)) = min |f ′ (x)h|. |h|=1

The smallest possible K ′ is the inner dilatation KI (f ) of f . It is easy to see by linear algebra that KO (f ) ≤ KI (f )n−1 and KI (f ) ≤ KO (f )n−1 . If max{KO (f ), KI (f )} ≤ K, f is called K-quasiregular .

To motivate the above definition, let us consider a holomorphic function f : U → C, where U ⊂ C is an open set. We write f as a mapping f = (u, v) : U → R2 , U ⊂ R2 , f (x, y) = u(x, y), v(x, y) .

Then u and v are harmonic real-valued functions in U and they satisfy the Cauchy-Riemann system of equations D1 u = D2 v D2 u = −D1 v, where D1 = ∂/∂x, D2 = ∂/∂y. For every (x, y) ∈ U , the differential f ′ (x, y) : R2 → R2 is a linear map whose matrix (with respect to the standard basis of the plane) is D1 u D 2 u D1 u D 2 u = . D1 v D 2 v −D2 u D1 u Hence (1.1)

|f ′ (x, y)|2 = det f ′ (x, y).

p-Laplace operator, quasiregular mappings, and Picard-type theorems

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The first trial definition for mappings f : U → Rn of a domain U ⊂ Rn , sharing some geometric and topological properties of holomorphic functions, could be mappings satisfying a condition (1.2)

|f ′ (x)|n = Jf (x),

x ∈ U.

However, it has turned out that, in dimensions n ≥ 3, a mapping f : U → Rn 1,n belonging to the Sobolev space Wloc (U ; Rn ) and satisfying (1.2) for a.e. x ∈ U is either constant or a restriction of a Möbius map. This is the so-called generalized Liouville theorem due to Gehring [12] and Reshetnyak [38]; see also the thorough discussion in [29]. Next candidate for the definition is obtained by replacing the equality (1.2) by a weaker condition (1.3)

|f ′ (x)|n ≤ K Jf (x) a.e. x ∈ U,

where K ≥ 1 is a constant. Note that Jf (x) ≤ |f ′ (x)|n holds for a.e. x ∈ U . Now there remains a question on the regularity assumption of such mapping f. Again there is some rigidity in dimensions n ≥ 3. Indeed, if a mapping f satisfying (1.3) is non-constant and smooth enough (more precisely, if f ∈ C k , with k = 2 for n ≥ 4 and k = 3 for n = 3), then f is a local homeomorphism. Furthermore, it then follows from a theorem of Zorich that such mapping f : Rn → Rn is necessarily a homeomorphism, for n ≥ 3; see [47]. We would also like a class of maps satisfying (1.3), with fixed K, to be closed under local uniform convergence. In order to obtain a rich enough class of mappings, it is thus necessary to weaken the regularity assumption from C k -smoothness. See [15], [5], and [32] for recent developments regarding smoothness and branching of quasiregular mappings. The basic analytic and topological properties of quasiregular mappings are listed in the following theorem by Reshetnyak; see [39], [41]. Theorem 1.2 (Reshetnyak’s theorem). Let U ⊂ Rn be a domain and let f : U → Rn be quasiregular. Then (1) f is differentiable a.e. and (2) f is either constant or it is discrete, open, and sense-preserving. Recall that a map g : X → Y between topological spaces X and Y is discrete if the preimage g −1 (y) of every y ∈ Y is a discrete subset of X and that g is open if gU is open for every open U ⊂ X. We also remark that a continuous discrete and open map g : X → Y is called a branched covering.

To say that f : U → Rn is sense-preserving means that the local degree µ(y, f, D) is positive for all domains D ⋐ U and for all y ∈ f D \ f ∂D. The local degree is an integer that tells, roughly speaking, how many times f wraps D around y. It can be defined, for example, by using cohomology groups with compact support. For the basic properties of the local degree, we refer to [41, Proposition I.4.4]; see also [11], [35], and [45]. For example, if f is differentiable at x0 with Jf (x0 ) 6= 0, then µ(f (x0 ), f, D) = sign Jf (x0 ) for sufficiently small

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connected neighborhoods D of x0 . Another useful property is the following homotopy invariance: If f and g are homotopic via a homotopy ht , h0 = f, h1 = g, such that y ∈ ht D \ ht ∂D for every t ∈ [0, 1], then µ(y, f, D) = µ(y, g, D). f y D f ∂D fD

The definition of quasiregular mappings extends easily to the case of continuous mappings f : M → N, where M and N are connected oriented Riemannian n-manifolds. Definition 1.3. A continuous mapping f : M → N is quasiregular (or a mapping 1,n of bounded distortion) if it belongs to the Sobolev space Wloc (M ; N ) and there exists a constant K ≥ 1 such that |Tx f |n ≤ KJf (x) for a.e. x ∈ M.

(1.4)

Here again Tx f : Tx M → Tf (x) N is the formal differential (or the tangent map) of f at x, |Tx f | is the operator norm of Tx f , and Jf (x) is the Jacobian determinant of f at x uniquely defined by (f ∗ volN )x = J(x, f )(volM )x almost everywhere. Note that Tx f can be defined for a.e. x by using partial derivatives of local expressions of f at x. The geometric interpretation of (1.4) is that Tx f maps balls of Tx M either to ellipsoids with controlled ratios of the semi-axes or Tx f is the constant linear map. Tx M Tf (x) N

Tx f

f

M

N

We assume from now on that M and N are connected oriented Riemannian n-manifolds.

2. A-harmonic functions It is well-known that the composition u◦f of a holomorphic function f : U → C and a harmonic function u : f U → R is a harmonic function in U . In other words, holomorphic functions are harmonic morphisms. Quasiregular mappings have a somewhat similar morphism property: If f : U → Rn is quasiregular and u is an n-harmonic function in a neighborhood of f U , then u◦f is a so-called A-harmonic


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function in U . In this section we introduce the notion of A-harmonic functions and recall some of their basic properties that are relevant for this survey. We denote by h·, ·i the Riemannian metric of M . Recall that the gradient of a smooth function u : M → R is the vector field ∇u such that h∇u(x), hi = du(x)h

for every x ∈ M and h ∈ Tx M.

The divergence of a smooth vector field V can be defined as a function div V : M → R satisfying LV ω = (div V ) ω,

where ω = volM is the (Riemannian) volume form and

(αt )∗ ω − ω t→0 t is the Lie derivative of ω with respect to V, and α is the flow of V. LV ω = lim

V

αt (x) x

α = flow of V

We say that a vector field ∇u ∈ L1loc (M ) is a weak gradient of u ∈ L1loc (M ) if Z Z (2.1) h∇u, V i = − u div V M

M

C0∞ (M ).

for all vector fields V ∈ Conversely, a function div V ∈ L1loc (M ) is a weak divergence of a (locally integrable) vector field V if (2.1) holds for all R u ∈ C0∞ (M ). Note that M div Y = 0 if Y is a smooth vector field in M with compact support. We define the Sobolev space W 1,p (M ) and its norm as

W 1,p (M ) = {u ∈ Lp (M ) : weak gradient ∇u ∈ Lp (M )}, 1 ≤ p < ∞, kuk1,p = kukp + k|∇u|kp .

Let G ⊂ M be open. Suppose that for a.e. x ∈ G we are given a continuous map Ax : Tx M → Tx M

such that the map x 7→ Ax (X) is a measurable vector field whenever X is. Suppose that there are constants 1 < p < ∞ and 0 < α ≤ β < ∞ such that and

hAx (h), hi ≥ α|h|p |Ax (h)| ≤ β|h|p−1

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for a.e. x ∈ G and for all h ∈ Tx M. In addition, we assume that for a.e. x ∈ G hAx (h) − Ax (k), h − ki > 0 whenever h 6= k, and whenever λ ∈ R \ {0}.

Ax (λh) = λ|λ|p−2 Ax (h)

1,p A function u ∈ Wloc (G) is called a (weak) solution of the equation

− div Ax (∇u) = 0

(2.2) in G if

Z

G

hAx (∇u), ∇ϕi = 0

for all ϕ ∈ C0∞ (G). Continuous solutions of (2.2) are called A-harmonic functions (of type p). By the fundamental work of Serrin [43], every solution of (2.2) has a continuous representative. In the special case Ax (h) = |h|p−2 h, A-harmonic functions are called p-harmonic and, in particular, if p = 2, we obtain the usual harmonic functions. The conformally invariant case p = n = the dimension of M is important in the sequel. In this case p-harmonic functions are called, of course, n-harmonic functions. 1,p A function u ∈ Wloc (G) is a subsolution of (2.2) in G if

− div Ax (∇u) ≤ 0 weakly in G, that is

Z

G

hAx (∇u), ∇ϕi ≤ 0

for all non-negative ϕ ∈ C0∞ (G). A function v is called supersolution of (2.2) if −v is a subsolution. The proofs of the following two basic estimates are straightforward once the appropriate test function is found. Therefore we just give the test function and leave the details to readers. Lemma 2.1 (Caccioppoli inequality). Let u be a positive solution of (2.2) (for a given fixed p) in G and let v = uq/p , where q ∈ R \ {0, p − 1}. Then p Z Z β|q| p p v p |∇η|p (2.3) η |∇v| ≤ α|q − p + 1| G G

for every non-negative η ∈ C0∞ (G).

Proof. Write κ = q − p + 1 and use ϕ = uκ η p as a test function. Remark 2.2. In fact, the estimate (2.3) holds for positive supersolutions if q < p − 1, q 6= 0, and for positive subsolutions if q > p − 1. The excluded case q = 0 above corresponds to the following logarithmic Caccioppoli inequality.


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Lemma 2.3 (Logarithmic Caccioppoli inequality). Let u be a positive supersolution of (2.2) (for a given fixed p) in G and let C ⊂ G be compact. Then Z Z p (2.4) |∇ log u| ≤ c |∇η|p C

G

for all η ∈ C0∞ (G), with η|C ≥ 1, where c = c(p, β/α). Proof. Choose ϕ = η p u1−p as a test function.

These two lemmas together with the Sobolev and Poincaré inequalities are used in proving Harnack’s inequality for non-negative A-harmonic functions by the familiar Moser iteration scheme. In the following |A| denotes the volume of a measurable set A ⊂ M. Theorem 2.4 (Harnack’s inequality). Let M be a complete Riemannian manifold and suppose that there are positive constants R0 , C, and τ ≥ 1 such that a volume doubling property |B(x, 2r)| ≤ C |B(x, r)|

(2.5)

holds for all x ∈ M and 0 < r ≤ R0 , and that M admits a weak (1, p)-Poincaré inequality  1/p Z Z (2.6) |v − vB | ≤ C r  |∇v|p  B

τB

for all balls B = B(x, r) ⊂ M, with τ B = B(x, τ r) and 0 < r ≤ R0 , and for all functions v ∈ C ∞ (B). Then there is a constant c such that (2.7)

sup u ≤ c inf u

B(x,r)

B(x,r)

whenever u is a non-negative A-harmonic function in a ball B(x, 2r), with 0 < r ≤ R0 . In particular, if the volume doubling condition (2.5) and the Poincaré inequality (2.6) hold globally, that is, without any bound on the radius r, we obtain a global Harnack inequality. We refer to [18], [9], and [16] for proofs of the Harnack inequality.

3. Morphism property and its consequences The very first step in developing the theory of quasiregular mappings is to prove, by direct computation, that quasiregular mappings have the morphism property in a special case where the n-harmonic function is smooth enough. Theorem 3.1. Let f : M → N be a quasiregular mapping (with a constant K) and let u ∈ C 2 (N ) be n-harmonic. Then v = u ◦ f is A-harmonic (of type n) in M, with (3.1)

n

Ax (h) = hGx h, hi 2 −1 Gx h,

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where Gx : Tx M → Tx M is given by ( Jf (x)2/n Tx f −1 (Tx f −1 )T h, if Jf (x) exists and is positive, Gx h = h, otherwise. The constants α and β for A depend only on n and K. Proof. Let us first write the proof formally and then discuss the steps in more detail. In the sequel ω stands for the volume forms in M and N. Let V ∈ C 1 (M ) be the vector field V = |∇u|n−2 ∇u. Since u is n-harmonic and C 2 -smooth, we have div V = 0. By Cartan’s formula we obtain d(V y ω) = d(V y ω) + V y (dω) = LV ω = (div V ) ω = 0

since dω = 0. Here Xy η is the contraction of a differential form η by a vector field X. Thus, for instance, V y ω is the (n − 1)-form

Hence

V y ω(·, . . . , ·) = ω(V, ·, . . . , ·). | {z } | {z }

(3.2)

df ∗ (V y ω) = f ∗ d(V y ω) = 0.

n−1

n−1

a.e.

On the other hand, we have a.e. in M f ∗ (V y ω) = W y f ∗ ω = W y (Jf ω) = Jf W y ω,

(3.3)

where W is a vector field that will be specified later (roughly speaking, f∗ W = V ). We obtain (3.4)

d(Jf W y ω) = 0,

or equivalently (3.5)

div(Jf W ) = 0

which can be written as (3.6) where A is as in the claim.

div Ax (∇v) = 0,

Some explanations are in order. When writing a.e.

f ∗ d(V y ω) = 0, we mean that for a.e. x ∈ U and for all vectors v1 , v2 , . . . , vn ∈ Tx M

f ∗ d(V y ω)(v1 , v2 , . . . , vn ) = d(V y ω)(f∗ v1 , f∗ v2 , . . . , f∗ vn ) = 0,

where f∗ = f∗,x = Tx f is the tangent mapping of f at x. The equality on the 1,n left-hand side of (3.2) holds in a weak sense since f ∈ Wloc (M ); see [39, p. 136]. ∞ This means that, for all n-forms η ∈ C0 (M ), Z Z ∗ (3.7) hf d(V y ω), ηi = hf ∗ (V y ω), δηi, M

M


125

where δ is the codifferential. Consequently, equations (3.4)–(3.6) are to be interpreted in weak sense. In particular, combining (3.2), (3.3), and (3.7) we get Z Z Z ∗ hJf W y ω, δηi = hf (V y ω), δηi = hf ∗ d(V y ω), ηi = 0 M

M

M

for all n-forms η ∈ C0∞ (M ), and so (3.4) holds in weak sense.

Let us next specify the vector field W. Let A = {x ∈ M : Jf (x) = det f∗,x 6= 0}. Hence f∗,x is invertible for all x ∈ A, and W = f∗−1 V in A. In M \ A, either Jf (x) does not exist, which can happen only in a set of measure zero, or Jf (x) ≤ 0. Quasiregularity of f, more precisely the distortion condition (1.4), implies that f∗,x = Tx f = 0 for almost every such x. Hence f∗,x = 0 for a.e. x ∈ M \ A. Setting W = 0 in M \ A, we obtain f ∗ (V y ω) = 0 = W y f ∗ ω a.e. in M \ A. Hence f ∗ (V y ω) = W y f ∗ ω a.e. in M, and so (3.3) holds. 3.1. Sketch of the proof of Reshetnyak’s theorem. We shall use Theorem 3.1 to sketch the proof of Reshetnyak’s theorem in a way that uses analysis, in particular, A-harmonic functions. First we recall some definitions concerning p-capacity. If Ω ⊂ M is an open set and C ⊂ Ω is compact, then the p-capacity of the pair (Ω, C) is Z (3.8) capp (Ω, C) = inf |∇ϕ|p , ϕ

Ω

where the infimum is taken over all functions ϕ ∈ C0∞ (Ω), with ϕ|C ≥ 1. A compact set C ⊂ M is of p-capacity zero, denoted by capp C = 0, if capp (Ω, C) = 0 for all open sets Ω ⊃ C. Finally, a closed set F is of p-capacity zero, denoted by capp F = 0, if capp C = 0 for all compact sets C ⊂ F. It is a well-known fact that a closed set F ⊂ Rn containing a continuum C cannot be of n-capacity zero. This can be seen by taking an open ball B containing C and any test function ϕ ∈ C0∞ (B), with ϕ|C = 1, and using a potential estimate Z Z |∇ϕ| |∇ϕ| |ϕ(x) − ϕ(y)| ≤ c dz + dz , x, y ∈ B, n−1 n−1 B |x − z| B |y − z| combined with a maximal function and covering arguments. Similarly, if C is ¯ ⊂ Ω \ C, then a continuum in a domain Ω and B is an open ball, with B ¯ capn (C, B; Ω) > 0, where Z ¯ capn (C, B; Ω) = inf |∇ϕ|p > 0, ϕ

Ω

the infimum being taken over all functions ϕ ∈ C ∞ (Ω), with ϕ|C = 1 and ¯ = 0. ϕ|B

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f is light. Suppose then that U ⊂ Rn is a domain and that f : U → Rn is a non-constant quasiregular mapping. We will show first that f is light which means that, for all y ∈ Rn , the preimage f −1 (y) is totally disconnected, i.e. each component of f −1 (y) is a point. Fix y ∈ Rn and define u : Rn \ {y} → R by 1 . u(x) = log |x − y|

Then u is C ∞ and n-harmonic in Rn \ {y} by a direct computation. By Theorem 3.1, v = u ◦ f, 1 , v(x) = log |f (x) − y| is A-harmonic in an open non-empty set U \ f −1 (y) and v(x) → +∞ as x → z ∈ f −1 (y). We set v(z) = +∞ for z ∈ f −1 (y).

To show that f is light we use the logarithmic Caccioppoli inequality (2.4). Suppose that C ⊂ f −1 (y) ∩ U is a continuum. Since f is non-constant and continuous, there exists m > 1 such that the set Ω = {x ∈ U : v(x) > m} is an ¯ ⊂ U. We choose another neighborhood D of C open neighborhood of C and Ω ¯ such that D ⊂ Ω is compact. Now we observe that vi = min{v, i} is a positive supersolution for all i > m. The logarithmic Caccioppoli inequality (2.4) then implies that Z ¯ ≤c 0 capn (C, B; since C is a continuum. Let MB = max log v. ¯ B

Now the idea is to use 1 v log }} k MB ¯ D) for every k ∈ N. We get a contradiction since as a test function for capn (C, B; ¯ D) ≤ k −n k∇ log vkLp (D) → 0 0 < capn (C, B; min{1, max{0,

as k → ∞. Thus f −1 (y) can not contain a continuum.

Differentiability a.e. Assume that f = (f1 , . . . , fn ) : U → Rn , U ⊂ Rn , is quasiregular. Then coordinate functions fj are A-harmonic again by Theorem 3.1, since functions x = (x1 , . . . , xn ) 7→ xj are n-harmonic. Now there are at least two ways to prove that f is differentiable almost everywhere. For instance, since each fj is A-harmonic, one can show by employing reverse Hölder inequality 1,p techniques that, in fact, f ∈ Wloc (U ), with some p > n. This then implies that f is differentiable a.e. in U ; see e.g. [3]. Another way is to conclude


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that f is monotone, i.e. each coordinate function fj is monotone, and therefore 1,n differentiable a.e. since f ∈ Wloc (U ); see [41]. The monotonicity of fj holds since A-harmonic functions obey the maximum principle. 1,n f is sense-preserving. Here one first shows that conditions f ∈ Wloc (U ) and Jf (x) ≥ 0 a.e. imply that f is weakly sense-preserving, i.e. µ(y, f, D) ≥ 0 for all domains D ⋐ U and for all y ∈ f D \ f ∂D. This step employs approximation of f by smooth mappings. Pick then a domain D ⋐ U and y ∈ f D \ f ∂D. Denote by Y the y-component of Rn \ f ∂D and write V = D ∩ f −1 Y . Since f is light, D \ f −1 (y) is non-empty. Thus we can find a point x0 ∈ f −1 (y) ∩ V. Next we conclude that the set {x ∈ V : Jf (x) > 0} has positive measure. Otherwise, since f is ACL and |f ′ (x)| = 0 a.e. in V , f would be constant in a ball centered at x0 contradicting the fact that f is light. Thus there is a point x in V where f is differentiable and Jf (x) > 0. Now a homotopy argument, using the differential of f at z, and µ(y, f, D) ≥ 0 imply that f is sensepreserving.

f is discrete and open. This part of the proof is purely topological. A sensepreserving light mapping is discrete and open by Titus and Young; see e.g. [41]. Further properties of f . Once Reshetnyak’s theorem is established it is possible to prove further properties for quasiregular mappings. We collect these properties to the following theorems and refer to the books [39] and [41] for the proofs. Theorem 3.2. Let f : M → N be a non-constant quasiregular map. Then

1. |f E| = 0 if and only if |E| = 0. 2. |Bf | = 0, where Bf is the branch set of f, i.e. the set of all x ∈ M where f does not define a local homeomorphism. 3. Jf (x) > 0 a.e. 4. The integral transformation formula Z Z (h ◦ f )(x)Jf (x)dm(x) = h(y)N (y, f, A)dm(y) A

N

holds for every measurable h : N → [0, +∞] and for every measurable A ⊂ M, where N (y, f, A) = card f −1 (y) ∩ A. 1,n 1,n 5. If u ∈ Wloc (N, R), then v = u ◦ f ∈ Wloc (M, R) and ∇v(x) = Tx f T ∇u(f (x)) a.e.

Furthermore, we have a generalization of the morphism property. Theorem 3.3. Let f : M → N be quasiregular and let u : N → R be an Aharmonic function (or a subsolution or a supersolution, respectively) of type n. Then v = u ◦ f is f # A-harmonic (a subsolution or a supersolution, respectively), where ( Jf (x)Tx f −1 Af (x) (Tx f −1 )T h), if Jf (x) exists and is positive, f # Ax (h) = |h|n−2 h, otherwise.

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The ingredients of the proof of Theorem 3.3 include, for instance, the locality of A-harmonicity, Theorem 3.2, and a method to ”push-forward” (test) functions; see e.g. [16] and [41].

4. Modulus and capacity inequalities Although the main emphasis of this survey is on the relation between quasiregular mappings and p-harmonic functions, we want to introduce also the other main tool in the theory of quasiregular mappings. Let 1 ≤ p < ∞ and let Γ be a family of paths in M. We denote by F(Γ) the set of all Borel functions ̺ : M → [0, +∞] such that Z γ

̺ds ≥ 1

for all locally rectifiable path γ ∈ Γ. We call the functions in F(Γ) admissible for Γ. The p-modulus of Γ is defined by Z Mp (Γ) = inf ̺p dm. ̺∈F (Γ)

M

There is a close connection between p-modulus and p-capacity. Indeed, suppose that Ω ⊂ M is open and C ⊂ Ω is compact. Let Γ be the family of all paths in Ω \ C connecting C and ∂Ω. Then

(4.1)

capp (Ω, C) = Mp (Γ).

The inequality capp (Ω, C) ≥ Mp (Γ) follows easily since ̺ = |∇ϕ| is admissible for Γ for each function ϕ as in (3.8). The other direction is harder and requires an approximation argument; see [41, Proposition II.10.2]. If p = n = the dimension of M , we call Mn (Γ) the conformal modulus of Γ, or simply the modulus of Γ. In that case Mn (Γ) is invariant under conformal changes of the metric. In fact, Mn (Γ) can be interpreted as follows: Define a new measurable Riemannian metric hh·, ·ii = ̺2 h·, ·i. Then, with respect to hh·, ·ii, the length of γ has a lower bound Z ℓhh·,·ii (γ) = ̺ds ≥ 1 γ

and the volume of M is given by Volhh·,·ii (M ) =

Z

̺n dm.

M

Thus we are minimizing the volume of M under the constraint that paths in Γ have length at least 1. The importance of the conformal modulus for quasiregular mappings lies in the following invariance properties; see [41, II.2.4, II.8.1]


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Theorem 4.1. Let f : M → N be a non-constant quasiregular mapping. Let A ⊂ M be a Borel set with N (f, A) := supy N (y, f, A) < ∞, and let Γ be a family of paths in A. Then (4.2)

Mn (Γ) ≤ KO (f ) N (f, A) Mn (f Γ).

Theorem 4.2 (Poletsky’s inequality). Let f : M → N be a non-constant quasiregular mapping and let Γ be a family of paths in M. Then (4.3)

Mn (f Γ) ≤ KI (f ) Mn (Γ).

The proof of (4.2) is based on the change of variable formula for integrals (Theorem 3.2 3.) and on Fuglede’s theorem. The estimate (4.3) in the converse direction is more useful than (4.2) but also much harder to prove; see [41, p. 39–50]. As an application of the use of p-modulus and p-capacity, we prove a Harnack’s inequality for positive A-harmonic functions of type p > n − 1. Assume that Ω ⊂ M is a domain, D ⋐ Ω another domain, and C ⊂ D is compact. For p > n − 1, we set λp (C, D) = inf Mp (Γ(E, F ; D)), E,F

where E and F are continua joining C and Ω \ D, and Γ(E, F ; D) is the family of all paths joining E and F in D. Theorem 4.3 (Harnack’s inequality, p > n − 1). Let Ω, D, and C be as above. Let u be a positive A-harmonic function in Ω of type p > n − 1. Then ¯ 1/p capp (Ω, D) MC ≤ c0 , (4.4) log mC λp (C, D)

where

MC = max u(x), x∈C

mC = min u(x), x∈C

and c0 = c0 (p, β/α).

E γ

Ω

C F

D

Proof. We may assume that MC > mC . Let ε > 0 be so small that MC − ε > mC + ε. Then the sets {x : u(x) ≥ MC − ε} and {x : u(x) ≤ mC + ε} contain continua E and F , respectively, that join C and Ω \ D. Write w=

log u − log(mC + ε) log(MC − ε) − log(mC + ε)

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and observe that w ≥ 1 in E and w ≤ 0 in F. Therefore |∇w| is admissible for Γ(E, F ; D) and hence Z |∇w|p ≥ Mp (Γ(E, F ; D)) ≥ λp (C, D). D

On the other hand,

Z

D

¯ |∇ log u|p dm ≤ c(p, β/α) capp (Ω, D)

by the logarithmic Caccioppoli inequality (2.4), and MC − ε ∇ log u = log ∇w. mC + ε Hence MC − ε ≤ c0 log mC + ε and (4.4) follows by letting ε → 0.

¯ 1/p capp (Ω, D) λp (C, D)

We can define λp (C, D) analogously for p ≤ n − 1, too. However, λp (C, D) vanishes for p ≤ n − 1. Consequently, Theorem 4.3 is useful only for p > n − 1. The idea of the proof is basically due to Granlund [13]. In the above form, (4.4) appeared first time in [17]. In general, it is difficult to obtain an effective lower ¯ However, if bound for λp (C, D) together with an upper bound for capp (Ω, D). n M = R and p = n, one obtains a global Harnack inequality by choosing C, D, and Ω as concentric balls.

5. Liouville-type results for A-harmonic functions We say that M is strong p-Liouville if M does not support non-constant positive A-harmonic functions for any A of type p. We have already mentioned that a global Harnack inequality max u ≤ c min u

B(x,r)

B(x,r)

holds for non-negative A-harmonic functions on B(x, 2r) with a (Harnack-)constant c independent of x, r, and u if M is complete and admits a global volume doubling condition and (1, p)-Poincaré’s inequality. It follows from the global Harnack inequality that such manifold M is strong p-Liouville. Example 5.1. 1. Let M be complete with non-negative Ricci curvature. Then it is well-known that M admits a global volume doubling property by the Bishop-Gromov comparison theorem (see [2], [8]). Furthermore, Buser’s isoperimetric inequality [6] implies that M also admits a (1, p)-Poincaré inequality for every p ≥ 1. Hence M is strong p-Liouville.


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2. Let Hn be the Heisenberg group. We write elements of Hn as (z, t), where z = (z1 , . . . , zn ) ∈ Cn and t ∈ R. Furthermore, we assume that Hn is equipped with a left-invariant Riemannian metric in which the vector fields ∂ ∂ + 2yj , ∂xj ∂t ∂ ∂ Yj = − 2xj , ∂yj ∂t ∂ T = , ∂t j = 1, . . . , n, form an orthonormal frame. Harnack’s inequality for nonnegative A-harmonic functions on Hn was proved in [18] by using Jerison’s version of Poincaré’s inequality. Jerison proved in [31] that (1,1)-Poincaré’s inequality holds for the horizontal gradient Xj =

∇0 u =

n X

((Xj u)Xj + (Yj u)Yj )

j=1

and for balls in so-called Carnot-Carathéodory metric. Since the Lp -norm of the Riemannian gradient is larger than that of the horizonal gradient, we have (1,1)-Poincaré’s inequality for the Riemannian gradient as well if geodesic balls are replaced by Carnot-Carathéodory balls or Heisenberg balls BH (r) = {(z, t) ∈ Hn : (|z|4 + t2 )1/4 < r} and their left-translations. Classically, a Riemannian manifold M is called parabolic if it does not support a positive Green’s function for the Laplace equation. Definition 5.2. We say that a Riemannian manifold M is p-parabolic, with 1 < p < ∞, if capp (M, C) = 0 for all compact sets C ⊂ M. Otherwise, we say that M is p-hyperbolic. Example 5.3. 1. A compact Riemannian manifold is p-parabolic for all p ≥ 1. 2. In the Euclidean space Rn we have precise formulas for p-capacities of balls: ( c rn−p , if 1 ≤ p < n, ¯ capp (Rn , B(r)) = 0, otherwise. Hence Rn is p-parabolic if and only if p ≥ n. 3. If the Heisenberg group Hn is equipped with the left-invariant Riemannian metric, we do not have precise formulas for capacities of balls. However, for r ≥ 1, ¯H (r)) ≈ r2n+2−p capp (Hn , B ¯H (r)) = 0 if p ≥ 2n + 2. Hence Hn is if 1 ≤ p < 2n + 2, and capp (Hn , B p-parabolic if and only if p ≥ 2n + 2.

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4. Any complete Riemannian manifold M with finite volume Vol(M ) < ∞ is p-parabolic for all p ≥ 1. This is easily seen by fixing a point o ∈ M ¯ r) = 1 and |∇ϕ| ≤ and taking a function ϕ ∈ C0∞ B(o, R) , with ϕ|B(o, c/(R − r). We obtain an estimate ¯ r) ≤ c Vol(M )/(R − r)p → 0 capp B(o, R), B(o,

as R → ∞. 5. Let M n be a Cartan-Hadamard n-manifold, i.e. a complete, simply connected Riemannian manifold of non-positive sectional curvatures and dimension n. If sectional curvatures have a negative upper bound KM ≤ −a2 < 0, then M is p-hyperbolic for all p ≥ 1. This follows since M n satisfies an isoperimetric inequality a Area(∂D) Vol(D) ≤ n−1 for all domains D ⋐ M, with smooth boundary; see [46], [7]. Another proof uses the Laplace comparison and Green’s formula. If p > 1, then v(x) = exp(−δd(x, o)) is a positive supersolution of the p-Laplace equation for some δ = δ(n, p) > 0 (see [20]). Hence the p-hyperbolicity of M also follows from the theorem below for p > 1. Theorem 5.4. Let M be a Riemannian manifold and 1 < p < ∞. Then the following conditions are equivalent: 1. M is p-parabolic. 2. Mp (Γ∞ ) = 0, where Γ∞ is the family of all paths γ : [0, ∞) → M such that γ(t) → ∞ as t → ∞. 3. Every non-negative supersolution of (5.1)

− div Ax (∇u) = 0

on M is constant for all A of type p. 4. M does not support a positive Green’s function g(·, y) for (5.1) for any A of type p and y ∈ M. Here γ(t) → ∞ means that γ(t) eventually leaves any compact set. For the proof of Theorem 5.4 as well as for the discussion below we refer to [17]. Let us explain what is Green’s function for (5.1). We define it first in a ”regular” domain Ω ⋐ M, where regular means that the Dirichlet problem for A-harmonic equation is solvable with continuous boundary data. For this notion, see [16]. We need a concept of A-capacity. Let C ⊂ Ω be compact, and assume for simplicity that Ω \ C is regular. Thus there exists a unique A-harmonic function in Ω \ C with continuous boundary values u = 0 on ∂Ω and u = 1 in C. Call u the A-potential of (Ω, C). We define Z capA (Ω, C) = hAx (∇u), ∇ui. Ω

Then

capA (Ω, C) ≈ capp (Ω, C)


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and furthermore, (5.2)

capA (Ω1 , C1 ) ≥ capA (Ω2 , C2 )

if C2 ⊂ C1 and/or Ω1 ⊂ Ω2 . Note that this property is obvious for variational capacities but capA is not necessary a variational capacity. The definition of Green’s function, and in particular its uniqueness when p = n, relies on the following observation. Lemma 5.5. Let Ω ⋐ M be a domain and let C ⊂ Ω be compact such that Ω \ C is regular. Let u be the A-potential of (Ω, C). Then, for every 0 ≤ a < b ≤ 1, capA ({u > a}, {u ≥ b}) =

capA (Ω, C) . (b − a)p−1

Definition 5.6. Suppose that Ω ⋐ M is a regular domain and let y ∈ Ω. A function g = g(·, y) is called a Green’s function for (5.1) in Ω if 1. g is positive and A-harmonic in Ω \ {y}, 2. limx→z g(x) = 0 for all z ∈ ∂Ω, 3. lim g(x) = capA (Ω, {y})1/(1−p) , x→y

which we interpret to mean limx→y g(x) = ∞ if p ≤ n, 4. for all 0 ≤ a < b < capA (Ω, {y})1/(1−p) , capA ({g > a}, {g ≥ b}) = (b − a)1/(1−p) .

Theorem 5.7. Let Ω ⋐ M be a regular domain and y ∈ Ω. Then there exists a Green’s function for (5.1) in Ω. Furthermore, it is unique at least if p ≥ n. Monotonicity properties (5.2) of A-capacity and the so-called Loewner property, i.e. capn C > 0 if C is a continuum, are crucial in proving the uniqueness when p = n. Indeed, we can show that on sufficiently small spheres S(y, r) ¯ r))1/(1−n) | ≤ c, x ∈ S(y, r), |g(x, y) − capA (Ω, B(y,

which then easily implies the uniqueness.

Next take an exhaustion of M by regular domains Ωi ⊂ Ωi+1 ⋐ M, M = ∪i Ωi . We can construct an increasing sequence of Green’s functions gi (·, y) on Ωi . Then the limit is either identically +∞ or g(·, y) := lim gi (·, y) i→∞

is a positive A-harmonic function on M \ {y}. In the latter case we call the limit function g(·, y) a Green’s function for (5.1) on M. We have the following list of Liouville-type properties of M (which may or may not hold for M ): (1) M is p-parabolic. (2) Every non-negative A-harmonic function on M is constant for every A of type p. (Strong p-Liouville.)

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(3) Every bounded A-harmonic function on M is constant for every A of type p. (p-Liouville.) (4) Every A-harmonic function u on M with ∇u ∈ Lp (M ) is constant for every A of type p. (Dp -Liouville.)

We refer to [17] for the proof of the following general result, and to [18] and [25] for studies concerning the converse directions. Theorem 5.8. (1) ⇒ (2) ⇒ (3) ⇒ (4). Next we discuss the close connection between the volume growth and pparabolicity. Suppose that M is complete. Fix a point o ∈ M and write V (t) = Vol B(o, t) . Theorem 5.9. Let 1 < p < ∞ and suppose that 1/(p−1) Z ∞ t dt = ∞, V (t) or Z ∞ dt = ∞. ′ V (t)1/(p−1) Then M is p-parabolic.

Proof. One can either construct a test function involving the integrals above, or use a p-modulus estimate for separating (spherical) rings. More precisely, write B(t) = B(o, t) and S(t) = S(o, t) = ∂B(o, t). For R > r > 0 and integers k ≥ 1, we write ti = r + i(R − r)/k, i = 0, 1, . . . , k. Then, by a well-known property of modulus, Mp

k−1 1/(1−p) X ¯ i+1 ) 1/(1−p) ; ¯ Mp Γ S(ti ), S(ti+1 ); B(t ≥ Γ S(r), S(R); B(R) i=0

¯ see e.g. [41, II.1.5]. Here Γ S(r), S(R); B(R) is the family of all paths joining ¯ S(r) and S(R) in B(R). For each i = 0, . . . , k − 1 we have an estimate ¯ i+1 ) ≤ (V (ti+1 ) − V (ti )) (ti+1 − ti )−p . Mp Γ S(ti ), S(ti+1 ); B(t

Hence (5.3)

k−1 1/(1−p) X ¯ ≥ Mp Γ S(r), S(R); B(R) i=0

V (ti+1 ) − V (ti ) ti+1 − ti

1/(1−p)

Thus the right-hand side of (5.3) tends to the integral Z R dt ′ 1/(p−1) r V (t) as k → ∞. We obtain an estimate Z R 1−p dt ¯ Mp Γ S(r), S(R); B(R) ≤ . ′ 1/(p−1) r V (t)

(ti+1 − ti ).


In particular, if

Z

135

∞

dt =∞ V ′ (t)1/(p−1) r for some r > 0, then M is p-parabolic. The converse is not true in general. That is, M can be p-parabolic even if 1/(p−1) Z ∞ t dt < ∞ V (t) or Z ∞ dt < ∞; V ′ (t)1/(p−1) see [44]. It is interesting to study when the converse is true. We refer to [19] for the proofs of the following two theorems. Theorem 5.10. Suppose that M is complete and admits a global doubling property and global (1, p)-Poincaré inequality for 1 < p < ∞. Then 1/(p−1) Z ∞ t dt < ∞. (5.4) M is p-hyperbolic if and only if V (t) In some cases, we can estimate Green’s functions: Theorem 5.11. Suppose that M is complete and has non-negative Ricci curvature everywhere. Let 1 < p < ∞. Then 1/(p−1) Z ∞ t dt < ∞. M is p-hyperbolic if and only if V (t) Furthermore, we have estimates for Green’s functions for (5.1) 1/(p−1) 1/(p−1) Z ∞ Z ∞ t t −1 c dt ≤ g(x, o) ≤ c dt V (t) V (t) 2r 2r

for every x ∈ ∂M (r), where M (r) is the union of all unbounded components of ¯ r). The constant c depends only on n, p, α, and β. M \ B(o, Theorem 5.10 follows also from the following sharper result; see [21]. Theorem 5.12. Suppose that M is complete and that there exists a geodesic ray γ : [0, ∞) → M such that for all t > 0, |B(γ(t), 2s)| ≤ c|B(γ(t), s)|,

whenever 0 < s ≤ t/4, and that Z

Bγ (t)



 |u − uBγ (t) |dm ≤ c 

Z

1/p

 |∇u|p dm

2Bγ (t)

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for all u ∈ C ∞ 2Bγ (t) , where Bγ (t) = B(γ(t), t/8). Then M is p-hyperbolic if 1/(p−1) Z ∞ t dt < ∞. |B(γ(t), t/4)| Theorem 5.12 can be applied to obtain the following. Theorem 5.13. Let M be a complete Riemannian n-manifold whose Ricci curvature is non-negative outside a compact set. Suppose that M has maximal volume growth ( V (t) ≈ rn ). Then M is p-parabolic if and only if p ≥ n. To our knowledge it is an open problem whether the equivalence (5.4) holds for a complete Riemannian n-manifold whose Ricci curvature is non-negative outside a compact set.

6. Liouville-type results for quasiregular mappings Here we give applications of the above results on n-parabolicity and various Liouville properties to the existence of non-constant quasiregular mappings between given Riemannian manifolds. Let us start with the Gromov-Zorich ”global homeomorphism theorem” that is a generalization of Zorich’s theorem we mentioned in the introduction; see [14], [48]. Theorem 6.1. Suppose that M is n-parabolic, n = dim M ≥ 3, and that N is simply connected. Let f : M → N be a locally homeomorphic quasiregular map. Then f is injective and f M is n-parabolic. Proof. We give here a very rough idea of the proof. First one observes that f M is n-parabolic (see Theorem 6.2 below), and so N \ f M is of n-capacity zero. Then one shows, again by using the n-parabolicity of M, that the set E of all asymptotic limits of f is of zero capacity. Consequently, E is of Hausdorff dimension zero. Recall that an asymptotic limit of f is a point y ∈ N such that f γ(t) → y as t → ∞ for some path γ ∈ Γ∞ in M. Removing E ∪ (N \ F M ) from N has no effect on the simply connectivity for dimensions n ≥ 3. That is, f M \ E remains simply connected. Thus one can extend uniquely any branch of local inverses of f and obtain a homeomorphism g : f M \ E → g(f M \ E) such that f ◦ g = id |(f M \ E). Finally, g can be extended to E to obtain the inverse of f. In [22] we generalized the global homeomorphism theorem for mappings of finite distortion under mild conditions on the distortion. See also [49] for a related result for locally quasiconformal mappings. Theorem 6.2. If N is n-hyperbolic and M is n-parabolic, then every quasiregular mapping f : M → N is constant.


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Proof. Suppose that f : M → N is a non-constant quasiregular mapping. Then f M ⊂ N is open. If f M 6= N, pick a point y ∈ ∂(N \ f M ) and let g = g(·, y) be the Green’s function on N for the n-Laplacian. Then g ◦ f is a nonconstant positive A-harmonic function on M which gives a contradiction with the n-parabolicity of M and Theorem 5.8. If f M = N, let u be a non-constant positive supersolution on N for the n-Laplacian. Then u ◦ f is a non-constant supersolution on M for some A of type n which is again a contradiction. Example 6.3. 1. If N is a Cartan-Hadamard manifold, with KN ≤ −a2 < 0, then every quasiregular mapping f : Rn → N is constant. 2. Let Hn be the Heisenberg group with a left-invariant Riemannian metric, then every quasiregular mapping f : R2n+1 → Hn is constant.

Theorem 6.4. Suppose that M is strong n-Liouville while N is not. Then every quasiregular map f : M → N is constant. Proof. If N is not strong n-Liouville, then it is n-hyperbolic by Theorem 5.8. Suppose that f : M → N is a non-constant quasiregular mapping. Then f M ⊂ N is open. If f M 6= N, choose a point y ∈ ∂(N \ f M ) and let g = g(·, y) be the Green’s function for n-Laplacian on N . Then g ◦ f is a non-constant positive A-harmonic function, with A of type n. This is a contradiction. If f M = N , we choose a non-constant positive n-harmonic function u on N and get a contradiction as above. Theorem 6.5. Let N be a Cartan-Hadamard n-manifold, with −b2 ≤ K ≤ −a2 < 0, and let M be a complete Riemannian n-manifold admitting a global doubling property and a global (1, n)-Poincaré inequality. Then every quasiregular mapping f : M → N is constant. Proof. By [20], N admits non-constant positive n-harmonic functions. Hence N is not strong n-Liouville. On the other hand, the assumptions on M imply that a global Harnack’s inequality for positive A-harmonic functions of type n holds on M. Thus M is strong n-Liouville, and the claim follows from Theorem 6.4. Theorem 6.6 (”One-point Picard”). Suppose that N is n-hyperbolic and M is strong n-Liouville. Then every quasiregular mapping f : M → N \ {y}, with y ∈ N, is constant. Proof. Suppose that f : M → N \ {y} is a non-constant quasiregular mapping. Then ∂(N \ f M ) 6= ∅. Choose a point z ∈ ∂(N \ f M ), and let g = g(·, z) be the Green’s function on N for the n-Laplacian. Then g ◦ f is a non-constant positive A-harmonic function for some A of type n leading to a contradiction.

7. Picard-type theorems The classical big Picard theorem states that a holomorphic mapping of the punctured unit disc {z ∈ C : 0 < |z| < 1} into the complex plane omitting two

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values has a meromorphic extension to the whole disc; see e.g. [1, Theorem 114]. In [40] Rickman proved a counterpart of Picard’s theorem for quasiregular mappings (Theorem 7.1) and its local version (Theorem 7.2) corresponding to the big Picard theorem. Theorem 7.1 ([40]). For each integer n ≥ 2 and each K ≥ 1 there exists a positive integer q = q(n, K) such that if f : Rn → Rn \ {a1 , . . . , aq } is Kquasiregular and a1 , . . . , aq are distinct points in Rn , then f is constant. Theorem 7.2 ([40]). Let G = {x ∈ Rn : |x| > s} and let f : G → Rn \ {a1 , . . . , aq } be a K-quasiregular mapping, where a1 , . . . , aq are distinct points in Rn and q = q(n, K) is the integer in Theorem 7.1. Then the limit lim|x|→∞ f (x) exists. In this section we consider corollaries and extensions of the big Picard theorem for quasiregular mappings. Although the short argument yielding Theorem 7.1 from Theorem 7.2 is wellknown, it seems that the following corollary employing the same argument has gone unnoticed in the literature. Corollary 7.3. Let K ≥ 1 and R > 0. Let f : Rn → Rn be a continuous mapping omitting at least q = q(n, K) points, where q(n, K) is as in Theorem 7.1. Then at least one of the following conditions fails: ¯ n (R) is K-quasiregular, (i) f |Rn \ B (ii) f B n (r) is open for some r > R. Proof. Suppose towards a contradiction that both conditions (i) and (ii) hold. By Theorem 7.2, the mapping f has a limit at the infinity. Hence we may ¯n → R ¯ n . Moreover, f is K-quasiregular extend f to a continuous mapping R n n ¯ \B ¯ (R). By composing f with a Möbius mapping if necessary, we may in R assume that f (∞) = ∞. Since f is a non-constant quasiregular mapping on ¯n \B ¯n \B ¯ n is open in R ¯ n , by (ii). ¯ n (R), f |R ¯ n (R) is an open mapping. Hence f R R ¯ n is both open and closed, f R ¯n = R ¯ n and f Rn = Rn . This contradicts Since f R the assumption that f omits q points. The claim follows. In [23] the authors consider quasiregular mappings of the punctured unit ball into a Riemannian manifold N . We say that N has at least q ends, if there exists a compact set C ⊂ N such that N \ C has at least q components which are not relatively compact. Such a component of N \C is called an end of M with respect to C. Let E be the set of ends of N , that is, E ∈ E is an end of N with respect to some compact set C ⊂ N . We compactify N with respect to its ends as follows. There is a natural partial order in E induced by inclusion. We call a maximal totally ordered subset of E an asymptotic end of N . The set of asymptotic ends ˆ = N ∪ ∂N . We endow N ˆ with a topology such of N is denoted by ∂N and N ˆ that the inclusion N ⊂ N is an embedding and for every e ∈ ∂N sets E ∈ e form a neighborhood basis at e. The main result is the following version of the big Picard theorem.


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Theorem 7.4 ([23, Theorem 1.3]). For every K ≥ 1 there exists q = q(K, n) such that every K-quasiregular mapping f : B n \{0} → N has a limit limx→0 f (x) ˆ if N has at least q ends. in N In the spirit of Corollary 7.3 we formulate the following consequence Theorem 7.4. Corollary 7.5. Given n ≥ 2 and K ≥ 1 there exists q˜ = q˜(n, K) such that the following holds. Suppose that M is compact, {z1 , . . . , zk } ⊂ M , where 1 ≤ k < q˜, and that N has at least q˜ ends. Let f : M \ {z1 , . . . , zk } → N be a continuous mapping and let Ωi be a neighborhood of zi for every 1 ≤ i ≤ k. Then at least one of the following conditions fails: (i) f is K-quasiregular in Ωi \ {zi } for every i, (ii) there exists a neighborhood Ω of M \ (Ω1 ∪ · · · ∪ Ωk ) such that f Ω is open. Proof. Suppose that both conditions are satisfied. For every 1 ≤ i ≤ k we fix a 2-bilipschitz chart ϕi : Ui → ϕi Ui at zi . We may assume that Ui ⊂ Ωi . n Every mapping f ◦ ϕ−1 i |ϕi (Ui \ zi ) is 2 K-quasiregular, and therefore it has a limit at ϕi (zi ) by Theorem 7.4 if N has at least q(n, 2n K) ends. Hence f has ˆ. a limit at every point zi . We extend f to a continuous mapping fˆ: M → N Denote M ′ = M \ {z1 , . . . , zk }. Since f is an open mapping, ∂f M ′ ∩ f M ′ = ∅. Furthermore, since M is compact, f M ′ ⊂ fˆM = fˆM.

Hence ∂f M ′ ⊂ fˆM \ f M ′ . Thus card(∂f M ′ ) ≤ card(fˆM \ f M ′ ) ≤ k and ˆ = N = f M ′ = fˆM. N This is a contradiction, since ˆ \ f M ′ ) = card(fˆM \ f M ′ ). card(fˆM \ f M ′ ) ≤ k < q ≤ card(N In [26] Holopainen and Rickman applied a method of Lewis ([33]) that relies on Harnack’s inequality to prove the following general version of Picard’s theorem on the number of omitted values of a quasiregular mapping. We say that a complete Riemannian n-manifold M belongs to the class M(m, ϑ), where m : (0, 1) → N and ϑ : (0, ∞) → (0, ∞) are given functions, if following two conditions hold: (m) for each 0 < λ < 1 every ball of radius r in M can contain at most m(λ) disjoint balls of radius λr, and (ϑ) M admits a global Harnack’s inequality for non-negative A-harmonic functions of type n with Harnack-constant ϑ(β/α), where α and β are the constants of A.

Theorem 7.6 ([26]). Given n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0, ∞) → (0, ∞) there exists q = q(n, K, m, ϑ) ≥ 2 such that the following holds. Suppose that M belongs to the class M(m, ϑ) and that N has at least q ends. Then every K-quasiregular mapping f : M → N is constant.

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Next we show that this theorem admits a local version. Suppose that M is complete. We say that an asymptotic end e of M is of type E(m, ϑ) if there exists E ∈ e such that (Em) for each 0 < λ < 1 every ball of radius r in E can contain at most m(λ) disjoint balls of radius λr, and (Eϑ) E admits a uniform Harnack inequality for non-negative A-harmonic functions of E of type n for balls B ⊂ E satisfying 4B ⊂ E. We also assume that the Harnack constant ϑ depends only on β/α, where α and β are the constants of A. We also say that an asymptotic end e of M is p-parabolic (with p ≥ 1) if there exists E ∈ e such that for every ε > 0 there exists E ′ ∈ e such that Mp (Γ(E ′ , M \ E; M )) < ε.

Furthermore, we say that an asymptotic end e of M is locally C-quasiconvex if for every E ∈ e there exists E ′ ∈ e, E ′ ⊂ E, such that each pair of points x, y ∈ E ′ can be joint by a path in E ′ of length at most Cd(x, y), where d is the Riemannian distance of M . Theorem 7.7. Let n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0, ∞) → (0, ∞). Then there exists q = q(n, K, m, ϑ) such that the following holds. Suppose that M is complete and e is an n-parabolic locally C-quasiconvex asymptotic end of M of type E(m, ϑ), and that N has at least q ends. Let E ∈ e and f : E → N be a K-quasiregular mapping. Then f has a limit at e. Corollary 7.8. Let n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0, ∞) → (0, ∞). Then there exists q = q(n, K, m, ϑ) such that the following holds. Suppose that a complete Riemannian n-manifold M has asymptotic ends {e1 , . . . , ek }, k < q, of type E(m, ϑ) which are all n-parabolic and locally C-quasiconvex, and that N has at least q ends. Let f : M → N be a continuous mapping and Ei ∈ ei for every 1 ≤ i ≤ k. Then at least one of the following conditions fails: (i) f is K-quasiregular in Ei for every i, (ii) there exists a neighborhood Ω of M \ (E1 ∪ · · · ∪ Ek ) such that f Ω is open. Proof. Suppose that both conditions hold. By Theorem 7.7, we may extend f ˆ →N ˆ . Since M ˆ is compact, we may follow the to a continuous mapping fˆ: M proof of Corollary 7.5. We need several lemmas in order to prove Theorem 7.7. Let us first recall the definition of a Harnack function. Let M be a Riemannian manifold. A continuous function u : M → R is called a Harnack function with constant θ if M (h, x, r) := sup h ≤ θ inf h B(x,r)

B(x,r)

holds in each ball B(x, r) whenever the function h is nonnegative in B(x, 2r), ¯ 2r) ⊂ M is compact. The has the form h = ±u + a for some a ∈ R, and B(x, original version of Lewis’ lemma is stated for Harnack functions. It is well known


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(see [16, 6.2]) that A-harmonic functions in the Euclidean setting are Harnack functions with some θ depending only on n and on the constants p, α, and β of A. In that case θ is called the Harnack constant of A. Lemma 7.9. Let e be an n-parabolic locally C-quasiconvex asymptotic end of a complete Riemannian n-manifold M . Suppose u : E → R, where E ∈ e, is a Harnack function with constant θ such that lim supx→e u(x) = ∞ and lim inf x→e u(x) < 0. Then for every C0 > 0 there exists a ball B = B(x0 , r0 ) ⊂ E such that (1) B(x0 , 100Cr0 ) ⊂ E, (2) u(x0 ) = 0, and (3) maxB u ≥ C0 . Proof. It is sufficient to modify the proof of [23, Lemma 2.1] as follows. Let E ′ ∈ e be such that E ′ ⊂ E and E ′ is C-quasiconvex. Let F ′ ⊂ M be a compact set such that E ′ is a component of M \ F ′ , fix o ∈ M , and let R0 > 0 be such ¯ R0 /2). that F ′ ⊂ B(o, Fix k ∈ N such that given r > R0 and x, y ∈ ∂B(o, r) ∩ E ′ there exists k balls Bi = B(xi , r/1000), 1 ≤ i ≤ k, in E such that (1) (2) (3) (4)

x ∈ B1 , y ∈ Bk , xi ∈ E ′ for every i, and Bi ∩ Bi+1 6= ∅ for every i ∈ {1, . . . , k − 1}.

Indeed, since Bi ∩B(o, R0 /2) = ∅, we have Bi ⊂ E, and since E ′ is C-quasiconvex, we may choose any k > 2000C. We may now apply the proof of [23, Lemma 5] almost verbatim. Lemma 7.10. Let e be an n-parabolic locally C-quasiconvex asymptotic end of a complete Riemannian n-manifold M and E ∈ e. If f : E → N is a quasiregular ˆ at e. mapping such that f E is n-hyperbolic, then f has a limit in N Proof. Suppose that f E is n-hyperbolic. We may assume that E is C-quasiconvex. If f has no limit at e, there exists a compact set F ⊂ N such that f E ′ ∩ F 6= ∅ for every E ′ ∈ e. Hence there exists a sequence (xk ) such that xk → e and f (xk ) → z ∈ N as k → ∞. Let (yk ) be another sequence such that yk → e as k → ∞. We show that the hyperbolicity of f E yields f (yk ) → z as k → ∞, which is a contradiction. For every k we fix a path αk : [0, 1] → E such that αk (0) = xk , αk (1) = yk , and ℓ(αk ) ≤ Cd(xk , yk ). Then capn (E, |αk |) → 0

as k → ∞. By Poletsky’s inequality (4.3) and (4.1),

capn (f E, f |αk |) ≤ KI (f ) capn (E, |αk |)

for every k. Suppose that f (yk ) 6→ z. Then, by passing to a subsequence if necessary, we may assume that d(f (yk ), z) ≥ δ > 0 for every k. Since

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d(f (αk (0)), f (αk (1))) ≥ δ/2 for large k, we have, by the n-hyperbolicity of f E, that capn (f E, f |αk |) ≥ ε > 0. for every k. This is a contradiction. The following lemma is a reformulation of [29, Lemma 19.3.2]. Lemma 7.11. Let E ⊂ M , let u : E → R be a non-constant Harnack function with constant θ, and let α : [a, b] → E be a path. If ℓ(α) ≤ k dist(|α|, u−1 (0) ∪ M \ E), then u has a constant sign on |α|. Furthermore, max u ≤ θk min u |α|

if u is positive on |α|, and

|α|

max u ≤ θ−k min u |α|

|α|

if u is negative on |α|. Proof. Since |α| is connected, every non-vanishing function on |α| has constant sign. We may assume without loss of generality that u is positive on |α|. Let a = a0 < a1 < . . . < ak = b be a partition of [a, b] such that ℓ(α|[ai , ai+1 ]) = ℓ(α)/k for every i = 0, 1, . . . , k − 1. For every i fix xi ∈ α([ai , ai+1 ]) such that ¯ i , ℓ(α)/(2k)). Furtherℓ(α|[ai , xi ]) = ℓ(α|[xi , ai+1 ]). Then α([ai , ai+1 ]) ⊂ B(x −1 more, B(xi , ℓ(α)/k) ⊂ E and B(xi , ℓ(α)/k) ∩ u (0) = ∅. Since α(ai+1 ) ∈ ¯ i , ℓ(α)/(2k)) ∩ B(x ¯ i+1 , ℓ(α)/(2k)) for every i = 1, . . . , k − 1, a repeated use B(x of Harnack’s inequality yields max|α| u ≤ θk min|α| u. Lemma 7.12 (Lewis’ lemma). Let M , e, E, and u be as in Theorem 7.7. Then for every C0 > 0 there exists a ball B = B(x0 , r0 ) ⊂ E such that (1) 6B ⊂ E, (2) u(x0 ) = 0, and (3) C0 ≤ max6B u ≤ θ6 maxB u. Proof. Let C0 > 0 and B(x0 , R) be as in Lemma 7.9. Let Z = u−1 (0) and ¯ 0 , 41R). For each x ∈ ZR we set rx = R − d(x, x0 )/41 and ZR = Z ∩ B(x S ¯x is compact and x 7→ maxB¯ u is continuous. Bx = B(x, rx ). Then F = x∈ZR B x Let a ∈ ZR be a point of maximum for this function. Thus max u ≥ max u ≥ C0 .

¯ B(a,r a)

¯ 0 ,R) B(x

As in [29, Lemma 19.4.1], we have that ¯ 6ra ) \ F ) ≥ 5ra . dist(Z, B(a, 6 ¯ Let y0 ∈ B(a, 6ra ) be such that u(y0 ) = max u ≥ C0 > 0. ¯ B(a,6r a)


143

If y0 ∈ F , then, by the maximal property of ball B(a, ra ),

max u = u(y0 ) ≤ max u = max u ≤ θ6 max u.

¯ B(a,6r a)

F

¯ B(a,r a)

¯ B(a,r a)

¯ 6ra ) be nearest to y0 in length metric. As B(a, ¯ ra ) ⊂ F If y0 6∈ F , let y1 ∈ F ∩B(a, it follows that ¯ ra )) ≤ 6ra − ra = 5ra . d(y0 , y1 ) ≤ dist(y0 , B(a, Let α : [0, 1] → E be a path of minimal length such that α(0) = y0 and α(1) = y1 . Then α[0, 1) ∩ F = ∅. Hence 5ra . 6 Thus ℓ(α) ≤ Cd(y0 , y1 ) ≤ 6C dist(Z, |α|). By Lemma 7.11, dist(Z, |α|) ≥

u(y0 ) ≤ max u ≤ θ6C min u ≤ θ6C u(y1 ) ≤ θ6C max u = θ6C max u. |α|

F

|α|

¯ B(a,r a)

Lemma 7.13 ([24],[26]). Let N be an n-parabolic Riemannian manifold. Suppose that C ⊂ N is compact such that N has q ends V1 , . . . , Vq with respect to C. Then there exist n-harmonic functions vj , j = 2, . . . , q, and a positive constant κ such that |vj | ≤ κ in C, |vj − vi | ≤ 2κ in V1 , sup vj = ∞,

(7.1) (7.2) (7.3)

V1

(7.4)

inf vj = −∞,

(7.5) (7.6) (7.7)

vj is bounded in Vk for k 6= 1, j, if vj (x) > κ, then x ∈ V1 , if vj (x) < −κ then x ∈ Vj .

Vj

Proof of Theorem 7.7. Suppose that a K-quasiregular mapping f : E → N has no limit at e. By Lemma 7.10, N is n-parabolic. Let C ⊂ N be a compact set such that N has q ends V1 , . . . , Vq with respect to C. For every j = 2, . . . , q let us fix an n-harmonic function vj with properties (7.1) - (7.7) given in Lemma 7.13. For every j = 2, . . . , q we set uj = vj ◦f . Then functions uj are A-harmonic in E. Next we show that (7.8)

lim sup uj (x) = +∞ and lim inf uj (x) = −∞, x→e

x→e

and hence they satisfy the assumptions of Lemma 7.9. This can be seen by observing that the sets {x ∈ N : vj (x) > c} and {x ∈ N : vj (x) < −c} are nonempty and open for every c > 0 and j = 2, . . . , q. By Lemma 7.10, f (E \ F ) intersects these sets for every compact F ⊂ M , and therefore (7.8) follows. By

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Lemma 7.12 there are sequences xi ∈ E and ri ∈ (0, ∞), i ∈ N, such that u2 (xi ) = 0, B(xi , 3ri ) ⊂ E, M (u2 , xi , 3ri ) ≤ θ6 M (u2 , xi , ri /2),

and M (u2 , xi , ri /2) → ∞ as i → ∞. Let us fix an index i such that M (u2 , xi , ri /2) ≥ 4θκ, where θ > 1 is the Harnack constant of A and κ is the constant in Lemma 7.13. We write x = xi and r = ri . By (7.6), f B(x, r/2) ∩ V1 6= ∅. Thus, by (7.2), we have (7.9)

M (u2 , x, s) − 2κ ≤ M (uj , x, s) ≤ M (u2 , x, s) + 2κ

whenever s ≥ r/2. Next we conclude by using Harnack’s inequality that (7.10)

M (uj , x, r) ≤ (θ − 1)M (−uj , x, 2r)

for all j. Let us first show that uj (z) = 0 for some z ∈ B(x, r). Suppose on the contrary, that uj > 0 in B(x, r). Then uj (y) ≤ θuj (x) for all y ∈ B(x, r/2) by Harnack’s inequality. Since M (u2 , x, r/2) ≥ 4θκ, there exists y ∈ B(x, r/2) such that uj (y) > 2θκ by (7.9). Thus uj (x) > 2κ, and so x ∈ V1 . By (7.2), u2 (x) ≥ uj (x) − 2κ > 0 contradicting the assumption u2 (x) = 0. Therefore there exists z ∈ B(x, r) such that uj (z) = 0. Thus inf B(x,r) uj ≤ 0. Inequality (7.10) follows now from the calculation M (uj , x, r) = sup uj = sup uj − inf uj + inf uj B(x,r)

≤ θ inf

B(x,r)

B(x,2r)

B(x,r)

uj − inf uj B(x,2r)

B(x,2r)

+ inf uj B(x,2r)

= θ inf uj + (1 − θ) inf uj B(x,r)

B(x,2r)

≤ −(θ − 1) inf uj = (θ − 1) sup (−uj ) B(x,2r)

B(x,2r)

= (θ − 1)M (−uj , x, 2r),

since uj − inf B(x,2r) uj ≥ 0 in B(x, 2r).

Inequalities (7.9) and (7.10), and the assumption M (u2 , x, r/2) ≥ 4θκ together yield the inequality (7.11) Indeed,

M (u2 , x, r) ≤ θM (−uj , x, 2r).

M (u2 , x, r) ≤ M (uj , x, r) + θ−1 M (u2 , x, r) ≤ (θ − 1)M (−uj , x, 2r) + θ−1 M (u2 , x, r), ¯ 2r) such that which is equivalent to (7.11). We fix zj ∈ B(x, (7.12)

M (−uj , x, 2r) = −uj (zj ).

The well-known oscillation estimate (see e.g. [16, 6.6]) osc uj ≤ c(ρ/r)γ osc uj

B(y,ρ)

B(y,r)


145

together with [24, Lemma 4.2] and (7.9) imply that osc uj ≤ c1 (ρ/r)γ M (u2 , x, 3r)

(7.13)

B(zj ,ρ)

for ρ ∈ (0, r). See [24, (5.5)] for details. Thus max uj =

¯ j ,ρ) B(z

osc uj + min uj

B(zj ,ρ)

¯ j ,ρ) B(z

≤ c1 (ρ/r)γ M (u2 , x, 3r) + uj (zj ) ≤ c1 (ρ/r)γ M (u2 , x, 3r) − θ−1 M (u2 , x, r),

by (7.13), (7.12), and (7.11). Since M (u2 , x, 3r) ≤ θ6 M (u2 , x, r), we obtain c1 (ρ/r)γ M (u2 , x, 3r) ≤ (2θ)−1 M (u2 , x, r)

by choosing ρ = (2θ7 c1 )−1/γ r. Hence

max uj ≤ −(2θ)−1 M (u2 , x, r) ≤ −2κ.

¯ j ,ρ) B(z

By (7.7), we conclude that f B(zj , ρ) ⊂ Vj and hence the balls B(zj , ρ) are disjoint. Since B(zj , ρ) ⊂ B(x, 3r), there can be at most m(ρ/3r) of them. Hence q has an upper bound that depends only on n, K, ϑ, and m.

8. Quasiregular mappings, p-harmonic forms, and de Rham cohomology The use of n-harmonic functions in studying Liouville-type theorems for quasiregular mappings f : M → N is restricted to the case, where N is non-compact. The reason for this restriction is simple: a compact Riemannian manifold does not carry non-constant p-harmonic functions. Therefore, in the case of a compact target manifold, we have to use p-harmonic forms. In this final section we discuss briefly p-harmonic and A-harmonic forms and their connections to quasiregular mappings. For detailed discussions on A-harmonic forms, see e.g. [27], [28], [29], [30], and [42]. For the connection of A-harmonic forms to quasiregular mappings, see e.g. [4], [29], and [34]. Riemannian metric of M induces an inner product to the exterior bundle Vℓ The ∗ T M for every ℓ ∈ {1, . . . , n}, see e.g. [29, 9.6] for details. We denote this inner product by h·, ·i and the corresponding norm by | · |. As usual, sections of V the bundle ℓ T ∗ M are called ℓ-forms. The Lp -space of measurable ℓ-forms is V denoted by Lp ( ℓ M ) and the Lp -norm is defined by Z 1/p p kξkp = |ξ| dx . M

V The local Lp -spaces of ℓ-forms are denoted by Lploc ( ℓ M ). The space of C ∞ V smooth ℓ-forms on M is denoted by C ∞ ( ℓ M ), and the space of compactly V supported C ∞ -smooth ℓ-forms by C0∞ ( ℓ M ).

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V V Let ℓ ∈ {1, . . . , n−1} and p > 1. Let A : ℓ T ∗ M → ℓ T ∗ M be a measurable bundle map such that there exists positive constants a and b satisfying

hA(ξ) − A(ζ), ξ − ζi ≥ a(|ξ| + |ζ|)p−2 |ξ − ζ|2 , |A(ξ) − A(ζ)| ≤ b(|ξ| + |ζ|)p−2 |ξ − ζ|, and A(tξ) = t|t|p−2 A(ξ) V for all ξ, ζ ∈ ℓ Tx∗ M , t ∈ R, and for almost every x ∈ M . We also assume that V x 7→ Ax (ω) is a measurable ℓ-form for every measurable ℓ-form ω : M → ℓ T ∗ M . We say that an ℓ-form ξ is A-harmonic (of type p) on M if ξ is a weakly closed d,p Vℓ continuous form in Wloc ( M ) and satisfies equality (8.1) (8.2) (8.3)

δ(A(ξ)) = 0

weakly, that is,

Z

hA(ξ), dϕi = 0 V d,p Vℓ for all ϕ ∈ C0∞ ( ℓ−1 M ). Here Wloc ( M ) is the partial Sobolev space of p Vℓ d,p Vℓ ℓ-forms. A form ω ∈ Lloc ( M ) is in the space Wloc ( M ) if the distribup Vℓ+1 tional exterior derivative dω exists and dω ∈ Lloc ( M ). The global space Vℓ d,p Vℓ d,p W ( M ) is defined similarly. A form ω ∈ Wloc ( M ) is weakly closed if d,p Vℓ−1 dω = 0 and weakly exact if ω = dτ for some τ ∈ Wloc ( M ). Apart from minor differences between conditions (8.1)-(8.3) and the corresponding conditions in Section 2, we can say that A-harmonic functions correspond to A-harmonic weakly exact 1-forms. Let f : M → N be a quasiregular mapping. Since f is almost everywhere n/ℓ V differentiable, we may define the pull-back f ∗ ξ of the form ξ ∈ Lloc ( ℓ N ) by M

(f ∗ ξ)x = (Tx f )∗ ξf (x) . n/ℓ V By the quasiregularity of f , f ∗ ξ ∈ Lloc ( ℓ M ). Furthermore, d(f ∗ ξ) = f ∗ (dξ) 1,n/ℓ V 1,n/ℓ V 1,n/ℓ V if ξ ∈ Wloc ( ℓ N ). Hence f ∗ ξ ∈ Wloc ( ℓ M ) for ξ ∈ Wloc ( ℓ M ). The quasiregularity of f also yields that the pull-back f ∗ ξ of an (n/ℓ)-harmonic ℓ-form V is A-harmonic. Similarly to the case of A-harmonic functions, A : ℓ T ∗ M → Vℓ ∗ T M is defined by A(η) = hG∗ η, ηi(n/ℓ)−2 G∗ η, where T Gx = Jf (x)2/n (Tx f )−1 (Tx f )−1 a.e. .

Recently in [4] Bonk and Heinonen studied cohomology of quasiregularly elliptic manifolds using p-harmonic forms. A connected Riemannian manifold is called K-quasiregularly elliptic if it receives a non-constant K-quasiregular mapping from Rn . The main result of [4] is the following theorem.

Theorem 8.1 ([4, Theorem 1.1]). Given n ≥ 2 and K ≥ 1 there exists a constant C = C(n, K) > 1 such that dim H ∗ (N ) ≤ C for every K-quasiregularly elliptic closed n-manifold N .


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As the Picard-type theorem 7.6, also this theorem has a local counterpart. Theorem 8.2 ([34, Theorem 2]). Given n ≥ 2 and K ≥ 1 there exists a constant C ′ = C ′ (n, K) > 1 such that every K-quasiregular mapping f : B n \{0} → N has a limit at origin if N is closed, connected, and oriented Riemannian n-manifold with dim H ∗ (N ) ≥ C ′ . We close this section with a sketch of the proof of Theorem 8.2. The following theorem on exact A-harmonic forms is essential in the proof. For details, see [34]. Theorem 8.3. Let n ≥ 3 and let η be a weakly exact A-harmonic ℓ-form, ℓ ∈ ¯ n such that {2, . . . , n − 1}, on Rn \ B Z (8.4) |η|n/ℓ = ∞. ¯ n (2) Rn \B

Then there exists γ = γ(n, a, b) > 0 such that Z 1 |η|n/ℓ > 0. (8.5) lim inf γ r→∞ r ¯ n (2) B n (r)\B Here a and b are as in (8.1) and (8.2). Sketch of the proof of Theorem 8.2. Let us first consider some exceptions. For Riemannian surfaces the result is classical and follows from the uniformization theorem and the measurable Riemann mapping theorem, see [34, Theorem 3]. For n ≥ 3 we may give a bound for the first cohomology using a well-known result of Varopoulos on the fundamental group and n-hyperbolicity. For details, see [34, Theorem 4]. Hence we may restrict our discussion to dimensions n ≥ 3 and to cohomology dimensions ℓ ≥ 2. Let n ≥ 3 and 2 ≤ ℓ ≤ n − 1, and suppose that f : B n \ {0} → N does not have a limit at the origin. Without changing the notation we precompose f with ¯ n ) = B n \ {0}. Let us a sense-preserving Möbius mapping σ such that σ(Rn \ B ℓ now show that dim H (N ) is bounded from above by a constant depending only on n and K. We fix p-harmonic ℓ-forms ξi generating H ℓ (N ), with p = n/ℓ. This can be done by a result of Scott [42]. Furthermore, we may assume that forms ξi are uniformly separated and uniformly bounded in Lp , that is, kξi − ξj kp ≥ 1 and kξi kp = 1 for every i and j.

A local version [34, Theorem 6] of the value distribution result of Mattila and Rickman yields that Z Z ∗ n/ℓ (8.6) |f ξ| ∼ Jf B n (r)\B n (2)

B n (r)\B n (2)

for large radii r. Using Theorem 8.3 and a decomposition technique due to Rickman, we find a radius R and a decomposition of the annulus B n (R) \ B n (2)

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into domains quasiconformally equivalent to B n in such a way that we have a ¯ n (2) with properties quasiregular embedding ψ : B n → Rn \ B 1/4 Z Z (8.7) Jf & Jf ψB n (1/2)

and (8.8)

Z

ψB n

B n (R)\B n (2)

Jf .

Z

Jf .

B n (R)\B n (2)

Combining (8.6) with (8.7) and (8.8), we have that forms ϕ∗ f ∗ ξi are uniformly bounded in Lp (B n ) and uniformly separated in Lp (B n (1/2)). By compactness, the number of forms is bounded by a constant depending on data. Remark 8.4. The use of A-harmonic forms in the proof of Theorem 8.2 is very similar to their use in the proof of Theorem 8.1. Also Theorem 8.3 corresponds to a theorem of Bonk and Heinonen ([4, Theorem 1.11]).

References [1] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. [2] R. Bishop and R. Crittenden, Geometry of manifolds, Pure Appl. Math. 15, Academic Press, New York, 1964. [3] B, Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in Rn , Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257–324. [4] M. Bonk and J. Heinonen, Quasiregular mappings and cohomology, Acta Math. 186 (2001), 219–238. [5] M. Bonk and J. Heinonen, Smooth quasiregular mappings with branching, Publ. Math. ´ Inst. Hautes Etudes Sci. 100 (2004), 153–170. ´ [6] P. Buser, A note on the isoperimetric constant, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 213–230. [7] I. Chavel, Riemannian geometry: A modern introduction, Cambridge University Press, 1993. [8] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15–53. [9] T. Coulhon, I. Holopainen, and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic p-Laplacian with applications to Picard type theorems, Geom. Funct. Anal. 11 (2001), 1139– 1191. [10] A. Eremenko and J. L. Lewis, Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), 361–375. [11] I. Fonseca and W. Gangbo, Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications, Clarendon Press, Oxford, 1995. [12] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. [13] S. Granlund, Harnack’s inequality in the borderline case, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 159–163. [14] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser Boston Inc., Boston, 1999; Structures métriques pour les


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variétés riemanniennes. Lecture notes by P. Pansu et J. Lafontaine, Cedic Nathan, Paris, 1981. J. Heinonen, The branch set of a quasiregular mapping, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 691–700. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford - New York - Tokyo, 1993. I. Holopainen, Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 74 (1990), 1–45. I. Holopainen, Positive solutions of quasilinear elliptic equations on Riemannian manifolds, Proc. London Math. Soc. (3) 65 (1992), 651–672. I. Holopainen, Volume growth, Green’s function, and parabolicity of ends, Duke Math. J. 97 (1999), 319-346. I. Holopainen, Asymptotic Dirichlet problem for the p-Laplacian on Cartan-Hadamard manifolds, Proc. Amer. Math. Soc. 130 (2002), 3393–3400. I. Holopainen and P. Koskela, Volume growth and parabolicity, Proc. Amer. Math. Soc. 129 (2001), 3425–3435. I. Holopainen and P. Pankka, Mappings of finite distortion: Global homeomorphism theorem, Ann. Acad. Sci. Fenn. Math. 29 (2004), 59–80. I. Holopainen and P. Pankka, A big Picard type theorem for quasiregular mappings into manifolds with many ends, Proc. Amer. Math. Soc. 130 (2005), 1143-1150. I. Holopainen and S. Rickman, A Picard type theorem for quasiregular mappings of Rn into n-manifolds with many ends, Rev. Mat. Iberoamericana 8 (1992), 131-148. I. Holopainen and S. Rickman, Classification of Riemannian manifolds in nonlinear potential theory, Potential Anal. 2 (1993), 37–66. I. Holopainen and S. Rickman, Ricci curvature, Harnack functions, and Picard type theorems for quasiregular mappings in Analysis and topology, World Sci. Publishing, River Edge, NJ, 1998, 315–326. T. Iwaniec, p-harmonic tensors and quasiregular mappings, Ann. of Math. 136 (1992), 589–624. T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29–81. T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2001. T. Iwaniec, C. Scott, and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. 177 (1999), 37–115. D. Jerison, The Poincaré inequality for vector fields satisfying H¨ ormander’s condition, Duke Math. J. 53 (1986), 503–523. R. Kaufman, J. T. Tyson, and J.-M. Wu, Smooth quasiregular maps with branching in ´ Rn , Publ. Math. Inst. Hautes Etudes Sci. 101 (2005), 209–241. J. L. Lewis, Picard’s theorem and Rickman’s theorem by way of Harnack’s inequality, Proc. Amer. Math. Soc. 122 (1994), 199–206. P. Pankka, Quasiregular mappings from a punctured ball into compact manifolds, Conform. Geom. Dyn. (to appear). T. Rado and P. V. Reichelderfer, Continuous transformations in analysis, Die Grundlehren der mathematische Wissenschaften 75, Springer-Verlag, Berlin, 1955. Yu. G. Reshetnyak, Estimates of the modulus of continuity for certain mappings, Sibirsk. Mat. Z. 7 (1966), 1106–1114. Yu. G. Reshetnyak, Spatial mappings with bounded distortion, Sibirsk. Mat. Z. 8 (1967), 629–659.

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[38] Yu. G. Reshetnyak, Liouville’s conformal mapping theorem under minimal regularity hypotheses, Sibirsk. Mat. Z. 8 (1967), 835–840. [39] Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs 73, Amer. Math. Soc., Providence, RI, 1989. [40] S. Rickman, On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100-117. [41] S. Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26, Springer-Verlag, Berlin-Heidelberg-New York, 1993. [42] C. Scott, Lp theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995), 2075–2096. [43] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. [44] N. Varopoulos, Potential theory and diffusion on Riemannian manifolds in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. 2, Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983, 821–837. [45] J. Väis¨ alä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. 392 (1966), 1–10. [46] S. T. Yau, Isoperimetric constants and the first eigenvalue of a compact manifold, Ann. ´ Sci. Ecole Norm. Sup. 8 (1975), 159–171. [47] V. A. Zorich, The theorem of M. A. Lavrent’ev on quasiconformal mappings in space, Mat. Sb. 74 (1967) 417–433. [48] V.A. Zorich, Quasiconformal immersions of Riemannian manifolds and a Picard type theorem, Functional Analysis and Its Appl. 34 (2000), 188-196. [49] V.A. Zorich, Asymptotics of the admissible growth of the coefficient of quasiconformality at infinity and injectivity of immersions of Riemannian manifolds, Publ. Inst. Math. (Beograd) (N.S.) 75(89) (2004), 53–57. Ilkka Holopainen Address: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland E-mail: [email protected] Pekka Pankka Address: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland E-mail: [email protected]


Hyperbolic-type metrics Henri Lind´ en Abstract. The article is a status report on the contemporary research of hyperbolic-type metrics, and considers progress in the study of the classes of isometry- and bilipschitz mappings with respect to some of the presented metrics. Also, the Gromov hyperbolicity question is discussed. Keywords. Hyperbolic-type metric, intrinsic metric, isometry problem, bilipschitzmapping, Gromov hyperbolic space. 2000 MSC. Primary 30F45, Secondary 30C65,53C23.


151

2. The metrics

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3. Isometries and bilipschitz-mappings

157

4. Gromov hyperbolicity

160

References

163

1. Introduction In geometric function theory there are many different distance functions around, which — to a greater or lesser degree — resemble the classical hyperbolic metric. Some of these are defined by geometric means, some by implicit formulas, and many by integrating over certain weight functions. What all these metrics have in common, is that they are defined in some proper subdomain D ( Rn , and are strongly affected by the geometry of the domain boundary. Thus we should actually speak of families of metrics {dD }D(Rn , since the metric looks different in each domain, even though the defining formula might be the same. In the literature, however, one usually abuses notation and speaks only of “the metric d”, which we will do here also. The metrics typically have negative curvature, ie. the geodesics, if they exist, avoid the boundary. Most of the metrics described here also have an invariance property in the sense that (1.1)

dD (x, y) = df (D) (f (x), f (y)),

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for mappings f belonging to some fixed class, say similarities, Möbius transformations, or conformal mappings. Many of the metrics, especially those with simple explicit formulas, have been developed as tools for estimating other, more hard-to-handle metrics, such as the quasihyperbolic metric, which is probably the one most commonly used metric presented in this text. It has found applications in many branches of analysis, and is a very natural generalization of the classical hyperbolic metric to any domain D and dimension n ≥ 2. It has some flaws though, in most cases one cannot compute it, and actually very little is known about the metric itself. The difficulty of explicit computation is typical also for some other metrics, and for this reason we have a lot of “similar” metrics around, which in many cases are equivalent to each other; a handy feature, if one metric is suited for your study, but the other is not. Here we will try to give a survey on some of these metrics.

2. The metrics The classical starting point is the hyperbolic geometry developed by Poincaré and Lobachevsky in the early 19:th century. Poincaré used the unit ball as domain for his model, and Lobachevsky used the half space. These models turned out to be equivalent in the sense that Möbius transformations between them are isometries. 2.1. Definition. Let D ∈ {Hn , Bn }, and define a weight (or density) function w : D → R by 2 1 , for D = Hn and w(z) = w(z) = , for D = Bn . dist(z, ∂D) 1 − |z|2 Then the hyperbolic length ℓρ (γ) of a curve γ is defined by Z (2.2) ℓρ (γ) = ℓρ,D (γ) = w(z) |dz|, γ

where |dz| denotes the length element. After this, the hyperbolic distance ρD is defined for all x, y ∈ D by Z ρD (x, y) = inf ℓρ,D (γ) = inf (2.3) w(z) |dz|, γ∈Γxy

γ∈Γxy

γ

where Γxy is the family of all rectifiable curves joining x and y within D.

The above method to define metrics is frequently used. In fact, to get a completely new metric, the only thing that needs to be changed is the weight function. After that, the length and the new distance function are defined as in (2.2) and (2.3), respectively. The benefit of defining a metric d like this is that it will automatically be intrinsic, in other words, it will be its own inner metric ˆ This means that d. (2.4) dD (x, y) = dˆD (x, y) := inf ℓd,D (γ). γ∈Γxy

Hyperbolic-type metrics

153

2.5. Geodesics. When a metric is defined in the way described above, one might ask how to find the curve γ ∈ Γxy giving the desired infimum (which — if it is found — is in fact a minimum). In general, this can be far from trivial, even if such a curve exists. Curves minimizing the distance in this way are called geodesics or geodesic segments. Another way of characterizing a geodesic, is that it satisfies the triangle inequality with equality, ie. the curve γ ∈ Γxy is a geodesic, if for all u, v, w ∈ |γ| properly ordered, we have dD (u, w) = dD (u, v) + dD (v, w). We denote by JdD [x, y] the geodesic segment between x and y in (D, d). This segment may, however, not be unique, and no particular choice is made here. A metric space in which geodesic segments exist between any two given points, is called a geodesic metric space. If, in addition, the geodesic is unique, the space is totally geodesic. Naturally a geodesic metric is always intrinsic. 2.6. Hyperbolic metric in G. It is also possible to define the hyperbolic metric in a general simply connected subdomain G of the plane, since by the Riemann mapping theorem there exists a conformal mapping f : G → f G = B2 . Then the metric density is defined by ρG (z) = ρB2 (f (z))|f ′ (z)|. From the Schwarz lemma it follows that ρG is independent of the choice of f . We then define the hyperbolic metric hG by (2.3) using the density ρG . This definition automatically gives the hyperbolic metric the invariance property of (1.1) for the class of conformal mappings. Note, that while in the classical cases we use the traditional notation ρBn and ρHn for the hyperbolic metric, in general domains we use hG . Also, note that when the dimension n ≥ 3, every conformal mapping is a Möbius mapping, so it is not possible to extend the definition to general simply connected domains like above. In fact, for n ≥ 3 the hyperbolic metric is defined only in Bn and Hn . The hyperbolic metric is well understood, and the geodesic flow is known. In fact, in the classical models Bn and Hn the geodesics are known to be circular arcs orthogonal to the boundary, and in other domains the geodesics simply are induced by the conformal mapping. Moreover, for the classical cases there are explicit formulas to calculate the value of the hyperbolic metrics in terms of euclidean distances. For a comprehensive study on the classical cases, see the book by Beardon [Be1]. The hyperbolic metric in an arbitrary domain has been studied by F. Gehring, K. Hag and A. Beardon, see eg. the articles [Be3] and [GeHa1].

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Bn y*

H

n

y

x

y x x*

x*

y*

Figure 1: Hyperbolic geodesics in Bn and Hn . One way to calculate the hyperbolic distance, is to use the absolute cross-ratio defined by |a − c||b − d| , a, b, c, d ∈ Rn . |a, b, c, d| = |a − b||c − d| One can prove that, if C is the circle containing JρBn [x, y] or JρHn [x, y] and {x∗ , y ∗ } = C ∩ ∂Bn or {x∗ , y ∗ } = C ∩ ∂Hn in the same order as in Figure 1, then

(2.7)

ρBn (x, y) = log |x∗ , x, y, y ∗ | = ρHn (x, y).

Other explicit formulas have also been derived, see the book [Be1]. 2.8. The Apollonian metric. The formula in (2.7) makes one wonder whether a similar approach could be generalized to any domain D ( Rn . It turns out that this is very much possible; the Apollonian distance in a domain D is defined by |z − x| |w − y| (2.9) , αD (x, y) = sup log |z − y| |w − x| z,w∈∂D for all x, y ∈ D. This is a metric, unless the boundary is the subset of a circle or a line, in which case it is only a pseudo-metric, ie. the metric axiom d(x, y) = 0 ⇒ x = y need not hold.

Geometrically the Apollonian metric can be thought of in the following way: an Apollonian circle (or sphere, when n ≥ 3) with respect to the pair (x, y), is a set n |z − x| Bx,y,q = z ∈ R =q . |z − y| Then the Apollonian metric is αD (x, y) = log qx qy ,

where qx and qy are the ratios of the largest possible balls Bx,y,qx and By,x,qy still contained in D.


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D w

Bx x z

y

By Figure 2: The Apollonian balls approach. The Apollonian metric is invariant in Möbius mappings in the sense of (1.1). It is an easy exercise in geometry to show that in the case D = Hn the points z and w are actually the points x∗ and y ∗ in (2.7), and thus ρHn = αHn . The Apollonian metric has been studied in [GeHa2] and [Se], but especially by P. Hästö and Z. Ibragimov in a series of articles, see e.g. [Hä1],[Hä2],[HäIb] and [Ib]. The Apollonian metric is in a way a convenient construction with a clear geometric interpretation, but as a shortcoming it has its lack of geodesics. In the article [HäLi] some work is done to overcome this problem, by introducing the half-Apollonian metric, defined by |x − z| (2.10) ηD (x, y) = sup log , |y − z| z∈∂D for all x, y ∈ D. The geometric intuition here is the same as for the Apollonian metric, Indeed, instead of log qx qy we have ηD (x, y) = log max{qx , qy }.

This metric is a only similarity invariant, but instead it has more geodesics than the Apollonian metric. It is also bilipschitz equivalent to the Apollonian metric, in fact 1 αD (x, y) ≤ ηD (x, y) ≤ αD (x, y). 2 2.11. The quasihyperbolic metric. The quasihyperbolic metric is perhaps the most well-known and frequently used of the metrics considered here. It was developed by F. Gehring and his collaborators in the 70’s. It is defined by the method of 2.1 using 1 w(z) = , z∈D dist(z, ∂D) as weight function. It is immediate that for D = Hn the quasihyperbolic metric coincides with the hyperbolic metric ρHn . The quasihyperbolic metric is invariant under the class of similarity mappings.

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The quasihyperbolic metric is well-behaved in many senses: the weight function is quite simple and it is a natural generalization of the hyperbolic metric. Also, it is known to be geodesic for any domain D ( Rn [GeOs]. One of the shortcomings of the metric is that in general the geodesics are not easy to determine. Besides the half-space Hn , the geodesics are known in the punctured space Rn \ {z} and in the ball Bn , see [MaOs]. Recently the geodesics were determined also for the punctured ball Bn \ {0}, and planar angular domains see [Li1].

Sϕ = {(r, θ) | 0 < θ < ϕ}, 0 < ϕ < 2π,

2.12. Distance-ratio metrics. As the quasihyperbolic metric cannot be explicitly evaluated in the case of general domains, a typical way to overcome this problem is to approximate it by another metric, often one of the distance-ratio metrics or j-metrics. (Actually, by their construction also the Apollonian and half-Apollonian metrics could be described as “distance-ratio metrics”). There are two versions of these. The first, introduced by F. Gehring, is defined by |x − y| |x − y| e (2.13) jD (x, y) = log 1 + 1+ , x, y, ∈ D. dist(x, ∂D) dist(y, ∂D) The other one is defined by |x − y| . x, y, ∈ D. (2.14) jD (x, y) = log 1 + dist(x, ∂D) ∧ dist(y, ∂D) is a modification due to M. Vuorinen. The two metrics have much in common, but also important differences, which will be discussed further in Sections 2 and 3. Both are similarity invariant, and can be used to estimate the quasihyperbolic metric. The metrics satisfy the relation jD (x, y) ≤ e jD (x, y) ≤ 2 jD (x, y), x, y ∈ D. The lower bound for the quasihyperbolic metric is given by the inequality jD (x, y) ≤ kD (x, y)

proved in [GePa], which holds for points x, y in any proper subdomain D. The upper bound holds for so called uniform domains, which is a wide class of domains introduced in [MaSa]. 2.15. Definition. A domain D ( Rn is called uniform or A-uniform, if there exists a number A ≥ 1 such that the inequality holds for all x, y ∈ D.

kD (x, y) ≤ A jD (x, y)

There are many definitions for uniform domains around, see eg. [Ge], so often many “nice” domains can be shown to be uniform by other means, and so one has access to the inequality in 2.15. However, typically very little can be said about the constant A. These matters have been studied in [Li1].


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The j-metric defined in (2.14) has another important connection to the quasihyperbolic metric. The quasihyperbolic metric is namely the inner metric of the j-metric, in the sense of (2.4). In other words kD (x, y) = inf ℓj,D (γ). γ∈Γxy

Since the j-metric fails to be intrinsic, it cannot be geodesic either. In fact, the j-metric has geodesics only in some special cases, see [HäIbLi, 3.7]. Very little is known about the geodesic segments of the e j-metric, although it can be conjectured that there is not much of them either.

3. Isometries and bilipschitz-mappings As pointed out earlier, most of the hyperbolic-type metrics defined in this article satisfy some kind of invariance property, that is, they satisfy the equality (1.1) for some class of mappings f . Typically this invariance property follows almost directly from the definition of the metric, for instance, it is easy to see from the formulas (2.9) and (2.10) that the Apollonian metric is Möbius-invariant and the half-Apollonian metric is similarity invariant. The interesting question mostly regards the other implication. Is the class of “natural candidates” the only mappings which give isometries in the metric in question? And what are the “near-isometries”, that is, the bilipschitz mappings? There are still many open ends regarding these questions, though some progress has been made recently. 3.1. Definition. Let D and D′ = f (D) be domains such that equipped with distances dD and dD′ they are metric spaces. Then a continuous mapping f : D → D′ is said to be L-bilipschitz in (or with respect to) the metric d if for all x, y ∈ D we have 1 dD (x, y) ≤ dD′ (f (x), f (y)) ≤ L dD (x, y) L for some L ≥ 1. If the above inequality holds with L = 1, f is a d-isometry. 3.2. “One-point” and “two-point” metrics. In general, the hyperbolictype metrics can be divided into length-metrics, defined by means of integrating a weight function, and point-distance metrics. The point-distance metrics may again be classified by the number of boundary points used in their definition. So for instance the j-metric and the half-Apollonian metric would be ‘one-point metrics”, whereas the e j, and the Apollonian metrics are “two-point metrics”. Actually also the length metrics can be characterized in the same way, by looking at their weight function. Then the quasihyperbolic metric is a one-point metric. An example of a two-point length metric is the so called Ferrand metric n σD , see [Fe1]. It is defined for a domain D ( R with card ∂D ≥ 2, using the weight function |a − b| , a,b∈∂D |x − a||x − b|

wD (x) = sup

x ∈ D \ {∞}.

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This metric is Möbius invariant and coincides with the hyperbolic metric on Hn and Bn . Moreover, it is bilipschitz equivalent to the quasihyperbolic metric by the inequality (3.3)

kD (x, y) ≤ σD (x, y) ≤ 2 kD (x, y),

x, y ∈ D.

Naturally one would expect the one-point point-distance metrics to be the easiest ones to study. In fact, much can be said about these metrics when it comes to the isometry question. The half-Apollonian metric has recently been studied in [HäLi]. A point x ∈ D is called circularly accessible if there exists a ball B ⊂ G such that x ∈ ∂B. If x is circularly accessible by two distinct balls whose surfaces intersect at more than one point, it is called a corner point, otherwise a regular point. 3.4. Theorem. Let D ( Rn be a domain which has at least n regular boundary points which span a hyperplane. Then f : D → Rn is a homeomorphic η-isometry if and only if it is a similarity mapping. Furthermore, it was shown that Möbius mappings are in fact 2-bilipschitz with respect to ηD . For the j-metric, some results can be found in [HäIbLi], and in fact in a slightly more general setting. The implications for the j-metric can be expressed as follows; 3.5. Corollary. Let D ( Rn . Then f : D → Rn is a j-isometry if and only if (1) f is a similarity, or (2) D = Rn \ {a} and, up to similarity, f is the inversion in a sphere centered at a. Since ˆjD = kD , it immediately follows that every isometry of the j-metric is an isometry of the quasihyperbolic metric, of course in this case that does not provide us with very much new information. However, a similar relation is true for the Seittenranta metric δD defined in [Se] by |x − y||a − b| δD (x, y) = log 1 + sup , x, y ∈ D, a,b∈∂D |a − x||b − y| which is also studied in [HäIbLi]. Namely, here we have that δˆD = σD , so we directly see that this is a Möbius invariant metric. In [Se] it is proved that at least Euclidean bilipschitz mappings are bilipschitz with respect to δ. The converse is not true, as can be shown by the counterexample f : B2 \ {0} → B2 \ {0}, f (x) = |x| · x. However, in [Se] it was shown that every bilipschitz δ-mapping is a quasiconformal mapping, and that every δ-isometry is conformal with respect to the Euclidean metric (and thus Möbius for n ≥ 3). In [HäIbLi] it was shown that also for n = 2 in fact the δ-isometries are exactly the Möbius mappings.


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For the j-metric there are still many open problems regarding the bilipschitz question. It is well known (see [Vu]), that an Euclidean L-bilipschitz mapping is L2 -bilipschitz with respect to the j (and k) metric. For the Apollonian metric the isometry and bilipschitz questions have been studied by several authors. The work was started by Beardon in [Be2], and continued by Gehring and Hag in [GeHa2] where they studied Apollonian bilipschitz mappings. They proved the following theorem. 3.6. Theorem. Let D ( R2 be a quasidisk and f : D → D′ be an Apollonian bilipschitz mapping. (1) If D′ is a quasidisk, then f is quasiconformal in D and f = g|D , where 2 2 g : R → R is quasiconformal. (2) If f is quasiconformal in D, then D′ is a quasidisk and f = g|D , where 2 2 g : R → R is quasiconformal. In [Hä2] the above property (1) was generalized to hold also for n ≥ 3. In the same article also a condition was introduced which determines when a Euclidean bilipschitz mapping is also Apollonian bilipschitz. In the article [HäIb] it is shown that for n = 2 the Apollonian isometries are exactly restrictions of Möbius mappings. For the quasihyperbolic metric the question regarding the isometries has long been open. In [MaOs] it was shown that every kD -isometry is a conformal mapping. A similar proof gives the same result for Ferrand’s metric σD . However, in [Hä3] it is shown that if the boundary of the domain is regular enough (C 3 , or C 2 unless the domain is either strictly convex or has strictly convex complement), then the quasihyperbolic isometries are exactly the similarity mappings. 3.7. Conformal modulus. We conclude by introducing two new metrics which are particularly interesting regarding the question of bilipschitz mappings. Let Γ n be a family of curves in R . By F(Γ) we denote the family of admissible functions, n that is, non-negative Borel-measurable functions ρ : R → R such that Z ρ ds ≥ 1 γ

for each locally rectifiable curve γ ∈ Γ. The n-modulus or the conformal modulus of Γ is defined by Z ρn dm,

M(Γ) = Mn (Γ) = inf

ρ∈F (Γ)

Rn

where m is the n-dimensional Lebesgue measure. It is a conformal invariant, i.e. if f : G → G′ is a conformal mapping and Γ is a curve family in G, then M(Γ) = M(f Γ). n

For E, F, G ⊂ R we denote by ∆(E, F ; D) the family of all closed nonn constant curves joining E and F in D, that is, γ : [a, b] → R belongs to ∆(E, F ; D) if one of γ(a), γ(b) belongs to E and the other to F , and furthermore γ(t) ∈ D for all a < t < b.

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Now we will define two new conformal invariants in the following way. For n x, y ∈ D ( R λD is defined by λD (x, y) = inf M ∆(Cx , Cy ; D) , Cx ,Cy

where Cz = γz [0, 1) and γz : [0, 1] → D is a curve such that z ∈ |γz | and γz (t) → ∂D when t → 1 and z = x, y. Correspondingly, µD (x, y) = inf M ∆(Cxy , ∂D; D) , Cxy

where Cxy is such that Cxy = γ[0, 1] and γ is a curve with γ(0) = x and γ(1) = y.

It is not difficult to show that both quantities µD and λD are conformal invariants, and that µD is a metric (often called the modulus metric) when cap ∂D > 0, 1/(1−n) see [Gá]. λD is not a metric, but λ∗D = λD introduced in [Fe2] is, as long as the boundary of the domain has more then two points. One of the interesting feature regarding these metrics is that both are easily seen — by their definitions — to be conformal invariants. Moreover, the following can be shown (see [Vu, 10.19]); 3.8. Theorem. If f : D → D′ = f D is a quasiconformal mapping, then (1) µD (x, y)/L ≤ µf D (f (x), f (y) ≤ L µD (x, y), (2) λ∗D (x, y)/L1/(n−1) ≤ λ∗f D (f (x), f (y)) ≤ L1/(n−1) λ∗D (x, y) hold for all x, y ∈ D, where L = max{KI (f ), KO (f )} is the maximal dilatation of f . It is not known if the class of bilipschitz mappings with respect to µ or λ∗ includes any other than quasiconformal mappings.

4. Gromov hyperbolicity One way of telling “how hyperbolic” a metric in fact is, is to study whether it satisfies hyperbolicity in the sense of M. Gromov. Classically such spaces have been studied in the geodesic case, and then a space is said to be Gromov δ-hyperbolic if for all triples of geodesics Jd [x, y], Jd [y, z] and Jd [x, z] we have that dist(w, Jd [y, z] ∪ Jd [z, x]) ≤ δ for all w ∈ Jd [x, y], i.e. if all geodesic triangles are δ-thin.

4.1. The Gromov product. In non-geodesic spaces, however, we are constrained to use the definition involving the Gromov product. This can be defined for two points x, y ∈ D with respect to a base point w by setting 1 (x|y)w = d(x, w) + d(y, w) − d(x, y) . 2


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A space is then said to be Gromov δ-hyperbolic if it satisfies the inequality (x|z)w ≥ (x|y)w ∧ (y|z)w − δ

for all x, y, z ∈ D and a base point w ∈ D. A space is said to be Gromov hyperbolic if it is Gromov δ-hyperbolic for some δ. Sometimes one wants to use the equivalent definition for Gromov hyperbolicity (4.2) d(x, z) + d(y, w) ≤ d(x, w) + d(y, z) ∨ d(x, y) + d(z, w) + 2δ.

Recently the study of Gromov hyperbolicity has become quite popular, and even hyperbolicity results on particular metrics in geometric function theory have been developed by a number of authors. A systematic study of the different metrics is made easier by the fact that Gromov hyperbolicity is preserved by certain classes of mappings, so called rough isometries. We say that two metrics d and d′ are roughly isometric if there exists a positive constant C such that d(x, y) − C ≤ d′ (x, y) ≤ d(x, y) + C.

It is immediately clear from the definition (4.2) that roughly isometric metrics are Gromov hyperbolic in the same domains. Moreover, we say that two metrics are (A, C)-quasi-isometric if there is A ≥ 1, C ≥ 0 such that A−1 d(x, y) − C ≤ d′ (x, y) ≤ A d(x, y) + C.

Also quasi-isometries (and thus bilipschitz mappings) are known to preserve Gromov hyperbolicity, provided that the spaces are geodesic. Naturally we would want the hyperbolic metric itself to be Gromov hyperbolic also, and in fact it is, with constant δ = log 3, as is shown in [CoDePa]. One of the more interesting and general results is one from the comprehensive study of M. Bonk, J. Heinonen and P. Koskela [BoHeKo], where it is shown that for a uniform domain D the space (D, kD ) is always Gromov hyperbolic. For many of the other metrics Gromov hyperbolicity is easily proved or disproved using the results from [Hä4]. Namely, it turns out that the e j-metric is Gromov hyperbolic in every proper subdomain of Rn , whereas the j-metric is Gromov hyperbolic only in Rn \ {a}. Then, using inequalities jD (x, y) − log 3 ≤ ηD (x, y) ≤ jD (x, y),

and

˜jD (x, y) − log 9 ≤ αD (x, y) ≤ ˜jD (x, y),

αD (x, y) ≤ δD (x, y) ≤ αD (x, y) + log 3 we immediately get some results by rough isometry, that is, the results in Table 1 regarding the Apollonian, half-Apollonian and Seittenranta metrics. For proving Gromov hyperbolicity of the Ferrand metric one can use geodesity, Gromov hyperbolicity of the quasihyperbolic metric, and the bilipschitz equivalence in (3.3). Finally, for the µ and λ∗ metrics positive results regarding Gromov hyperbolicity are shown in [Li2].

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4.3. Theorem. The metric space (Bn , λ∗Bn ) is Gromov δ-hyperbolic, with Gromov constant 1 1 1−n 1 ωn−1 1−n 64 64 + 4 log λ + 4(log 2 + n − 1) , log ≤ log δ ≤ 21 ωn−1 n 2 3 2 2 3

where ωn−1 denotes the (n − 1)-dimensional surface area of S n−1 and λn is the Grötzsch constant. Also, any simply connected proper subdomain D ( R2 is Gromov δ-hyperbolic with respect to the metric λ∗G , where log 5462 ≈ 1.3696. δ≤ 2π

4.4. Theorem. The metric space (Bn , µBn ) is Gromov δ-hyperbolic, with Gromov constant δ ≤ 2n−1 cn log 12, where cn is the spherical cap inequality constant, see [Vu]. Especially, every simply connected domain D ( R2 is Gromov hyperbolic with 2 log 12 ≈ 1.5819. δ≤ π 4.5. Theorem. The metric space (Rn \ {z}, λ∗Rn \{z} ) is Gromov hyperbolic, with 1 1 n−1 n−1 δ ≤ 2ωn−1 log 18λ2n ≤ 2ωn−1 log 72 + 2n − 2 . As the below table indicates, the j-metric and the half-Apollonian metric are the only metrics of the ones discussed here which fail to be Gromov hyperbolic in most cases. These results indicate that these metrics are in a way “too easy”, or have too little structure for satisfying Gromov hyperbolicity. On the other hand, in other contexts that is one of their strongest features, as has been seen in earlier sections.

kD hD αD ηD jD ˜jD δD σD λ∗D µD

Domain condition

Proved where

D uniform n = 2 all domains defined, n ≥ 3, D = Bn , Hn All domains D ( Rn Only D = Rn \ {z}, δ = log 9 Only D = Rn \ {z}, δ = log 9 All domains D ( Rn All domains D ( Rn D uniform, for D = Bn δ = log 3 D = Bn , Rn∗ , n = 2 simply conn. domains D = Bn , n = 2 simply conn. domains

[BoHeKo] [CoDePa] and conf. invariance [H¨ a4] and rough isometry [H¨ a4],[H¨ aLi] [H¨ a4] [H¨ a4] [H¨ a4],[Se] [Fe1],[BoHeKo] [Li2] [Li2]

Table 1: Gromov hyperbolicity of some metrics.


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References [Be1] [Be2]

[Be3] [BoHeKo] [CoDePa]

[Fe1]

[Fe2] [G´ a] [Ge] [GeHa1]

[GeHa2]

[GeOs] [GePa] [H¨ a1] [H¨ a2] [H¨ a3] [H¨ a4] [H¨ aIb] [H¨ aIbLi] [H¨ aLi] [Ib] [Li1]

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[MaOs] [MaSa] [Se] [Vu]

H. Lindén

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ń: Gromov hyperbolicity of certain invariant metrics. In preparation. H. Linde Available as preprint in Reports Dept. Math. Stat. Univ. Helsinki 409, (2005), University of Helsinki. G. Martin and B. Osgood: The quasihyperbolic metric and the associated estimates on the hyperbolic metric. J. Anal. Math. 47 (1986), 37–53. O. Martio and J. Sarvas: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, (1978/79), 383–401. P. Seittenranta: M¨ obius-invariant metrics. Math. Proc. Camb. Phil. Soc. 125 (1999), 511–533. M. Vuorinen: Conformal geometry and quasiregular mappings. Lecture Notes in Mathematics, Vol. 1319 Springer-Verlag, Berlin, 1988.

Henri Lindén Address: P.O.Box 68, 00014 University of Helsinki, FINLAND

E-mail: [email protected]


Geometric properties of hyperbolic geodesics W. Ma and D. Minda Abstract. In the unit disk D hyperbolic geodesic rays emanating from the origin and hyperbolic disks centered at the origin exhibit simple geometric properties. The goal is to determine whether analogs of these geometric properties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simply connected region Ω. According to whether the simply connected region Ω is a subset of the unit disk D, the complex plane C or the extended complex plane (Riemann sphere) C∞ = C ∪ {∞}, the geometric properties are measured relative to the background geometry on Ω inherited as a subset of one of these classical geometries, hyperbolic, Euclidean and spherical. In a simply connected hyperbolic region Ω ⊂ C hyperbolic polar coordinates possess global Euclidean properties similar to those of hyperbolic polar coordinates about the origin in the unit disk if and only if the region is Euclidean convex. For example, the Euclidean distance between travelers moving at unit hyperbolic speed along distinct hyperbolic geodesic rays emanating from an arbitrary common initial point is increasing if and only if the region is convex. A simple consequence of this is the fact that the two ends of a hyperbolic geodesic in a convex region cannot be too close. Exact analogs of this Euclidean separating property of hyperbolic geodesic rays hold when Ω lies in either the hyperbolic plane D or the spherical plane C∞ . Keywords. hyperbolic metric, hyperbolic geodesics, hyperbolic disks, Euclidean convexity, hyperbolic convexity, spherical convexity. 2000 MSC. Primary 30F45; Secondary 30C55.


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3. Hyperbolic polar coordinates in a disk or half-plane

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4. Hyperbolic polar coordinates in simply connected regions

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6. Euclidean convex regions

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7. Spherical geometry

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Version October 19, 2006. The second author was supported by a Taft Faculty Fellowship.

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1. Introduction The results in this expository paper are adapted from [16] and [17] and concern geometric properties of hyperbolic geodesics in a simply connected hyperbolic region Ω and, to a lesser extent, geometric properties of hyperbolic disks. These two references contain many results not mentioned here and as well as the details that are not presented in this largely expository article. In particular, proofs not given in this article can be found in these two references. There are three different cases to consider according to whether the region Ω is a subset of the hyperbolic plane D, the Euclidean plane C, or the spherical plane C∞ = C ∪ {∞}. Two geometries on the region Ω will be considered. First, the intrinsic hyperbolic geometry on Ω and, second, the geometry that Ω inherits as a subset of the hyperbolic, Euclidean or spherical plane. Here is a rough description of the types of behavior of hyperbolic geodesics that we will consider. Fix a point w0 ∈ Ω. For θ ∈ R, let ρ(w0 , Ω) denote the hyperbolic geodesic ray emanating from w0 that has unit Euclidean tangent eiθ at w0 and let w0 (s, θ) be the hyperbolic arc length parametrization of this geodesic. Under what conditions does the point w0 (s, θ) move monotonically away from w0 when s increases? Here motion away from w0 is measured relative to the background distance. For example, if Ω lies in the Euclidean plane, this means the Euclidean distance |w0 (s, θ) − w0 | should increase with s. The second type of behavior we consider is whether the background distance between distinct geodesic rays increases as points move along these rays. In the Euclidean case we inquire whether the Euclidean distance |w0 (s, θ1 ) − w0 (s, θ2 )| increases with s when eiθ1 6= eiθ2 . Intuitively, one can think of two travelers departing from w0 at the same time along different hyperbolic geodesic rays and traveling at unit hyperbolic speed along the geodesics and asking whether the travelers separate monotonically in the Euclidean sense. Finally, we investigate the shape of hyperbolic circles relative to the background geometry. The main concern is whether hyperbolic circles are convex curves relative to the background geometry. Hyperbolic rays emanating from a point w0 together with hyperbolic circles centered at w0 form the coordinate grid for hyperbolic polar coordinates in Ω, so our work can be interpreted as studying geometric properties of the hyperbolic polar coordinate grid relative to the background geometry. A descriptive outline of the paper follows. Hyperbolic polar coordinates in the unit disk are defined in Section 2, while Section 3 extends hyperbolic polar coordinates to any Euclidean disk or half-plane. Simple Euclidean properties of the hyperbolic polar coordinate grid in any disk or half-plane are established as the model for future investigations. Hyperbolic polar coordinates for a simply

Geometric properties of hyperbolic geodesics

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connected region are introduced in Section 4. Loosely speaking hyperbolic polar coordinates can be transferred from the unit disk to a simply connected region Ω by using the Riemann Mapping Theorem; a conformal map f : D → Ω is a hyperbolic isometry. Characterizations of Euclidean convex univalent functions are discussed in Section 5. in Section 6 these characterizations are used to establish Euclidean properties of hyperbolic polar coordinates in Euclidean convex regions and to show that these properties characterize Euclidean convex regions. The remainder of the paper is devoted to analogs of these results in the spherical and hyperbolic planes. The spherical plane is introduced in Section 7 along with the notion of a spherically convex region. The results for regions in the spherical plane parallels the Euclidean context. Spherically convex univalent functions are presented in Section 8. The reader should note the number of parallels between spherically convex univalent functions and Euclidean convex univalent functions. The results for spherically convex univalent functions seem more involved than those for Euclidean convex univalent functions; the more complicated nature of formulas relating to spherically convex univalent functions is due to the fact that the spherical metric has curvature 1 while the Euclidean metric has curvature 0. Nonzero curvature causes the appearance of extra terms. Applications of some results for spherically convex functions to the behavior of the hyperbolic coordinate grid in a spherically convex region are given in Section 9. Section 10 considers the behavior of the hyperbolic polar coordinate grid for hyperbolically convex regions in the unit disk. Because of the strong similarity with the previous cases for Euclidean convexity and spherical convexity, we present a concise discussion of the results. The reader should note that some theorems for hyperbolically convex univalent functions formally differ from those for spherically convex univalent functions by certain sign changes; these alterations in sign are due to the fact that the hyperbolic plane has curvature −1 while the spherical plane has curvature 1. The brief final section directs the reader to some other situations in function theory in which there are parallel results for the hyperbolic, Euclidean and spherical planes.

2. Hyperbolic polar coordinates in the unit disk We begin by recalling the unit disk as a model of the hyperbolic plane. The hyperbolic metric on the unit disk D = {z : |z| < 1} is λD (z)|dz| =

2|dz| . 1 − |z|2

The hyperbolic metric has curvature −1; that is, −

△ log λD (z) = −1, λ2Ω (z)

where z = x + iy and ∆=

∂2 ∂2 ∂2 + = 4 ∂x2 ∂y 2 ∂z∂ z¯

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denotes the usual Laplacian. For any piecewise smooth curve γ in D the hyperbolic length of γ is given by Z ℓD (γ) = λD (z)|dz|. γ

The hyperbolic distance between z, w ∈ D is defined by dD (z, w) = inf ℓD (γ),

where the infimum is taken over all piecewise smooth paths γ in D that join z and w. In fact, −1 a − b . dD (a, b) = 2 tanh 1 − ¯ba

The group A(D) of conformal automorphisms of the unit disk is the set of holomorphic isometries of the hyperbolic metric and also of the hyperbolic distance. A path γ joining z to w is called a hyperbolic geodesic arc if dD (z, w) = ℓD (γ). The (hyperbolic) geodesic through z and w is C ∩ D, where C is the unique Euclidean circle (or straight line) that passes through z and w and is orthogonal to the unit circle ∂D. If γ is any piecewise smooth curve joining z to w in D, then the hyperbolic length of γ is dD (z, w) if and only if γ is the arc of C in D that joins z and w. A hyperbolic disk in the unit disk is DD (a, r) = {z : dD (a, z) < r}, where a ∈ D is the hyperbolic center and r > 0 is the hyperbolic radius. A hyperbolic disk in D is Euclidean disk with closure contained in D. In fact, DD (a, r) is the Euclidean disk with center c and radius R, where a 1 − tanh2 (r/2) (1 − |a|2 ) tanh(r/2) c= and R = . 1 − |a|2 tanh2 (r/2) 1 − |a|2 tanh2 (r/2) For more details about hyperbolic geometry on the unit disk the reader should consult [1].

Hyperbolic polar coordinates on the unit disk relative to a specified pole or center are defined as follows. Fix a point a in D, called the pole or center for polar coordinates based at a. For θ in R let ρθ (a, D) = ρθ (a) denote the unique hyperbolic geodesic ray emanating from a that is tangent to the Euclidean unit vector eiθ at a. For θ = 0 the Euclidean unit tangent vector is 1 and ρ0 (a) is called the horizontal hyperbolic geodesic emanating from a because the unit tangent vector at a is horizontal. Of course, ρθ+2nπ (a) = ρθ (a) for all n in Z. Let s 7→ za (s, θ), 0 ≤ s < +∞, be the hyperbolic arc length parametrization of ρθ (a). This means (2.1)

∂za (s, θ) eiΘ(s,θ) = , ∂s λD (za (s, θ))

where eiΘ(s,θ) is a Euclidean unit tangent to ρθ (a) at the point za (s, θ). For fixed θ the point za (s, θ) moves along the geodesic ray ρθ (a) with unit hyperbolic speed. Two hyperbolic geodesic rays with distinct unit tangent vectors at a are disjoint except for their common initial point and D = ∪{ρθ : 0 ≤ θ < 2π}. For each z in D \ {a} there is a unique geodesic ray ρθ (a) with 0 ≤ θ < 2π that contains z, so


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there exist unique s > 0 and θ in [0, 2π) with za (s, θ) = z. The hyperbolic polar coordinates of the point z relative to the center or pole at a are the ordered pair (s, θ), where za (s, θ) = z. The first coordinate, s = dD (a, z), is the hyperbolic distance from a to z and the second polar coordinate, θ, is the angle between the horizontal hyperbolic geodesic ray ρ0 (a) and the ray ρθ (a) that contains z at the pole a. The hyperbolic circle with hyperbolic center a and hyperbolic radius s is cD (a, s) = {z : dD (a, z) = s}. Note that each geodesic ray ρθ (a) is orthogonal to every hyperbolic circle cΩ (a, s). Thus, the coordinate grid for hyperbolic polar coordinates based at a consists of hyperbolic geodesics emanating from a and hyperbolic circles centered at a. In terms of hyperbolic polar coordinates λ2D (z)(dx2 + dy 2 ) = ds2 + sinh2 (s)dθ2 . For a = 0, ρθ (0) is the radial segment [0, eiθ ) with hyperbolic arc length parametrization z0 (s, θ) = tanh(s/2)eiθ and (2.2)

eiθ 1 − |z0 (s, θ)|2 ∂z0 (s, θ) = = z0 (s, θ). ∂s λD (z0 (s, θ)) 2|z0 (s, θ)|

Hyperbolic polar coordinates about the origin can be transported to any other center in the unit disk by a hyperbolic isometry. Recall that each conformal automorphism of D is an isometry of the hyperbolic metric and the hyperbolic distance. For a ∈ D the Möbius transformation f (z) = (z + a)/(1 + a ¯z) is a conformal automorphism of D that sends the origin to a and f ′ (0) = (1−|a|2 ) > 0. The fact that f ′ (0) > 0 insures that f (ρθ (0)) = ρθ (a) for all θ ∈ R and so za (s, θ) = f (z0 (s, θ)) provides an explicit hyperbolic arc length parametrization of ρa (θ): tanh(s/2)eiθ + a za (s, θ) = . 1+a ¯ tanh(s/2)eiθ Trivially, the Euclidean distance from a = 0 to z0 (s, θ) is an increasing function of s for each fixed θ and the Euclidean distance between z0 (s, θ1 ) and z0 (s, θ2 ) is an increasing function of s when eiθ1 6= eiθ2 . It is plausible that these Euclidean properties remain valid for any center a ∈ D. Rather than investigating these assertions for the special case of the unit disk, we wait to consider the analogous questions in any disk or half-plane. Also, hyperbolic circles centered at the origin are Euclidean circles.

3. Hyperbolic polar coordinates in a disk or half-plane We let ∆ denote any Euclidean disk or half-plane when it is not necessary to distinguish between the cases; otherwise, we use D for a Euclidean disk and H for a Euclidean half-plane. Given ∆ there is a Möbius transformation f that maps ∆ onto the unit disk. Then the hyperbolic metric on ∆ is given by λ∆ (z) = λD (f (z))|f ′ (z)|.

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This defines the hyperbolic density λ∆ independent of the Möbius map of ∆ onto the unit disk. If D = {z : |z − a| < r}, then λD (z)|dz| = If H is any half-plane, then

2r|dz| . r2 − |z − a|2

λH (z)|dz| =

|dz| , d(z, ∂H)

where d(z, ∂H) denotes the Euclidean distance from z to the boundary of H. In particular, for the upper half-plane H = {z : Im (z) > 0}, λH (z)|dz| =

|dz| . Im (z)

Because Möbius transformations map circles onto circles, hyperbolic geodesics in a disk or half-plane are arcs of circles orthogonal to the boundary. Also, hyperbolic disks are Euclidean disks with closure contained in the disk or halfplane. Any Möbius map from ∆ onto D is an isometry from ∆ with the hyperbolic metric to D with the hyperbolic metric. See [1] for details. Hyperbolic polar coordinates are defined on ∆ analogous to the definition for the unit disk. Fix a point w0 in ∆. For θ in R let ρθ (w0 , ∆) denote the unique hyperbolic geodesic ray emanating from w0 that is tangent to eiθ at w0 . ρ0 (w0 , ∆) is called the horizontal hyperbolic geodesic emanating from w0 since its unit tangent vector at w0 is horizontal. When w0 and ∆ are fixed, we often write ρθ in place of ρθ (w0 , ∆). Of course, ρθ+2nπ = ρθ for all n in Z. Let s 7→ w0 (s, θ), 0 ≤ s < +∞, be the hyperbolic arc length parametrization of ρθ . This means (3.1)

eiΘ(s,θ) ∂w0 (s, θ) = , ∂s λ∆ (w0 (s, θ))

where eiΘ(s,θ) is a Euclidean unit tangent to ρθ at the point w0 (s, θ). Because ∆ = ∪{ρθ : 0 ≤ θ < 2π}, for each w in ∆ \ {w0 } there is a unique geodesic ray ρθ , 0 ≤ θ < 2π, that contains w. Hence, there exist unique s > 0 and θ in [0, 2π) with w0 (s, θ) = w. The hyperbolic polar coordinates for the point w relative to the center or pole at w0 are (s, θ). The coordinate s = d∆ (w0 , w) is the hyperbolic distance from w0 to w and θ is the angle between the horizontal hyperbolic geodesic ray ρ0 and the ray ρθ at w0 . The hyperbolic circle with hyperbolic center w0 and hyperbolic radius s is c∆ (w0 , s) = {w : d∆ (w0 , w) = s}. The coordinate grid for hyperbolic polar coordinates consists of hyperbolic geodesics emanating from w0 and hyperbolic circles centered at w0 . If f : D → ∆ is the Möbius transformation with f (0) = w0 and f ′ (0) > 0, then w0 (s, θ) = f (z0 (s, θ)). As we noted in the preceding section when a point in the unit disk moves away from the origin along a hyperbolic geodesic, the Euclidean distance from the origin increases and points along distinct geodesics separate monotonically in the Euclidean sense. In fact these properties hold for any disk or half-plane and for any center of hyperbolic polar coordinates.


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Theorem 3.1. Let ∆ be any Euclidean disk or half-plane in C and w0 ∈ ∆. (a) For each θ ∈ R, |w0 (s, θ) − w0 | is increasing for s ≥ 0. (b) For eiθ2 6= eiθ1 , |w0 (s, θ1 ) − w0 (s, θ2 )| is an increasing function of s ≥ 0. Proof. If f : D → ∆ is a Möbius mapping with f (0) = w0 and f ′ (0) > 0, then w0 (s, θ) = f (z0 (s, θ)). Suppose az + b , cz + d where ad − bc = 1. Because ∆ is a Euclidean disk or half-plane, ∞ does not lie in ∆. Consequently, −d/c, the preimage of ∞, cannot lie in D; equivalently, |c| ≤ |d|. Since ad − bc = 1, this implies d 6= 0. Also, w0 = f (0) = b/d. f (z) =

(a) If D(s) = log |w0 (s, θ) − w0 | = log |f (z0 (s, θ)) − w0 |, then by using (2.2) we obtain f ′ (z0 (s, θ)) ∂z0 (s, θ) D′ (s) = Re f (z0 (s, θ)) − w0 ∂s 2 z0 (s, θ)f ′ (z0 (s, θ)) 1 − |z0 (s, θ)| (3.2) . Re = 2|z0 (s, θ)| f (z0 (s, θ)) − w0 From

f ′ (z) =

1 (cz + d)2

we obtain

and f (z) − w0 =

z , d(cz + d)

zf ′ (z) d = . f (z) − w0 cz + d

Then for z ∈ D (3.3)

Re

d¯ cz¯ + |d|2 zf ′ (z) = Re >0 f (z) − w0 |cz + d|2

because |c| ≤ |d| and |z| < 1. Thus, (3.3) and (3.2) imply D(s) is increasing for s ≥ 0, so |w0 (s, θ) − w0 | is increasing for s ≥ 0.

(b) We assume −π/2 ≤ θ1 = −θ < 0 < θ2 = θ ≤ π/2; the general case can be reduced to this situation by performing a rotation. If then

E(s) = log |w0 (s, θ) − w0 (s, −θ)| = log |f (z0 (s, θ)) − f (z0 (s, −θ))|,

(s,θ) − f ′ (z0 (s, −θ)) ∂z0 (s,−θ) f ′ (z0 (s, θ)) ∂z0∂s ∂s E (s) = Re . f (z0 (s, θ)) − f (z0 (s, −θ)) Because of (2.2) and |z0 (s, θ)| = |z0 (s, −θ)|, we obtain z0 (s, θ)f ′ (z0 (s, θ)) − z0 (s, −θ)f ′ (z0 (s, −θ)) 1 − |z0 (s, θ)|2 ′ Re . (3.4) E (s) = 2|z0 (s, θ)| f (z0 (s, θ)) − f (z0 (s, −θ)) ′

Direct calculation produces

d2 − c2 ζz zf ′ (z) − ζf ′ (ζ) = . f (z) − f (ζ) (cz + d)(cζ + d)

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Set t = c/d. Then zf ′ (z) − ζf ′ (ζ) 1 − t2 ζz = f (z) − f (ζ) (1 + tz)(1 + tζ) 1 1 − tζ 1 − tz + . = 2 1 + tζ 1 + tz Because (1 − w)/(1 + w) has positive real part for w ∈ D and |tζ|, |tz| < 1, we conclude that for all z, ζ ∈ D ′ zf (z) − ζf ′ (ζ) (3.5) Re > 0. f (z) − f (ζ)

Hence, (3.4) and (3.5) imply E ′ (s) > 0 for s ≥ 0, so that |w0 (s, θ) − w0 (s, −θ)| is an increasing function of s ≥ 0.

4. Hyperbolic polar coordinates in simply connected regions A region Ω in the complex plane C is hyperbolic if C \ Ω contains at least two points. The hyperbolic metric on a hyperbolic region Ω is denoted by λΩ (w)|dw| and is normalized to have curvature △ log λΩ (w) = −1. − λ2Ω (w) If f : D → Ω is any holomorphic universal covering projection, then the density λΩ of the hyperbolic metric is determined from 2 . (4.1) λΩ (f (z))|f ′ (z)| = 1 − |z|2 For a, b in Ω the hyperbolic distance between these points is Z dΩ (a, b) = inf λΩ (w)|dw|, δ

where the infimum is taken over all piecewise smooth paths δ in Ω joining a and b. A path γ connecting a and b is a hyperbolic geodesic arc if Z dΩ (a, b) = λΩ (w)|dw|. γ

A hyperbolic geodesic always exists, but need not be unique when Ω is multiply connected. Given a ∈ Ω and r > 0, DΩ (a, r) = {z ∈ Ω : dΩ (a, z) < r} is the hyperbolic disk with hyperbolic center a and hyperbolic radius r. When Ω is simply connected, any conformal mapping f : D → Ω is an isometry from the hyperbolic metric on D to the hyperbolic metric on Ω. In this case f maps hyperbolic geodesics onto hyperbolic geodesics and hyperbolic disks onto hyperbolic disks. If Ω is multiply connected, then a covering f is only a local isometry, not an isometry.


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Suppose Ω is a simply connected hyperbolic region, w0 ∈ Ω and f : D → Ω is the unique conformal mapping with f (0) = w0 and f ′ (0) > 0. We can relate hyperbolic polar coordinates on Ω with pole at w0 to those on D with pole at the origin by using f . Tangent vectors for geodesic rays can be expressed in terms of this conformal mapping. Because f is an isometry from the hyperbolic metric on D to the hyperbolic metric on Ω and f ′ (0) > 0, w0 (s, θ) = f (z0 (s, θ)) is the hyperbolic arc length parametrization of ρθ (w0 , Ω) and the tangent vector to ρθ (w0 , Ω) is f ′ (z0 (s, θ))eiθ ∂w0 (s, θ) = . ∂s λD (z0 (s, θ))

(4.2) Thus,

f ′ (0)eiθ ∂w0 (0, θ) = , ∂s 2 so that s 7→ w0 (s, θ) is parallel to eiθ at w0 . By making use of (4.1) we find (4.3)

∂w0 (s, θ) f ′ (z0 (s, θ)) eiθ ei(ϕ(s,θ)+θ) = ′ = , ∂s |f (z0 (s, θ))| λΩ (f (z0 (s, θ))) λΩ (f (z0 (s, θ)))

0 (s,θ)) where eiϕ(s,θ) = |ff ′ (z . If arg f ′ (z) denotes the unique branch of the argument (z0 (s,θ))| of f ′ that vanishes at w0 , then ϕ(s, θ) = arg f ′ (z0 (s, θ)). From (3.1) and (4.3) we obtain ′

(4.4)

eiΘ(s,θ) = ei(ϕ(s,θ)+θ) .

In a similar manner, hyperbolic disks in Ω are the images of hyperbolic disks in D; explicitly, if f : D → Ω is a conformal map with f (0) = w0 , then f (DD (0, r)) = DΩ (w0 , r).

5. Euclidean convex univalent functions Several characterizations of Euclidean convex univalent functions are needed for our investigation of hyperbolic polar coordinates. We recall two classical characterizations of Euclidean convex and starlike univalent functions. First, a locally univalent holomorphic function f defined on D is a conformal map onto a Euclidean convex region if and only if [2, p 42] (5.1)

1 + Re

zf ′′ (z) ≥0 f ′ (z)

for z ∈ D. Second, if f (0) = w0 , then a holomorphic function f defined on D maps D conformally onto a region starlike with respect to w0 if and only if [2, p 41] (5.2) for z ∈ D.

Re

zf ′ (z) ≥0 f (z) − w0

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Theorem 5.1. Suppose f is holomorphic and locally univalent on D. f is Euclidean convex univalent on D if and only if zf ′ (z) − ζf ′ (ζ) >0 (5.3) Re f (z) − f (ζ) for all z, ζ in D. Proof. We present the short proof. Suppose f is Euclidean convex univalent on D. Then ([27] and [30]) z+ζ 2zf ′ (z) >0 − (5.4) Re f (z) − f (ζ) z − ζ

for z, ζ in D. If we interchange the roles of z and ζ in (5.4) and then add the two inequalities, we obtain zf ′ (z) − ζf ′ (ζ) 2 Re > 0, f (z) − f (ζ) which is equivalent to (5.3).

Conversely, suppose (5.3) holds for all z, ζ in D. Since zf ′ (z) − ζf ′ (ζ) zf ′′ (z) =1+ ′ , ζ→z f (z) − f (ζ) f (z) lim

we obtain (5.1). Hence, f is Euclidean convex univalent on D. Theorem 5.2. If f is a normalized, f (0) = 0 and f ′ (0) = 1, Euclidean convex univalent function on D and θ ∈ (0, π/2], then cos θ 2|z| sin θ 2|z| sin θ 1 + |z| iθ −iθ (5.5) . ≤ |f (e z) − f (e z)| ≤ 1 + 2|z| cos θ + |z|2 1 − |z|2 1 − |z|

The lower bound is best possible for all θ ∈ (0, π/2] and the upper bound is sharp for θ = π/2. Proof. We sketch the idea of the proof. Fix θ in (0, π/2] and consider the function f (eiθ z) − f (e−iθ z) f (eiθ z) − f (e−iθ z) g(z) = = . eiθ − e−iθ 2i sin θ From eiθ zf ′ (eiθ z) − e−iθ zf ′ (e−iθ z) zg ′ (z) = , g(z) f (eiθ z) − f (e−iθ z) Theorem 5.1 implies that g is starlike with respect to the origin on D because (5.2) holds with w0 = 0. If ∞ X an z n , f (z) = z + n=2

then

g(z) = z +

∞ X sin nθ n=2

sin θ

an z n .


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As f is convex univalent, |a2 | ≤ 1 [2]. Hence, |g ′′ (0)| sin 2θ = |a2 | ≤ 2 cos θ. 2 sin θ Then [3] gives the inequalities in (5.5).

Corollary 5.3. If f is a normalized, f (0) = 0 and f ′ (0) = 1, Euclidean convex univalent function on D, then for ϕ ∈ (0, π/2] 2r 2r (5.6) ≤ |f (reiϕ ) − f (−reiϕ )| ≤ . 2 1+r 1 − r2 These bounds are sharp. Example 5.4. If K(z) = z/(1 − z), then

e−iθ z eiθ z − 1 − eiθ z 1 − e−iθ z (2i sin θ)z = . 1 − (2 cos θ)z + z 2

K(eiθ z) − K(e−iθ z) =

This shows that the lower bound in (5.5) is sharp for K(z) when z = −r, r is in (0, 1), for any θ in (0, π/2]. For θ = π/2 the upper bound in (5.5) is sharp for the function K when z = ir, r in (0, 1). Also, 2r K(r) − K(−r) = 1 − r2 and 2ir , K(ir) − K(−ir) = 1 + r2 so both bounds in (5.6) are sharp.

6. Euclidean convex regions We establish various Euclidean properties for hyperbolic polar coordinates in Euclidean convex regions; in fact, these Euclidean properties characterize convex regions. Throughout this section we employ the notation of Section 4. In particular, f will always denote a conformal map of D onto Ω with f (0) = w0 and f ′ (0) > 0. We show that for each fixed θ, the point w0 (s, θ) moves monotonically away from w0 in the Euclidean sense. We give sharp upper and lower bounds on |w0 (s, θ) − w0 | in terms of s and λΩ (w0 ). Also, in any convex region Ω distinct hyperbolic geodesic rays separate monotonically in the Euclidean sense; this means that for eiθ2 6= eiθ1 , the distance |w0 (s, θ1 ) − w0 (s, θ2 )| is an increasing function of s. We give sharp upper and lower bounds on the difference |w0 (s, θ1 )−w0 (s, θ2 )|. These (and other) Euclidean properties of hyperbolic polar coordinates characterize convex regions. For example, a classical result of Study [29] implies that for every w0 ∈ Ω each hyperbolic circle cΩ (w0 , s) is a Euclidean convex curve when Ω is convex. The result of Study asserts that if f is a Euclidean convex univalent function,

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then f ({z : |z| < r}) is Euclidean convex for 0 < r < 1. Conversely, if every hyperbolic circle is Euclidean convex, then Ω is an increasing union of Euclidean convex regions and so is Euclidean convex. Theorem 6.1. Let Ω be a simply connected hyperbolic region in C. (a) If Ω is Euclidean convex and w0 ∈ Ω, then for each θ in R, |w0 (s, θ) − w0 | is an increasing function of s and es − 1 1 − e−s ≤ |w0 (s, θ) − w0 | ≤ . (6.1) λΩ (w0 ) λΩ (w0 ) These bounds are best possible. (b) Suppose that for every w0 in Ω and for each θ in R, |w0 (s, θ) − w0 | is an increasing function of s. Then Ω is Euclidean convex. The proof of Theorem 6.1 is given in [16]. Example 6.2. For the upper half-plane H, λH (w) = 1/Im(w). Then for w0 = i, w0 (s, π/2) = i + i(es − 1), w0 (s, −π/2) = i − i(1 − e−s ) and 1/λH (i) = 1, so the upper and lower bounds are best possible. Theorem 6.3. Suppose Ω is a simply connected hyperbolic region in C. (a) If Ω is Euclidean convex, w0 ∈ Ω and eiθ2 6= eiθ1 , then |w0 (s, θ1 ) − w0 (s, θ2 )| is an increasing function of s ≥ 0 and 2 sin θ tanh s (6.2) ≤ |w0 (s, θ1 ) − w0 (s, θ2 )|λΩ (a) ≤ 2es cos θ sin θ sinh s, 1 + cos θ tanh s where θ = (θ2 − θ1 )/2. (b) If for some w0 in Ω and all eiθ2 6= eiθ1 , |w0 (s, θ1 ) − w0 (s, θ2 )| is an increasing function of s ≥ 0, then Ω is Euclidean convex. Proof. We sketch the proof of (a). First, by translating Ω if necessary, we may assume w0 = 0. Next, by rotating Ω about the origin if needed, we may assume −π/2 ≤ θ1 = −θ < 0 < θ2 = θ ≤ π/2. Then |w0 (s, θ) − w0 (s, −θ)| = |f (z(s, θ)) − f (z(s, −θ))|.

All of the quantities involved in the theorem are invariant under translation and rotation. If then

E(s) = log |w0 (s, θ) − w0 (s, −θ)| = log |f (z(s, θ)) − f (z(s, −θ))|,

− f ′ (z(s, −θ)) ∂z(s,−θ) f ′ (z(s, θ)) ∂z(s,θ) ∂s ∂s . E (s) = Re f (z(s, θ)) − f (z(s, −θ)) Because of (2.2) and |z(s, θ)| = |z(s, −θ)|, we obtain z(s, θ)f ′ (z(s, θ)) − z(s, −θ)f ′ (z(s, −θ)) 1 − |z(s, θ)|2 ′ Re . (6.3) E (s) = 2|z(s, θ)| f (z(s, θ)) − f (z(s, −θ)) Suppose Ω is Euclidean convex. Then f is a Euclidean convex univalent function and so (5.3) implies E ′ (s) > 0. Hence, |w0 (s, θ) − w0 (s, −θ)| is an increasing function of s ≥ 0. Next, we establish (6.2). The function f /f ′ (0) is a normalized ′


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Euclidean convex univalent function so (5.5) with r = tanh(s/2) gives the bounds (6.2) for θ2 = θ and θ1 = −θ since f ′ (0) = 2/λΩ (a). Corollary 6.4. Suppose Ω 6= C is a Euclidean convex region and w0 is a point of Ω. Then |w0 (s, θ) − w0 (s, θ + π)| is increasing and (6.4)

tanh(s) ≤ |w0 (s, θ) − w0 (s, θ + π)|

λΩ (a) ≤ sinh(s). 2

Both bounds are sharp for a half-plane. Proof. This is the special case of the theorem in which θ2 = θ + π and θ1 = θ. It corresponds to two hyperbolic geodesic rays emanating from w0 in opposite directions. The lower bound in (6.2) has a simple geometric consequence. It gives lim |w0 (s, θ1 ) − w0 (s, θ2 )| ≥

s→+∞

2 tan(θ/2) . λΩ (w0 )

For any Euclidean convex region this shows that the ‘ends’ of distinct hyperbolic geodesic rays emanating from w0 cannot be too close. In particular, (6.4) implies that the two ends of a single hyperbolic geodesic cannot be closer than 2/λΩ (w0 ) for any point w0 on the geodesic. This inequality is sharp for the upper halfplane; we only consider the special case in which θ1 = 0 and θ2 = π. Consider w0 = ib, where b > 0. Then λH (w0 ) = 1/b. For the hyperbolic geodesic γ through ib that meets R in ±b, lim |w0 (s, 0) − w0 (s, π)| = 2b =

s→+∞

2 . λH (w0 )

7. Spherical geometry We discuss the geometry of the spherical plane C∞ with the chordal distance χ, the spherical metric σ(z) |dz| and the induced spherical distance dσ .

The extended complex plane C∞ is sometimes called the Riemann sphere because stereographic projection transforms C∞ into the unit sphere. Let S be the unit sphere {x ∈ R3 : ||x|| = 1} in R3 , and let n = (0, 0, 1) be the ‘north pole’. The stereographic projection ϕ of C∞ onto S is defined as follows. We regard the complex plane C as a subset of R3 by identifying z = x + iy with the point (x, y, 0). For z in C, the line through z = (x, y, 0) and n meets S at n and at a second point ϕ(z). This defines ϕ on C, and we set ϕ(∞) = n. It is easy to see that if z = x + iy ∈ C, then 2x 2y |z|2 − 1 ϕ(x + iy) = , , . |z|2 + 1 |z|2 + 1 |z|2 + 1

Observe that ϕ(0) = (0, 0, −1), the south pole, and that ϕ(z) = z = (x, y, 0) if and only if |z| = 1.

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The chordal distance χ is obtained by the following procedure. Use ϕ to transfer points in C∞ to S; then measure the Euclidean distance in R3 between the image points. Thus, χ is defined by χ(z, w) = ||ϕ(z) − ϕ(w)||;

explicitly,

2|z − w| , χ(z, w) = p (1 + |z|2 )(1 + |w|2 )

2 χ(z, ∞) = p . (1 + |z|2 )

This interpretation of χ immediately shows that it is a distance function on C∞ . Also, the metric space (C∞ , χ) is homeomorphic to S with the restriction of the Euclidean metric; so (C∞ , χ) is compact and connected. The spherical metric on C∞ is given by σ(w)|dw| = it has curvature

2|dw| ; 1 + |w|2

∆ log σ(w) = 1. σ 2 (w) The spherical distance on C∞ derived from this metric is −1 z − w ≤ π. dσ (z, w) = 2 tan 1 + wz ¯ The chordal and spherical metrics are related to each other by the formula 1 χ(z, w) = 2 sin dσ (z, w) . 2 −

From 2θ/π ≤ sin θ ≤ θ when 0 ≤ θ ≤ π/2, we obtain

(2/π)dσ (z, w) ≤ χ(z, w) ≤ dσ (z, w),

so the two distances induce the same topology on C∞ . Note that χ(z, w) dσ (z, w) lim = σ(z) = lim . w→z |z − w| w→z |z − w| We present a complete description of the isometries of the spherical plane. The orientation preserving conformal isometries of the spherical plane form a group. All of the following groups are identical, (1) the group of conformal isometries of the chordal distance; (2) the group of conformal isometries of the spherical distance; (3) the group of conformal isometries of the spherical metric; (4) the group of Möbius maps of the form az − c¯ , |a|2 + |c|2 = 1. z 7→ cz + a ¯ The orientation-preserving isometries of R3 that fix the origin are the rotations of R3 , and these are represented by the group SO(3) of 3 × 3 orthogonal matrices with determinant one. The group SO(3) is conjugate to the group ϕ−1 SO(3)ϕ


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which acts on C∞ ; the four identical groups above are equal to ϕ−1 SO(3)ϕ. For this reason the isometries of the spherical plane are sometimes called rotations. For antipodal z, w ∈ C∞ , that is, w = −1/¯ z , any of the infinitely many great circular arcs connecting z and w is a spherical geodesic. If z, w ∈ C∞ are not antipodal, then the unique spherical geodesic arc is the shorter arc between z and w of the unique great circle through z and w. Just as one studies convex regions in the Euclidean plane it is natural to study convex regions in the spherical plane. A simply connected region Ω on C∞ is called spherically convex (relative to spherical geometry on C∞ ) if for each pair of z, w ∈ Ω every spherical geodesic connecting z and w also lies in Ω. If Ω is spherically convex and contains a pair of antipodal points, then Ω = C∞ . A meromorphic and univalent function f defined on D is called spherically convex if its image f (D) is a spherically convex subset of C∞ . A number of authors have studied spherically convex functions; for example, [6], [8], [11], [15], [19], [21] and [25].

8. Spherically convex univalent functions In our discussion of Euclidean properties of hyperbolic geodesics, characterizations of Euclidean convex functions played a crucial role. Therefore, it is not surprising that characterizations of spherically convex functions play an important role in investigating spherical properties of hyperbolic geodesics. One such characterization obtained by Mejia and Minda [19] is ) ( zf ′′ (z) 2zf ′ (z)f (z) ≥0 − (8.1) Re 1 + ′ f (z) 1 + |f (z)|2 for all z in D; also see [8]. Sometimes it is difficult to use (8.1) because it contains the nonholomorphic term 2zf ′ (z)f (z)/(1 + |f (z)|2 ). One way to overcome this difficulty is to establish two-variable characterizations for spherically convex functions which are holomorphic in one of the two variables and are analogous to Theorem 5.1 We now state two-variable characterizations for spherically convex functions that will be applied to investigate properties of hyperbolic polar coordinates on spherically convex regions and to derive other results for spherically convex functions. Theorem 8.1. Let f be meromorphic and locally univalent in D. Then f is spherically convex if and only if ) ( z+ζ 2zf ′ (z)f (ζ) 2zf ′ (z) >0 − − (8.2) Re f (z) − f (ζ) z − ζ 1 + f (ζ)f (z) for all z, ζ in D.

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Proof. Here we prove only the sufficiency. Observe that (8.2) is the spherical analog of (5.4). Let (8.3)

p(z, ζ) =

z+ζ 2zf ′ (z)f (ζ) 2zf ′ (z) − − . f (z) − f (ζ) z − ζ 1 + f (ζ)f (z)

We show that if f satisfies the inequality (8.2), then (8.1) holds for all z ∈ D, which characterizes spherically convex functions [19]. The assumption is that Re {p(z, ζ)} > 0 for z, ζ ∈ D. Since (8.4)

p(z, z) = 1 +

f is spherically convex.

zf ′′ (z) 2zf ′ (z)f (z) − , f ′ (z) 1 + |f (z)|2

Corollary 8.2. Suppose f is meromorphic and locally univalent in D. Then f is spherically convex if and only if ) ( zf ′ (z) − ζf ′ (ζ) zf ′ (z)f (ζ) + ζf ′ (ζ)f (z) − >0 (8.5) Re f (z) − f (ζ) 1 + f (ζ)f (z) for all z, ζ in D. Proof. Note that (8.5) follows from (8.2) in the same manner that (5.3) was derived from (5.4). Conversely, suppose (8.5) holds for all z, ζ in D. Set (8.6)

zf ′ (z) − ζf ′ (ζ) zf ′ (z)f (ζ) + ζf ′ (ζ)f (z) q(z, ζ) = − . f (z) − f (ζ) 1 + f (ζ)f (z)

Then Re {q(z, z)} > 0 for all z in D. As q(z, z) = 1 +

zf ′′ (z) 2zf ′ (z)f (z) − , f ′ (z) 1 + |f (z)|2

the inequality (8.1) holds. Therefore, f is spherically convex. As we pointed out earlier, we cannot easily derive properties of spherically convex functions from (8.1) since it contains a nonholomorphic term. With Theorem 8.1, this difficulty is overcome in some cases. If f is spherically convex, then p(z, ζ) is holomorphic for in z ∈ D, has positive real part, and so satisfies 1 + |z|2 2|z| (8.7) p(z, ζ) − 1 − |z|2 ≤ 1 − |z|2 .

The nonholomorphic function p(z, z) (see (8.4)) still satisfies the inequality (8.7), which holds for the well known class consisting of holomorphic functions p(z) in D with p(0) = 1 and Re {p(z)} > 0. Note that 2 1 + |z| ≤ 2|z| p(z, z) − 2 1 − |z| 1 − |z|2

also characterizes spherically convex functions and implies the inequality (8.1), see [11].


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This idea can be used to derive a number of results for spherically convex functions. First, recall that a holomorphic and univalent function f in D with f (0) = f ′ (0) − 1 = 0 is called starlike of order β ≥ 0 if Re {zf ′ (z)/f (z)} > β in D. Using Theorem 8.1, we show that spherically convex functions are closely related to starlike functions. Theorem 8.3. If f (z) is spherically convex with f (0) = 0, then for every ζ ∈ D, Fζ (z) = is starlike of order 1/2.

f (z) − f (ζ) zζ f (ζ) (z − ζ) 1 + f (ζ)f (z)

Proof. Direct calculations yield 2zFζ (z) − 1 = p(z, ζ). Fζ (z) Theorem 8.1 implies the result. Since Re {F (z)/z} > 1/2 and F (z)2 /z is starlike if F is starlike of order 1/2 (see [26, p. 49]), we get the following results as corollaries of Theorem 8.3. Corollary 8.4. If f (z) is spherically convex with f (0) = 0, then for every ζ ∈ D,    ζ  1 f (z) − f (ζ) > Re  f (ζ) (z − ζ) 1 + f (ζ)f (z)  2 for all z in D.

Corollary 8.5. If f (z) is spherically convex with f (0) = 0, then for every ζ ∈ D, Fζ2 (z) zζ 2 = z f (ζ)2

is starlike in D.

(f (z) − f (ζ))2 2 2 (z − ζ) 1 + f (ζ)f (z)

Mejia and Pommerenke [21] obtained a number of results for spherically convex functions by observing that f (z) is (Euclidean) convex when f (z) is spherically convex and f (0) = 0. We now provide the sharp order of Euclidean convexity for spherically convex functions that fix the origin. Corollary 8.6. Let f (z) = αz + a2 z 2 + . . ., 0 < α < 1, be spherically convex. Then for all z in D 2 α zf ′′ (z) √ > . Re 1 + ′ f (z) 1 + 1 − α2 This result is best possible for each α.

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Example √8.7. For 0 < α ≤ 1, the spherical half-plane, or hemisphere, Ωα = w : |w − 1 − α2 /α| < 1/α is spherically convex and αz √ kα (z) = 1 − 1 − α2 z maps D conformally onto Ωα . For the function kα , 2 zkα′′ (z) α √ inf Re 1 + ′ :z∈D = . kα (z) 1 + 1 − α2 Next, we give the sharp lower bound on Re {a2 f (z)} for normalized spherically convex functions f (z) = αz+a2 z 2 +. . . . Similar results hold for Euclidean convex functions [4] and hyperbolically convex functions [14]. Theorem 8.8. Let f (z) = αz + a2 z 2 + . . . , 0 < α ≤ 1, be spherically convex. Then for all z in D √ Re {a2 f (z)} ≥ 1 − α2 − 1 − α2 . This result is best possible for all α.

It is easy to see that for the spherically √ convex functions kα (z), the infimum of Re {a2 kα (z)} over z ∈ D is 1 − α2 − 1 − α2 , so the lower bound is sharp for all α ∈ (0, 1].

9. Spherically convex regions Now, we establish certain properties of hyperbolic polar coordinates in spherically convex regions. It is convenient to use the density of the hyperbolic metric relative to the spherical metric; that is, 1 λΩ (w)|dw| = (1 + |w|2 )λΩ (w). µΩ (w) = σ(w)|dw| 2 Then λΩ (w)|dw| = µΩ (z)σ(w)|dw| and µΩ is invariant under all rotations of the sphere. Theorem 9.1. Suppose Ω is a hyperbolic region in C∞ . (a) If Ω is spherically convex and w0 ∈ Ω, then dσ (w0 (s, θ), w0 ) is an increasing function of s for all θ in R. Moreover, the sharp bounds tanh(s/2) 1 p ≤ tan dσ (w(s, θ), w0 ) 2 2 µΩ (w0 ) + tanh(s/2) µΩ (w0 ) − 1 tanh(s/2) p ≤ . µΩ (w0 ) − tanh(s/2) µ2Ω (w0 ) − 1

hold. (b) If dσ (w0 (s, θ), w0 ) is an increasing function of s for each w0 in Ω and all θ in R, then Ω is spherically convex. The proof of Theorem 9.1 is given in [17].


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Example 9.2. Consider the function kα defined in Example 8.7. For w0 = 0, µΩα (w0 ) = 1/α, α tanh(s/2) √ w0 (s, 0) = 1 − tanh(s/2) 1 − α2 √

2

is the hyperbolic arc length parametrization of [0, 1+ α1−α ), and the upper bound is equal to α tanh(s/2) 1 √ = tan dσ (w0 (s, 0), 0). 2 1 − tanh(s/2) 1 − α2 This shows that the upper bound is sharp. Similarly, −α tanh(s/2) √ w0 (s, π) = 1 + tanh(s/2) 1 − α2 √

2

is the hyperbolic arc length parametrization of ( −1+ α1−α , 0], and the lower bound is equal to 1 α tanh(s/2) √ = tan dσ (w0 (s, π), 0). 2 2 1 + tanh(s/2) 1 − α Hence, the lower bound is also sharp. Theorem 9.3. Suppose Ω is a hyperbolic region in C∞ . (a) If Ω is spherically convex and w0 ∈ Ω, then dσ (w0 (s, θ1 ), w0 (s, θ2 )) is an increasing function of s whenever eiθ2 6= eiθ1 . (b) If there exists w0 ∈ Ω such that dσ (w0 (s, θ1 ), w0 (s, θ2 )) is an increasing function of s whenever eiθ2 6= eiθ1 , then Ω is spherically convex. The reader can consult [17] for a proof of Thorem 9.3. Geometrically, Theorem 9.3(a) indicates that in a spherically convex region Ω, two hyperbolic geodesics starting off in different directions from a point w0 in Ω will spread farther apart relative to the spherical distance. If Ω is spherically convex, then so is every hyperbolic disk and conversely. This follows from the analog of Study’s theorem for spherically convex functions; see [19].

10. Hyperbolic geometry In this section, we indicate similar monotonicity properties for hyperbolic polar coordinates in hyperbolically convex regions. Because of the numerous similarities with the Euclidean and spherical cases, we present even fewer details in this situation. It is convenient to introduce the notation λΩ (w)|dw| 1 νΩ (w) = = (1 − |w|2 )λΩ (w) λD (w)|dw| 2 for the density of the hyperbolic metric of a region Ω ⊂ D relative to the background hyperbolic metric λD (w)|dw|. A simply connected region Ω in D is called hyperbolically convex (relative to the background hyperbolic geometry on D) if for all points z, w ∈ Ω the arc of

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the hyperbolic geodesic in D connecting z and w also lies in Ω. A holomorphic and univalent function f defined on D with f (D) ⊂ D is called hyperbolically convex if its image f (D) is a hyperbolically convex subset of D. Hyperbolically convex functions have been studied by a number of authors [7], [8], [12], [13], [14], [17], [20], [22]. The related concept of hyperbolically 1-convex functions was investigated in [9]. There are known characterizations of hyperbolically convex functions. For example, a holomorphic and locally univalent function f with f (D) ⊂ D is hyperbolically convex if and only if [12] ( ) zf ′′ (z) 2zf ′ (z)f (z) + (10.1) Re 1 + ′ ≥0 f (z) 1 − |f (z)|2 for all z in D. Mejia and Pommerenke [22] (also see [14]) showed that a holomorphic and locally univalent function f with f (D) ⊂ D is hyperbolically convex if and only if ) ( z+ζ 2zf ′ (z)f (ζ) 2zf ′ (z) >0 − + (10.2) Re f (z) − f (ζ) z − ζ 1 − f (ζ)f (z) for all z, ζ in D. This is the hyperbolic analog of (8.2). Similar to the proof of Corollary 8.2, we obtain the following characterization from (10.2). Theorem 10.1. A holomorphic and locally univalent function f with f (D) ⊂ D is hyperbolically convex if and only if ) ( zf ′ (z) − ζf ′ (ζ) zf ′ (z)f (ζ) + ζf ′ (ζ)f (z) >0 + (10.3) Re f (z) − f (ζ) 1 − f (ζ)f (z) for all z, ζ in D. These two-point characterizations can be used to derive monotonicity properties of hyperbolic polar coordinates on hyperbolically convex regions in D. Theorem 10.2. Let Ω ⊂ D. (a) If Ω is hyperbolically convex and w0 ∈ Ω, then dD (w0 (s, θ), w0 ) is an increasing function of s for all θ in R. Moreover, we have the following sharp bounds: 2 tanh(s/2) q νΩ (w0 )(1 + tanh(s/2)) + νΩ2 (w0 )(1 + tanh(s/2))2 − 4 tanh(s/2)

1 ≤ tanh dD (w0 (s, θ), w0 ) 2 2 tanh(s/2) q ≤ . 2 2 νΩ (w0 )(1 − tanh(s/2)) + νΩ (w0 )(1 − tanh(s/2)) + 4 tanh(s/2)

(b) If dD (w0 (s, θ), w0 ) is an increasing function of s for each w0 in Ω and all θ in R, then Ω is hyperbolically convex.


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Example 10.3. For 0 < α ≤ 1, the hyperbolic half-plane √ 1 − α2 1 Hα = D \ w : w + ≤ α α is hyperbolically convex and

Kα (z) =

1−z+

p

2αz

(1 − z)2 + 4α2 z

maps D conformally onto Hα . When w0 = 0, νHα (0) = 1/α, w0 (s, 0) = Kα (tanh(s/2)) is the hyperbolic arc length parametrization of [0, 1), and the upper bound is equal to 2α tanh(s/2) 1 p = tanh dD (w0 (s, 0), 0). 2 2 2 1 − tanh(s/2) + (1 − tanh(s/2)) + 4α tanh(s/2)

This shows that the upper bound is sharp. Similarly, w0 (s, π) = kα (− tanh(s/2)) √ −1− 1−α2 , 0], and the lower bound is the hyperbolic arc length parametrization of ( α is equal to 1 + tanh(s/2) +

p

2α tanh(s/2)

1 = tanh dD (w0 (s, π), 0). 2 (1 + tanh(s/2))2 − 4α2 tanh(s/2)

Thus, the lower bound is also sharp.

The proof of Theorem 10.4 below is analogous to the proof of Theorem 9.3; the characterization (10.3) for hyperbolically convex functions is used in place of (8.2). Theorem 10.4. Suppose Ω ⊂ D. (a) If Ω is hyperbolically convex and w0 ∈ Ω, then dD (w0 (s, θ1 ), w0 (s, θ2 )) is an increasing function of s for all eiθ2 6= eiθ1 . (b) If there exists w0 ∈ Ω such that dD (w0 (s, θ1 ), w0 (s, θ2 )) is an increasing function of s whenever eiθ2 6= eiθ1 , then Ω is hyperbolically convex. Ω ⊂ D is hyperbolically convex if and only if every hyperbolic disk DΩ (w0 , r) is hyperbolically convex as a subset of D. This is a direct consequence of the analog of Study’s Theorem for hyperbolically convex functions; see [20] and [12].

11. Concluding remarks Relative to the background geometry (hyperbolic, Euclidean, or spherical) hyperbolic geodesics and hyperbolic disks have similar behavior in convex regions. Moreover, there are numerous similarities between conformal maps of the unit disk onto convex regions in each of the three geometries, although formulas for spherical or hyperbolic convexity can be more complicated than those for Euclidean convexity because the spherical plane and the hyperbolic plane have nonzero curvature. It is possible to obtain results for Euclidean convexity as a limit of corresponding results for spherical convexity; for example, see

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Kim-Minda [6] for an illustration of the method. In the same manner Euclidean results can be obtained as the limit of hyperbolic convexity results. Holomorphic functions defined on the unit disk can be viewed as maps from D to the Euclidean plane. Bounded holomorphic functions on the unit disk can be regarded as maps from D to the hyperbolic plane, provided they are scaled to be bounded by one. Finally, meromorphic functions on D can be considered as maps into the spherical plane. Sometimes connections between classical results can be made by adopting this geometric view. This paper showed the close connection between Euclidean convexity, spherical convexity and hyperbolic convexity. Also, by adopting this geometric viewpoint it is possible to recognize there should be analogs of classical results for holomorphic functions for maps into the other two geometries. Some other function theory papers that relate to comparisons between hyperbolic, Euclidean and spherical geometry are [5], [23], [24], [25].

References [1] A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Proceedings of the International Workshop on Quasiconformal Mappings and their Applications, Narosa Publishing House, India, (2006), 159-206. [2] P. Duren, Univalent functions, Grundlehern Math. Wiss. 259, Springer, New York 1983. [3] M. Finkelstein, Growth estimates for convex functions, Proc. Amer. Math. Soc. 18 (1967), 412-418. [4] R. Fournier, J. Ma and S. Ruscheweyh, Convex univalent functions and omitted values, Approximation Theory, 21 (1998), 225-241. [5] S. Kim and D. Minda, A geometric approach to two-point comparisons for hyperbolic and euclidean geometry on convex regions, J. Korean Math. Soc., 36 (1999), 1169-1180. [6] S. Kim and D. Minda, The hyperbolic metric and spherically-convex regions, J. Math. Kyoto Univ., 41 (2001), 285-302. [7] S. Kim and T. Sugawa, Characterizations of hyperbolically convex regions, J. Math. Anal. Appl. 309 (2005), 37-51. [8] W. Ma, D. Mejia and D. Minda, Distortion theorems for hyperbolically and spherically k-convex functions, Proc. of an International Conference on New Trends in Geometric Function Theory and Application, R. Parvathan and S. Ponnusamy (editors), World Scientific, Singapore, 1991, 46-54. [9] W. Ma, D. Mejia and D. Minda, Hyperbolically 1-convex functions, Ann. Polon. Math., 84 (2004), 185-202. [10] W. Ma and D. Minda, Euclidean linear invariance and uniform local convexity, J. Austral. Math. Soc. 52 (1992), 401-418. [11] W. Ma and D. Minda, Spherical linear invariance and uniform local spherical convexity, Current Topics in Geometric Function Theory, H. M. Srivastava and S. Owa (editors), World Scientific, Singapore, 1993, 148-170. [12] W. Ma and D. Minda, Hyperbolically convex functions, Ann. Polonici Math. 60 (1994), 81-100. [13] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Australian Math. Soc., 58 (1995), 73-93. [14] W. Ma and D. Minda, Hyperbolically convex functions II, Ann. Polonici Math. 71 (1999), 273-285. [15] W. Ma and D. Minda, Two-point distortion theorems for spherically-convex functions, Rocky Mtn. J. Math., 30 (2000), 663-687.


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[16] W. Ma and D. Minda, Euclidean properties of hyperbolic polar coordinates, Comput. Methods Funct. Theory, to appear. [17] W. Ma and D. Minda, Geometric properties of hyperbolic polar coordinates, submitted. [18] D. Mejia and D. Minda, Hyperbolic geometry in k-convex regions, Pacific J. Math., 141 (1990), 333-354. [19] D. Mejia and D. Minda, Hyperbolic geometry in spherically k-convex regions, Computational Methods and Function Theory, Proceedings, Valparaiso, Chili, S. Ruscheweyh, E. B. Saff, L. C. Salinas and R. S. Varaga (editors), Lecture Notes in Mathematics, Vol. 1435, Springer-Verlag, New York, 1990, 117-129. [20] D. Mejia and D. Minda, Hyperbolic geometry in hyperbolically k-convex regions, Rev. Columbiana Math., 25 (1991), 123-142. [21] D. Mejia and Ch. Pommerenke, On spherically convex functions, Michigan Math. J. 47 (2000), 163-172. [22] D. Mejia and Ch. Pommerenke, On hyperbolically convex functions, J. Geom. Anal. 10 (2000), 365-378. [23] D. Minda, Bloch constants, J. Analyse Math. 4 (1982), 54-84. [24] D. Minda, Estimates for the hyperbolic metric, Kodai Math. J., 8 (1985), 249-258. [25] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math., 49 (1987), 90-105. [26] R. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l’Universite de Montréal, Montréal, 1982. [27] T. Sheil-Small, On convex univalent functions, J. London Math. Soc. 1 (1969), 483-492. [28] D. J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, Cambridge 1950. [29] E. Study, Konforme Abbildung einfachzusammenhängerder Bereiche, Vorlesungen u ¨ber ausgewählte Gegenstande, Heft 2, Leipzig und Berlin 1913. [30] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775-777. W. Ma E-mail: [email protected] Address: School of Integrated Studies, Pennsylvania College of Technology, Williamsport, PA 17701, USA D. Minda E-mail: [email protected] Address: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA


Quasiminimizers and Potential Theory Olli Martio Abstract. Quasiminimizers are almost minimizers of variational integrals. Although quasiminimizers do not form a sheaf and do not provide a unique solution to the Dirichlet problem it is shown that they form an interesting basis for a potential theory. Quasisuperminimizers and their Poisson modifications are considered as well as their convergence properties. Special attention is devoted to the theory on the real line. Keywords. quasiminimizers, quasisuperminimizers, quasisuperharmonic functions. 2000 MSC. Primary: 31C45; Secondary: 35J20, 35J60.


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2. Case n = 1

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3. Properties of quasiminimizers

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4. Quasisuperminimizers, Poisson modifications and regularity

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5. More about n = 1

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6. Quasisuperharmonic functions

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7. Appendix 1

200

8. Appendix 2

202

References

205

1. Introduction Quasiminimizers minimize a variational integral only up to a multiplicative constant. More precisely, let Ω ⊂ Rn be an open set, K ≥ 1 and 1 ≤ p < ∞. In the case of the p-Dirichlet integral, a function u belonging to the Sobolev space 1,p Wloc (Ω) is a (p, K)-quasiminimizer or a K-quasiminimizer, if Z Z p (1.1) |∇u| dx ≤ K |∇v|p dx Ω′

Version October 19, 2006.

Ω′

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for all functions v ∈ W 1,p (Ω′ ) with v − u ∈ W01,p (Ω′ ) and for all open sets Ω′ with a compact closure in Ω. A 1-quasiminimizer, called a minimizer, is a weak solution of the corresponding Euler equation (1.2)

div(|∇u|p−2 ∇u) = 0.

Clearly being a weak solution of (1.2) is a local property. However, being a Kquasiminimizer is not a local property as one-dimensional examples easily show. This indicates that the theory for quasiminimizers differs from the theory for minimizers and that there are some unexpected difficulties. Quasiminimizers have been previously used as tools in studying the regularity of minimizers of variational integrals, see [GG1–2]. The advantage of this approach is that it covers a wide range of applications and that it is based only on the minimization of the variational integrals instead of the corresponding Euler equation. Hence regularity properties as Hölder continuity and Lp -estimates are consequences of the quasiminimizing property. It is an important fact that nonnegative quasiminimizers satisfy the Harnack inequality, see [DT]. Instead of using quasiminimizers as tools, the objective of these lectures is to show that quasiminimizers have a fascinating theory themselves. In particular, they form a basis for nonlinear potential theoretic model with interesting features. From the potential theoretic point of view quasiminimizers have several drawbacks: They do not provide unique solutions of the Dirichlet problem, they do not obey the comparison principle, they do not form a sheaf and they do not have a linear structure even when the corresponding Euler equation is linear. However, quasiminimizers form a wide and flexible class of functions in the calculus of variations under very general circumstances. Observe that the quasiminimizing condition (1.1) applies not only to one particular variational integral but the whole class of variational integrals at the same time. For example, if a variational kernel F (x, ∇u) satisfies (1.3)

α|h|p ≤ F (x, h) ≤ β|h|p

for some 0 < α ≤ β < ∞, then the minimizers of Z F (x, ∇u) dx

are quasiminimizers of the p-Dirichlet integral Z (1.4) |∇u|p dx.

Hence the potential theory for quasiminimizers includes all minimizers of all variational integrals similar to (1.4). The essential feature of the theory is the control provided by the bounds in (1.3). For example, the coordinate functions of a quasiconformal or, more generally, quasiregular mapping are quasiminimizers of the n-Dirichlet integral Z |∇u|n dx

Quasiminimizers and Potential Theory

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in all dimensions n = 2, 3, . . . . Recently quasiminimizers have been considered in metric measure spaces. This means that a metric space (X, d) is equipped with a Borel measure µ which satisfies some standard assumptions like the doubling property. The Sobolev space W 1,p is replaced by the so called Newtonian space N 1,p which for Rn and the Lebesgue measure reduces to W 1,p . We do not consider metric spaces here although most of the results hold in this case under appropriate conditions. For this theory see [KM2].

2. Case n = 1 For n = 1 the definition (1.1) can be written in the following form: Let (a, b) 1,p be an open interval in R and u ∈ Wloc (a, b). Then u is a (p, K)-quasiminimizer, or K-quasiminimizer for short, if for all closed intervals [c, d] ⊂ (a, b) Zd

(2.1)

′ p

|u | dx ≤ K

c

whenever u − v ∈

Zd

|v ′ |p dx

c

W01,p (c, d).

Now affine functions are minimizers, i.e. 1-quasiminimizers, for every p ≥ 1. This fact can be easily deduced from the one dimensional version of (1.2) if p > 1. For p = 1 this is trivial. Moreover, affine functions are the only minimizers for p > 1. Thus choosing v(x) = α(x − c) + β where α = (u(d) − u(c))/(d − c), β = u(c)

we see that u − v ∈ W01,p (c, d) and (2.1) yields (2.2)

Zd

|u′ |p dx ≤ K

c

|u(d) − u(c)|p , |d − c|p−1

see [GG2]. The inequality (2.2) gives another definition for a K-quasiminimizer u: the function u is a locally absolutely continuous function in (a, b) that satisfies (2.2) on each subinterval [c, d] of (a, b). Observe that u ∈ W 1,p (c, d) in a bounded open interval (c, d) means that u is absolutely continuous on [c, d] with (2.3)

Zd

|u′ |p dx < ∞.

c

If u ∈ W 1,p (c, d) and u − v ∈ W01,p (c, d), then v ∈ W 1,p (c, d) and v(c) = 1,p u(c), v(d) = u(d). Functions u ∈ Wloc (a, b) are simply locally absolutely continuous functions on (a, b) such that (2.3) holds in each subinterval [c, d] ⊂ (a, b). We leave the following lemma as an exercise.

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Lemma 2.4. Suppose that u is a (p, K)-quasiminimizer in (a, b). Then u is a monotone function. If p > 1 then u is either strictly monotone or constant. The following lemma is more difficult to prove. It does not hold for p = 1. Lemma 2.5. Let u be a (p, K)-quasiminimizer, p > 1, in an interval (a, b). If b < ∞, then u has a continuous extension to b and (2.2) holds in all intervals [c, d] ⊂ (a, b]. Proof. We may assume that u is increasing, b = 1, [0, 1] ⊂ (a, b] and u(0) = 0. Fix 0 ≤ c < t < 1. Now ! p1 ! p1 Zt Zt Zt ≤ (t − c)(p−1)/p u′p dx u′ dx ≤ (t − c)(p−1)/p u′p dx c

≤ K

1 p

(t − c) t

p−1 p

p−1 p

c Zt

0

′

u dx = K

1 p

0

c 1− t

p−1 p

Zt

u′ dx

0

where we have used the Hölder inequality and (2.2). Next we choose c = 1 − 1 (2p K) 1−p . Then 0 < c < 1 and letting t ∈ (c, 1) we obtain p−1 p−1 c p 1 1− < (1 − c) p = 1 . t 2K p The above inequalities yield Zt Zt Zc Zt Zc 1 u′ dx u′ dx = u′ dx + u′ dx ≤ u′ dx + 2 0

0

c

and hence u(t) = u(t) − u(0) =

0

0

Zt

u′ dx ≤ 2

0

Zc

u′ dx = 2u(c).

0

Since u is increasing, letting t → 1 we obtain

u(b) = u(1) = lim u(t) ≤ 2u(c) < ∞ t→1

and the last assertion of the lemma now follows from (2.2). Lemma 2.5 shows that the natural domain of definition for a 1-dimensional quasiminimizer is the closed interval [a, b]. Example 2.6. The function u(x) = xα , α > 1/2, is a (2, K)-quasiminimizer in [0, ∞) for K = α2 /(2α − 1). This is a rather easy computation. Note that u(x) = x1/2 is not a (2, K)-quasiminimizer in [0, ∞) (and in (0, ∞)) since u′ does not belong to L2 (0, 1). We will consider one dimensional quasiminimizers again in Chapter 5. The one dimensional case was first studied in [GG2].


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3. Properties of quasiminimizers We start with a basic regularity property. Theorem 3.1. Suppose that u is a (p, K)-quasiminimizer in Ω ⊂ Rn , p > 1. If 0 < r < R are such that the ball B(x, 2R) ⊂ Ω, then osc(u, B(x, r)) ≤ C(r/R)α osc(u, B(x, R))

where C < ∞ and α ∈ (0, 1] depend on p, n and K only.

In particular Theorem 3.1 implies that u is locally Hölder continuous in Ω. Another important property is the Harnack inequality. Theorem 3.2. Let u be as in Theorem 3.1 and u ≥ 0. Then in each ball B(x, R) such that B(x, 2R) ⊂ Ω sup u ≤ C inf u B(x,R)

B(x,R)

where the constant C depends on p, n and K only. 1,p Since quasiminimizers are functions in Wloc (Ω) only, Theorem 3.1 also means that they can be made continuous after redefinition on a set of measure zero.

For p > n, and hence for all p > 1 for n = 1, Theorem 3.1 follows from the Sobolev embedding lemma. Note that for p = 1 = n quasiminimizers are continuous but they need not be Hölder continuous. We do not prove Theorems 3.1 and 3.2 here. The proof for Theorem 3.2 in the case n = 1 is relatively simple, see [GG2]. For the proof of Theorem 3.1 the De Giorgi method can be used. The basic tool is the Sobolev type inequality !1/t !1/p Z Z − |u|t dx − |∇u|p dx ≤ cr B(x,r)

B(x,r)

for functions u ∈ W01,p (B(x, r)) where t > p. The main difficulty is to prove that u is locally essentially bounded. For the proof see [GG1], [GG2] and [KS]. In the paper [KS] metric measure spaces are considered and hence the regularity proof of [KS] uses minimal assumptions. In the general case n ≥ 2 the proof for Theorem 3.2 is rather complicated, see [DT] and [KS]. The proof makes use of the Krylov–Safonov covering argument [KSa]. Very recently it has turned out that the Moser method can be employed to prove Theorems 3.1 and 3.2 for quasiminimizers even in metric measure spaces, see [Ma]. In Potential Theory the Harnack inequality and Harnack’s principle are essentially equivalent. From Theorem 3.1 and 3.2 it easily follows (p > 1): Suppose that (ui ) is an increasing sequence of K-quasiminimizers in a domain Ω. If lim ui (x0 ) < ∞ at some point x0 ∈ Ω, then lim ui is a K-quasiminimizer. We will return to the proof of this fact in the next chapter and in Appendix 2.

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4. Quasisuperminimizers, Poisson modifications and regularity 1,p Let Ω be an open set in Rn . A function u ∈ Wloc (Ω) is called a (p, K)quasisuperminimizer, or a K-quasisuperminimizer, if Z Z p |∇u| dx ≤ K |∇v|p dx (4.1) Ω′

Ω′

1,p holds for all open Ω′ ⊂⊂ Ω and all v ∈ Wloc (Ω) such that v ≥ u a.e. in Ω′ and v − u ∈ W01,p (Ω′ ).

Remarks 4.2. (a) A 1-quasisuperminimizer is called a superminimizer. (b) A superminimizer is a supersolution of the p-harmonic equation ∇ · (|∇u|p−2 ∇u) = 0,

i.e. u satisfies

Z

|∇u|p−2 ∇u · ∇ϕ dx ≥ 0

Ω

for all non–negative ϕ ∈ C0∞ (Ω), see [HKM] for this theory. Observe that for p = 2 every superharmonic (in the classical sense) function u is a superminimizer 1,2 provided that u belongs to Wloc (Ω), however, a superharmonic function need not 1,2 belong to Wloc (Ω). For n = 2 the classical example is u(x) = − log |x| which is superharmonic in R2 but does not belong to W 1,2 (B(0, 1)). We return to this problem in Chapter 6. (c) The inequality (4.1) can be replaced by several other inequalities, for example Z Z p |∇u| dx ≤ K |∇v|p dx, ′

Ω′ \E

Ω′ \E

where E ⊂ Ω \ {u 6= v} is any measurable set. For the list of these conditions see [B] and [KM2]. A function u is called a K-quasisubminimizer if −u is a K-quasisuperminimizer.

Note that if u is a K-quasisuperminimimizer, then αu and u + β are Kquasisuperminimizers when α ≥ 0 and β ∈ R. However, the sum of two Kquasisuperminimizers need not be a K-quasisuperminimizer even in the case p = 2.

Lemma 4.3. Suppose that ui , i = 1, 2, are Ki -quasisuperminimizers in Ω. Then min(u1 , u2 ) is a K-quasisuperminimizer in Ω with K = min(K1 K2 , K1 + K2 ). Proof. We prove that u = min(u1 , u2 ) is a K-quasisuperminimizer with K ≤ K1 K2 ; this inequality is important in applications. The proof for K ≤ K1 + K2 is similar, see [KM2]. To this end let Ω′ ⊂⊂ Ω be an open set and v − u ∈ W01,p (Ω′ ), v = u in Ω \ Ω′ and v ≥ u. Now


Z

Z

p

|∇u| dx =

Ω′

195

Z

p

|∇u1 | dx +

|∇u2 |p dx

{u1 >u2 }∩Ω′

{u1 ≤u2 }∩Ω′

and write w = max(min(u2 , v), u1 ). Then w ≥ u1 a.e. in Ω′ , w − u1 ∈ W01,p (Ω′ ) and w = u1 if u1 > u2 . Thus the quasisuperharmonicity of u1 , see Remark 4.2 (c), yields Z Z p |∇u1 | dx ≤ K1 |∇w|p dx

{u1 ≤u2 }∩Ω′

{u1 ≤u2 }∩Ω′

Z

= K1

|∇v| dx + K1

Ω′

|∇v|p dx

{u1 ≤u2 }∩{vu2 }∩Ω′

{u1 ≤u2 }∩{v≥u2 }∩Ω′

Z

|∇u2 |p dx.

{u1 ≤u2 }∩{v≥u2 }∩Ω′

{u1 ≤u2 }∩{v 1, if and only if min(u, c) is a K-quasisuperminimizer for each c ∈ R.

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The Poisson modification is an important tool in Potential Theory. In the classical case p = 2 this means the following: Let u be superharmonic in Ω 1,2 and u ∈ Wloc (Ω) (this assumption is not actually needed). If Ω′ ⊂⊂ Ω is an open set let h be a minimizer (harmonic) in Ω′ with boundary values u, i.e. u − h ∈ W01,2 (Ω′ ). The Poisson modification of u in Ω′ is defined as h in Ω′ , ′ (4.7) P (u, Ω ) = u in Ω \ Ω′ . Then P (u, Ω′ ) is a superharmonic function in Ω, P (u, Ω′ ) ≤ u and P (u, Ω′ ) is harmonic in Ω′ . This theory works well for superminimizers for all p > 1, see [HKM].

For quasisuperminimizers the above method does not work as above. In particular there exists a K-quasisuperminimizer (even a K-quasiminimizer) u such that the function P (u, Ω′ ) in (4.7) is not a K ′ -quasisuperminimizer for any K ′ < ∞. The example is one dimensional and requires some computation. However, there are two replacements for P (u, Ω′ ). The first Poisson modification is a modification of (4.7). Let u be a Kquasisuperminimizer in Ω, p > 1, and Ω′ ⊂⊂ Ω an open set. Let h be the minimizer with boundary values u in Ω′ , i.e. u − h ∈ W01,p (Ω′ ). Observe that such a unique function h exists - this is a basic result in the theory of Sobolev spaces, see e.g. [HKM]. Let min(u, h) in Ω′ , ′ (4.8) P1 (u, Ω ) = u in Ω \ Ω′ . Theorem 4.9. [KM2] The function P1 (u, Ω′ ) is K-quasisuperminimizer in Ω. By the construction of P1 (u, Ω′ ), P1 (u, Ω′ ) ≤ u in Ω. Note also that if u is a K-quasiminimizer, then P1 (u, Ω′ ) is also a K-quasiminimizer in Ω′ by Corollary 4.4. Next we consider another possibility for a Poisson modification. For this we need to consider obstacle problems. The obstacle method is the most important method in the nonlinear potential theory. Let Ω ⊂ Rn be an open set and u ∈ W 1,p (Ω). Write Ku+ (Ω) = {v ∈ W 1,p (Ω) : v − u ∈ W01,p (Ω), v ≥ u a.e. in Ω}.

The class Ku− is defined similarly but v ≥ u is replaced by v ≤ u. The following result is well-known. Lemma 4.10. [HKM] The obstacle problem Z inf |∇v|p dx, p > 1, − v∈Ku (Ω)

Ω

−

has a unique solution u ∈ uous if u is continuous.

Ku− (Ω).

Moreover, u− is a subminimizer and contin-


197

A similar result holds for the class Ku+ (Ω) and the solution is a superminimizer.

Suppose now that u is a K-quasisuperminimizer in Ω and Ω′ ⊂⊂ Ω is open. Let u− be the solution to the Ku− (Ω′ )-obstacle problem. Define − u in Ω′ , ′ (4.11) P2 (u, Ω ) = u in Ω \ Ω′ . Theorem 4.12. The function P2 (u, Ω′ ) is a K-quasisuperminimizer in Ω, a subminimizer in Ω′ and a K-quasiminimizer in Ω′ . The proof for Theorem 4.12 is in Appendix 1.

Superharmonic functions in the classical potential theory are lower semicontinuous. It turns out that quasisuperminimizers can be defined pointwise and the resulting function is lower semicontinuous. Theorem 4.13. [KM2] Suppose that u is a K-quasisuperminimizer in Ω, p > 1. Then the function u∗ : Ω → (−∞, ∞] defined by u∗ (x) = lim ess inf u r→0 B(x,r)

is lower semicontinuous and u∗ = u a.e. (u and u∗ are the same function in 1,n Wloc (Ω)). The proof for Theorem 4.13 is based on the De Giorgi method that is used to prove a weak Harnack inequality !1/σ Z − uσ dx

≤ c ess inf u B(x,3r)

B(x,r)

for a K-quasisuperharmonic function u ≥ 0. Here B(x, 5r) ⊂ Ω and c and σ > 0 depend only on n, p > 1 and K. The following is Harnack’s principle for quasisuperminimizers. Theorem 4.14. Suppose that (ui ) is an increasing sequence of K-quasisuperminimizers in Ω, p > 1. If u = lim ui is such that either (i) u is locally bounded above or 1,p (ii) u ∈ Wloc (Ω), then u is a K-quasisuperminimizer. The proof for Theorem 4.14 is somewhat tedious. It is presented in Appendix 2.

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5. More about n = 1 We take a closer look at the case n = 1. Recall that a superminimizer is a 1-quasisuperminimizer. The next result is easy to prove, see [MS]. Lemma 5.1. Suppose that u : [a, b] → R is a superminimizer, p > 1. Then u is a concave function. ¿From Lemma 5.1 it follows that a superminimizer is a Lipschitz function in each interval [c, d] ⊂⊂ (a, b). It need not be a Lipschitz function in [a, b].

How regular are K-quasiminimizers and K-quasisuperminimizers? The answer is not known for n ≥ 2 but for n = 1 some exact answers have been obtained. Let 1 < p < ∞ and let ω : [a, b] → [0, ∞) be a weight in L1 (a, b). Set !−1 ! p1 Z Z − ω dx Gp (ω) = sup − ω p dx I

I

I

where the supremum is taken over all intervals I ⊂ [a, b]. If Gp (ω) < ∞, then ω is said to belong to the Gp -class of Gehring. For a non-constant quasiminimizer u in [a, b] set Kp (u) = sup [c,d]

Zd c

|u′ |p dx

(d − c)p−1 |u(d) − u(c)|p

where the supremum is taken over all intervals [c, d] ⊂ [a, b]. In other words, Kp (u) is the least constant in (2.2). The following lemma is immediate. Lemma 5.2. Let u : [a, b] → R be absolutely continuous and non-constant with u′ ≥ 0 a.e. Then u is Kp (u)-quasiminimizer with exponent p, p > 1, if and only 1 if u′ belongs to the Gp -class with Gp (u′ ) = Kp (u) p . A. Korenovskii [K] has determined the optimal higher integrability bound p0 = p0 (p, K) for a weight ω in the Gp -class. Hence Lemma 5.2 enables us to determinate the optimal integrability bound for the derivative of a Kp -quasiminimizer. Let γp,t : [p, ∞) → R, p > 1, t > 1, be the function x−p x p ( ), γp,t (x) = 1 − tp x x−1 and let p1 (p, t) ∈ (p, ∞) be the unique solution of the equation γp,t (x) = 0. For the properties of p1 (p, t) see [DS, Section 3]. Theorem 5.3. Suppose that u is a (p, K)-quasiminimizer, p ≥ 1, K ≥ 1, in 1 [a, b]. Then u′ ∈ Ls (a, b) for 1 ≤ s < p1 (p, K p ) if p > 1 and K > 1, u′ ∈ L1 (a, b) if p = 1 and u′ ∈ L∞ (a, b) if p > 1 and K = 1. All these integrability conditions are sharp.


199

Proof. Let first p > 1 and K > 1. We may assume that u is increasing. By 1 Lemma 5.2, Gp (u′ ) = Kp (u) p and from [K, Theorem 2] we conclude that u′ ∈ 1 Ls (a, b) for 1 ≤ s < p1 (p, Gp (u′ )) = p1 (p, Kp (u) p ). If p > 1 and K = 1, then u is affine and hence u′ ∈ L∞ (a, b). For p = 1, u′ trivially belongs to L1 (a, b). 1

To see that the bound α = p1 (p, K p ) is sharp for p > 1 and K > 1 it suffices to consider the interval [0, 1]. The function α−1 α u(x) = x α , x ∈ [0, 1], α−1 1

has the derivative u′ (x) = x− α and a direct computation shows that u′ belongs 1 to the Gehring class with Gp (u′ ) = K p , see [DS, Proposition 2.3]. By Lemma 5.2 u is a K-quasiminimizer. On the other hand, u′ does not belong to Lα (0, 1). This 1 shows that the open ended upper bound p1 (p, K p ) is sharp. For p = 1 the integrability of u′ cannot be improved since every increasing absolutely continuous function u is a 1-quasiminimizer. The theorem follows. Remark 5.4. For p = 2, 1

and hence

p1 (2, t) = 1 + t(t2 − 1)− 2 , t > 1, 1

1

1

p1 (2, K 2 ) = 1 + K 2 (K − 1)− 2 , K > 1. In [MS] the inverse functions of one dimensional quasiminimizers are also considered.

6. Quasisuperharmonic functions In the nonlinear potential theory superharmonic functions can be defined in many ways. The most natural definition uses the comparison principle, see (6.3) below. Let p > 1 and let Ω be an open set in Rn . A function u : Ω → (−∞, ∞] is said to be superharmonic, i.e. (p, 1)-superharmonic, if (6.1) u is lower semicontinuous, (6.2) u ≡ 6 ∞ in any component of Ω, ′ (6.3) for each open set Ω′ ⊂⊂ Ω and each minimizer h ∈ C(Ω ), i.e. (p, 1)quasiminimizer, the inequality u ≥ h on ∂Ω′ implies u ≥ h in Ω′ . For the theory of superharmonic functions in the nonlinear situation see [HKM]. Superharmonic functions can also be defined with the help of minimizers, see [HKM, Theorem 7.10] and [HKM, Corollary 7.20]. For other definitions see [B]. Lemma 6.4. Suppose that u : Ω → (−∞, ∞] satisfies (6.1) and (6.2). Then u is (p, 1)-superharmonic if and only if there is an increasing sequence (u∗i ) of (p, 1)-quasisuperminimizers, i.e. superminimizers, with u = lim u∗i . Here u∗i is the lower semicontinuous representative of a superminimizer ui .

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Note that given a superharmonic function u the sequence u∗i = min(u, i), i = 1, 2, . . ., is the required sequence. In view of Lemma 6.4 the following definition for (p, K)-quasisuperharmonicity is natural. Definition 6.5. Let Ω ⊂ Rn be an open set, p > 1 and K ≥ 1. A function u : Ω → (−∞, ∞] is said to be (p, K)-quasisuperharmonic if there is an increasing sequence of K-quasisuperminimizers ui in Ω such that lim u∗i = u and u 6≡ ∞ in each component of Ω. Lemma 6.6. Suppose that u is a (p, K)-quasisuperharmonic function in Ω and locally bounded above. Then u is a (p, K)-quasisuperminimizer. Proof. By the definition for quasisuperharmonicity there is an increasing sequence of quasisuperminimizers u∗i : Ω → (−∞, ∞) such that lim u∗i = u. From Theorem 4.14 it follows that u is a K-quasisuperminimizer as required. Note that a (p, K)-quasisuperharmonic function is automatically lower semicontinuous as a limit of an increasing sequence of lower semicontinuous functions. Lemma 6.7. Let u be a K-quasisuperharmonic function in Ω and h a (continuous) minimizer in Ω. Then min(u, h) is a K-quasisuperminimizer (and hence K-quasisuperharmonic) in Ω. Proof. Let u∗i be an increasing sequence of K-quasisuperminimizers in Ω such that u∗i → u. Now min(u∗i , h) is lower semicontinuous and by Corollary 4.4, min(u∗i , h) is a K-quasisuperminimizer. Since min(u∗i , h) ≤ h, it follows that min(u, h) = lim min(u∗i , h) is a K-quasisuperharmonic function. By Lemma 6.6, min(u, h) is a K-quasisuperminimizer. Theorem 6.8. Suppose that u : Ω → (−∞, ∞] satisfies (6.1) and (6.2). Then u is a K-quasisuperharmonic if and only if min(u, c) is a K-quasisuperminimizer for each c ∈ R. Proof. The only if part follows from Lemma 6.7. For the sufficiency choose c = i, i = 1, 2, . . .. Then min(u, i) is a lower semicontinuous K-quasisuperminimizer and it follows from Definition 6.5 that u is a K-quasisuperharmonic function. The theory for K-quasisuperharmonic functions is still in its infancy. However, the following result was proved in [KM2]: A set C ⊂ Rn is said to be (p, K)-polar, if there is a neighborhood Ω of C and a (p, K)-quasisuperharmonic function u in Ω such that u(x) = ∞ for each x ∈ C. Then C is a (p, K)-polar set if and only if the p-capacity of C is zero. It has been previously known that a set C ⊂ Rn is a (p, 1)-polar set if and only if the p-capacity of C is zero. Hence allowing the freedom due to K ≥ 1 adds nothing new to the structure of polar sets.

7. Appendix 1 Proof for Theorem 4.12. That P2 (u, Ω′ ) is a subminimizer in Ω′ follows from Lemma 4.9. In order to show that w = P2 (u, Ω′ ) is a K-quasisuperminimizer


201

in Ω let Ω′′ ⊂⊂ Ω be open and v a function such that v − w ∈ W01,p (Ω′′ ) and v ≥ w in Ω′′ . We set v = w in Ω \ Ω′′ . For the K-quasisuperminimizing property of w it suffices to show Z Z p (a) |∇w| dx ≤ K |∇v|p dx. Ω′′ ∩{w 1 and let E, F be subsets of Hn . Then 1 Mp (∆(E, F ; Hn )) ≥ Mp (∆(E, F )). 2 11111111111111111111111111 00000000000000000000000000 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 F11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 G 11111111111111111111111111 00000000000000000000000000 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 E11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111 00000000000000000000000000 11111111111111111111111111

h

Figure 5: Cylinder with bases E and F . Example 4.12. Let E ⊂ {x ∈ Rn : xn = 0} be a Borel set, h > 0, F = E + hen . We define a cylinder G with bases E, F by G = {x ∈ Rn : (x1 , . . . , xn−1 , 0) ∈ E, 0 < xn < h}. Then Mp (∆(E, F ; G)) = mn−1 (E)h1−p = m(G)h−p .

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Proof. Choose ρ ∈ F(Γ) where Γ = ∆(E, F ; G)) and let γy be the vertical segment from y ∈ E. Then γy ∈ Γ. We note that 1/p + (p − 1)/p = 1, and hence by Hölder’s inequality Z p−1 Z p Z Z p p−1 1≤ ρ ds ≤ 1 ds ρ ds = h ρp ds. γy

γy

γy

γy

This holds for all y ∈ E and hence by the Fubini theorem Z Z Z mn−1 (E) p p . ρ dm ≥ ρ ds dmn−1 ≥ hp−1 Rn E γy Since the above holds for any ρ ∈ F(Γ),

mn−1 (E) . hp−1 Next we choose ρ = 1/h inside G and ρ = 0 otherwise. Then ρ is admissible for Γ and Z mn−1 (E) . ρp dm = Mp (Γ) ≤ hp−1 Rn Mp (Γ) ≥

Remark 4.13. In Example 4.12 the modulus is invariant under similarity mappings if and only if p = n. This is the reason why the case p = n is so important in the theory of quasiconformal mappings. Later in this section we will show that M(Γ) is a conformal invariant. n

n

Ring domains. A domain G in R is called a ring, if R \ G has exactly two components. If the components are E and F , we denote the ring by R(E, F ). In general, it is difficult to calculate the modulus of a given path family. Next two lemmas give us an important tool, letting us to obtain effective upper and lower bounds for the modulus in many situations.

b

a

Figure 6: Spherical ring with 0 < a < b < ∞.

Introduction to quasiconformal mappings in n-space

249

n

Lemma 4.14. [21, 7.5] Let 0 < a < b < ∞, A = Bn (b) \ B (a) and ΓA = ∆ S n−1 (a), S n−1 (b); A . Then

b 1−n . M(ΓA ) = ωn−1 log a

Proof. Let ρ ∈ F(ΓA ). For each unit vector y ∈ S n−1 let γy : [a, b] → Rn the radial line segment defined by γy (s) = sy. As in Example 4.12 by Hölder’s inequality we obtain Z b Z n Z b n−1 1 n n−1 ρ(sy) s ds 1 ≤ ρ ds ≤ ds a s a γy Z b n−1 b = log ρ(sy)n sn−1 ds. a a

By integrating over y ∈ S n−1 , we have Z b n−1 ρn dm. (4.15) ωn−1 ≤ log a n R Taking the infimum over all admissible ρ yields b n−1 M(ΓA ). ωn−1 ≤ log a Next we define ρ(x) = 1/ |x| log(b/a) for x ∈ A, and ρ(x) = 0 otherwise. Clearly ρ is admissible for ΓA , and hence Z Z b 1−n b −n b 1 n . ds = ωn−1 log M(ΓA ) ≤ ρ dm = ωn−1 log a a a s Rn n

Lemma 4.16. [21, 7.8] Let x0 ∈ R and let Γ be the family of all nonconstant paths through x0 . Then M(Γ) = 0. Proof. If x0 = ∞, the claim follows immediately from Corollary 4.7. If x0 6= ∞, we let

Γk = {γ ∈ Γ : |γ| ∩ S n−1 (x0 , 1/k) 6= ∅}.

We may assume that x0 = 0. Then for all R > 1/k

n Γk > ∆R , where ∆R = ∆ S n−1 (1/k), S n−1 (R); Bn (R) \ B (1/k) ,

and by Lemma 4.4 and Lemma 4.14 we have 1−n R →0 M(Γk ) ≤ M(∆R ) = ωn−1 log 1/k

as R → ∞, and thus M(Γk ) = 0. On the other hand, because Γ = by Lemma 4.3 (3) X M(Γ) ≤ M(Γk ) = 0. k

S

k

Γk we have

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n

n

Modulus in conformal mappings. Let G ⊂ R and f : G → R be a continuous function. Suppose that Γ is a family of paths in G. Then Γ′ = {f ◦γ : γ ∈ Γ} is a family of paths in f (G). Γ′ is called the image of Γ under f . Theorem 4.17. [21, 8.1] If f : G → f (G) is conformal, then M(f (Γ)) = M(Γ) for all path families Γ in G. Proof. By Lemma 4.16 we may assume that the paths of Γ, f (Γ) do not go through ∞. Let ρ1 ∈ F(f (Γ)), and define ρ(x) = ρ1 f (x) |f ′ (x)|

for x ∈ G and ρ(x) = 0 otherwise. Because f is a conformal mapping (see [21, 5.6]), Z Z Z ′ ρ ds = ρ1 f (x) |f (x)| |dx| = ρ1 ds ≥ 1 γ

γ

f ◦γ

for every locally rectifiable γ ∈ Γ. It follows that ρ ∈ F(Γ), and Z Z Z Z n n n M(Γ) ≤ ρ1 dm = ρ dm = ρ1 f (x) |Jf (x)| dm = Rn

f (G)

G

ρn1 dm

Rn

for all ρ1 ∈ F(f (Γ)), and thus M(Γ) ≤ M(f (Γ)). The inverse inequality follows from the fact that f −1 is conformal.

Lemma 4.18. Let A ⊂ Hn , B ⊂ (∁Hn ), Γ = ∆(A, B), and let Γ1 = ∆(A, ∂Hn ), Γ2 = ∆(B, ∂Hn ).

Then

M(Γ) ≤ 2−n M(Γ1 ) + M(Γ2 ) . In particular, the equality holds if A = g(B), where g is the reflection in Hn . Proof. Let ρ1 ∈ F(Γ1 ) and ρ2 ∈ F(Γ2 ). We note that if γ ∈ Γ is a rectifiable path, then γ has subpaths γ1 , γ2 such that γ1 ∈ Γ1 , γ2 ∈ Γ2 . Thus Z Z 1 1 ρ1 ds + ρ2 ds. 1≤ 2 γ1 2 γ2

We define ρ = ρ1 /2 + ρ2 /2. Now ρ is an admissible function for the curve family Γ and hence Z ρn dm. M(Γ) ≤ Rn

We may assume that ρ1 (z) = 0 for z ∈ / Hn , and ρ2 (z) = 0 for z ∈ Hn . As ρ = ρ1 /2 + ρ2 /2, we obtain Z Z Z n −n n −n ρn2 dm ρ1 dm + 2 ρ dm = 2 n n n n R \H H R Z Z = 2−n ρn1 dm + ρn2 dm . Rn

Rn


251

It follows that

M(Γ) ≤ 2−n M(Γ1 ) + M(Γ2 ) . Next we consider the case A = g(B). Let ρ be an admissible function for the path family Γ and denote ρ ◦ g by ρ¯. Now the function ρ + ρ¯ on Hn , ρˆ = 0 on ∁Hn , is admissible for the path family Γ1 . By the inequality (a + b)n ≤ 2n−1 (an + bn ) (for a, b ≥ 0) and the fact that M(Γ1 ) = M(Γ2 ) it follows that Z Z 1 n M(Γ1 ) ≤ ρˆ dm = (ρ + ρ¯)n dm 2 Rn Rn Z Z n−2 n n n−1 ≤ 2 (ρ + ρ¯ )dm = 2 ρn dm. Rn

Rn

Hence,

n

M(Γ1 ) + M(Γ2 ) = 2M(Γ1 ) ≤ 2

Z

ρn dm,

Rn

for any ρ admissible for the curve family Γ. By taking infimum over all admissible ρ, the claim follows. Capacity of a condenser. A condenser in Rn is a pair E = (A, C), where A is open in Rn and C is a compact subset of A. The p-capacity of E is defined by Z |∇u|p dm, 1 ≤ p < ∞, (4.19) capp E = inf u

A

where the infimum is taken over all nonnegative functions u in ACLp (A) with compact support in A and u|C ≥ 1. The n-capacity of E is called the conformal capacity of E and denoted by capE.

C

A Figure 7: Condenser E = (A, C). Lemma 4.20. [22, 7.9] For all condensers (A, C) in Rn (4.21) cap(A, C) = M ∆(C, ∂A; A) .

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Sets of zero capacity. A compact set E in Rn is said to be of capacity zero, denoted capE = 0, if there exists a bounded set A with E ⊂ A and cap(A, E) = n n 0. A compact set E ⊂ R , E 6= R is said to be of capacity zero if E can be mapped by a Möbius transformation onto a bounded set of capacity zero. Otherwise E is said to be of positive capacity, and we write capE > 0. n

Spherical symmetrizations. Let L be a ray from x0 to ∞ and E ⊂ R be a compact set. We define spherical symmetrization of E in L as the set E ∗ satisfying the following conditions: (1) x0 ∈ E ∗ if and only if x0 ∈ E, (2) ∞ ∈ E ∗ if and only if ∞ ∈ E, (3) For r ∈ (0, ∞) the set E ∗ ∩ S n−1 (x0 , r) is a closed spherical cap centered on L with the same (n − 1)-dimensional Lebesgue measure as E ∩ S n−1 (x0 , r) for E ∩ S n−1 (x0 , r) 6= ∅ and ∅ otherwise. We note that E ∗ is always compact and connected if E is. 111 000 000 111 000 111 000 111 000 111 000 111

E 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

L *

E

11111111111 L 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

Figure 8: Spherical symmetrization. Theorem 4.22. If E ∗ is the spherical symmetrization of E in a ray L, then (1) m(E ∗ ) = m(E), and (2) mn−1 (∂E ∗ ) ≤ mn−1 (∂E). Proof. (Outline, [7, p.224]) By Fubini’s theorem Z ∞ Z ∞ ∗ ∗ n−1 m(E ) = mn−1 (E ∩S (x0 , r))dr = mn−1 (E ∩S n−1 (x0 , r))dr = m(E), 0

0

which gives the first part. To prove the second part, assume first that E is a polyhedron. Then for r ∈ (0, ∞) the Brunn–Minkowski inequality yields E ∗ (r) = {x : d(x, E ∗ ) ≤ r} ⊂ {x : d(x, E) ≤ r}∗ = E(r)∗ ,


253

and hence m(E ∗ (r)) − m(E ∗ ) mn−1 (∂E ) ≤ lim sup 2r r→0 m(E(r)) − m(E) = mn−1 (∂E). ≤ lim sup 2r r→0 ∗

The result for the general domains is obtained by approximating the boundary with polyhedrons. C

0 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

L0

C1

1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0

C 1*

*

L1

L0

C0

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111

0

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111

L1

Figure 9: Spherical symmetrization of a ring. Theorem 4.23. If R = R(C0 , C1 ) is a ring and if C0∗ and C1∗ are the sphrerical symmetrizations of C0 and C1 in opposite rays L0 , L1 , then R∗ = R(C0∗ , C1∗ ) is a ring with cap R∗ ≤ cap R. Proof. (Idea, [7, p.225]) Let u be a locally lipschitz function that is admissible for R. Choose u∗ such that {x : u∗ (x) ≤ t} = {x : u(x) ≤ t}∗ . Then u∗ is admissible for R∗ and from Theorem 4.22 we obtain Z Z ∗ ∗ n cap(R ) ≤ |∇u | dm ≤ |∇u|n dm. Rn

Rn

By taking the infimum over all admissible u the claim follows.

Canonical ring domains. The complementary components of the Grötzsch n uller ring ring RG,n (s) in Rn are B and [se1 , ∞], s > 1, and those of the Teichm¨ RT,n (s) are [−e1 , 0] and [se1 , ∞], s > 0. We define two special functions γn (s), s > 1 and τn (s), s > 0 by n γn (s) = M ∆(B , [se1 , ∞]) = γ(s), τn (s) = M ∆([−e1 , 0], [se1 , ∞]) = τ (s),

respectively. The subscript n is omitted if there is no danger of confusion. We shall refer to these functions as the Grötzsch capacity and the Teichm¨ uller capacity.

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Γ

Γ

∞ B

n

se 1

∞ −e1

0

se 1

Figure 10: Grötzsch ring RG,n (s) (left) and Teichm¨ uller ring RT,n (s) (right). Lemma 4.24. [22, 5.53] For all s > 1 γn (s) = 2n−1 τn (s2 − 1)

and that τn : (0, ∞) → (0, ∞) is a decreasing homeomorphism. Proof. (Idea) Apply Lemma 4.18 and an auxiliary Möbius transformation. Lemma 4.25. [22, 5.63(1)] Let s > 0. Then τ (s) ≤ γ(1 + 2s) = 2n−1 τ (4s2 + 4s) Proof. Let Γ = ∆(S n−1 (−e1 /2, 1/2), [se1 , ∞]). Then by Lemma 4.24 M(Γ) = γ(1 + 2s) = 2n−1 τ (4s2 + 4s).

By Lemma 4.4 τ (s) ≤ M(Γ). Lemma 4.26. [3, (8.65),(8.62)] The following estimates hold for τn (t), t > 0: λ √ √ 1−n n 1−n , τn (t) ≥ 2 ωn−1 log ( 1 + t + t) 2 and for γn (1/r), r ∈ (0, 1): √ λn 1 + 1 − r2 1−n λn 1−n ≥ ωn−1 log , γn (1/r) ≥ ωn−1 log 2r r where λn is the Grötzsch ring constant depending only on n.

The value of λn is known only for n = 2, namely λ2 = 4. For n ≥ 3 it is known that 20.76(n−1) ≤ λn ≤ 2en−1 . For more information on λn , see [3, p.169]. Lemma 4.27. (see [9, 2.31]) Let 0 < r0 < 1. Then M ∆(Bn (r), S n−1 ) ≥ γn (1/r) ≥ C(n, r0 )M ∆(Bn (r), S n−1 ) , for r0 > r > 0.

Proof. By Lemma 4.26, √ λn 1 + 1 − r2 1−n λn 1−n γn (1/r) ≥ ωn−1 log . ≥ ωn−1 log 2r r


We note that λn ≤ log r for 0 < r < r0 . Thus

with

255

1 log λn log , 1− log r0 r

1 1−n γn (1/r) ≥ C(n, r0 )ωn−1 log r n = C(n, r0 )M ∆(B (r), S n−1 ) , C(n, r0 ) =

log λn 1− log r0

1−n

.

The second inequality follows immediately from the fact that the line segment [0, r) is contained in the ball of radius r. Remark 4.28. Note that C(n, r0 ) → 1 as r0 → 0 in Lemma 4.27. Lemma 4.29. [22, 7.34] Let R = R(E, F ) be a ring in Rn , and let a, b ∈ E, c, ∞ ∈ F be distinct points. Then |a − c| M(∆(E, F )) ≥ τ . |a − b| Equality holds for E = [−e1 , 0], a = 0, b = −e1 , F = [se1 , ∞), c = se1 , d = ∞. It is not obvious from the definition how M(∆(E, F )), for nonempty E, F ∈ R , depends on the geometric setup and the structure of the sets E, F . The following lemma gives a lower bound for M(∆(E, F )) in the terms of d(E, F )/ min{d(E), d(F )}. n

Lemma 4.30. [22, 7.38] Let E, F be disjoint continua in Rn with d(E), d(F ) > 0. Then M(∆(E, F )) ≥ τ (4s2 + 4s) ≥ cn log(1 + 1/s)

where s = d(E, F )/ min{d(E), d(F )} and cn > 0 is a constant depending only on n.

This result can be improved to the following Lemma, which shows that M(∆(E, F )) and s = d(E, F )/ min{d(E), d(F )} are simultaneously small or large, provided that E, F are connected. Lemma 4.31. [9, 2.30] For n ≥ 2 there are homeomorphisms h1 , h2 of the positive real axis with the following property. If E, F are the components of the n complements of a nondegenerate ring domain in R , then h1 (s) ≤ M(∆(E, F )) ≤ h2 (s), where s = d(E, F )/ min{d(E), d(F )}.

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Spherical metric. The stereographic projection π : R → S n ( 12 en+1 , 12 ) is defined by x − en+1 (4.32) π(x) = en+1 + , x ∈ Rn ; π(∞) = en+1 . |x − en+1 |2 n

Stereographic projection is the restriction to R of the inversion in S n (en+1 , 1) n+1 in R . Since π −1 = π, it follows that π maps the Riemann sphere S n ( 21 en+1 , 12 ) n n onto R . The chordal metric q in R is defined by n

q(x, y) = |π(x) − π(y)|; x, y ∈ R .

(4.33)

Lemma 4.34. [22, 7.37] If R = R(E, F ) is a ring, then 1 M(∆(E, F )) ≥ τ (4.35) , q(E)q(F ) 4q(E, F ) (4.36) . M(∆(E, F )) ≥ τ q(E)q(F )

5. Quasiconformal mappings A homeomorphism f : G → Rn , n ≥ 2, of a domain G in Rn is called quasiconformal if f is in ACLn , and there exists a constant K, 1 ≤ K < ∞ such that |f ′ (x)|n ≤ K|Jf (x)|, |f ′ (x)| = max |f ′ (x)h|, |h|=1

′

a.e. in G, where f (x) is the formal derivative. The smallest K ≥ 1 for which this inequality is true is called the outer dilatation of f and denoted by KO (f ). If f is quasiconformal, then the smallest K ≥ 1 for which the inequality |Jf (x)| ≤ Kl(f ′ (x))n , l(f ′ (x)) = min |f ′ (x)h|, |h|=1

holds a.e. in G is called the inner dilatation of f and denoted by KI (f ). The maximal dilatation of f is the number K(f ) = max{KI (f ), KO (f )}. If K(f ) ≤ K, f is said to be K-quasiconformal. It is well-known that KI (f ) ≤ KOn−1 (f ),

KO (f ) ≤ KIn−1 (f ),

and hence KI (f ) and KO (f ) are simultaneously finite. Theorem 5.1. [21, 32.2,33.2] Let f : G → Rn be a quasiconformal mapping. Then (1) f is differentiable a.e., (2) f satisfies condition (N), i.e. if A ⊂ G and m(A) = 0, then m(f A) = 0. The next lemma gives another definition of quasiconformality. This definition is called the geometric definition, and it is very useful in applications. The proof for equivalence of these definitions is given in [21].


257

Lemma 5.2. A homeomorphism f : G → G′ is K-quasiconformal if and only if M(Γ)/K ≤ M(f (Γ)) ≤ KM(Γ)

for every path family Γ in G.

We may also give geometric definitions for the inner and outer dilatations. Again, we refer to [21] for the proofs for the equivalence of these definitions. n

Let G, G′ be domains in R and f : G → G′ be a homeomorphism. Then inner and outer dilatations of f are respectively M (f (Γ)) M (Γ) KI (f ) = sup , KO (f ) = sup , M (Γ) M (f (Γ)) where the suprema are taken over all path families Γ in G such that M (Γ) and M (f (Γ)) are not simultaneously 0 or ∞. The maximal dilatation of f is K(f ) = max{KI (f ), KO (f )}.

Theorem 5.3. [21, 13.2] Let f : G′ → G′′ , g : G → G′ be quasiconformal mappings. Then (1) (2) (3) (4) (5) (6)

KI (f −1 ) = KO (f ), KO (f −1 ) = KI (f ), K(f −1 ) = K(f ), KI (f ◦ g) ≤ KI (f )KI (g), KO (f ◦ g) ≤ KO (f )KO (g), K(f ◦ g) ≤ K(f )K(g).

Examples. (see [21, pp.49-50]) (1) A homeomorphism f : G → f G satisfying |x − y|/L ≤ |f (x) − f (y)| ≤ L|x − y|

for all x, y ∈ G is called L-bilipschitz. It is easy to see that L-bilipschitz maps are L2(n−1) -quasiconformal. (2) Let a 6= 0 be a real number, and let f (x) = |x|a−1 x. We can extend f to n n a homeomorphism f : R → R by defining f (0) = 0, f (∞) = ∞ for a > 0 and f (0) = ∞, f (∞) = 0 for a < 0. Then f is quasiconformal with KI (f ) = |a|, KO (f ) = |a|n−1 1−n KI (f ) = |a| , KO (f ) = |a|−1

if |a| ≥ 1, if |a| ≤ 1.

(3) Let (r, ϕ, z) be the cylindrical coordinates of a point x ∈ Rn , i.e. r ≥ 0, 0 ≤ ϕ ≤ 2π, z ∈ Rn−2 , and   x1 = r cos ϕ, x2 = r sin ϕ,  xj = zj−2 for 3 ≤ j ≤ n.

The domain Gα , defined by 0 < ϕ < α, is called a wedge of angle α, α ∈ (0, 2π). Let 0 < α ≤ β < 2π. The folding f : Gα → Gβ , defined by f (r, ϕ, z) = (r, βϕ/α, z),

is quasiconformal with KI (f ) = β/α, KO (f ) = (β/α)n−1 .

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6. An application of the modulus technique As an application, we give a bound for how close to a point α the values attained by a quasiconformal mapping on a sequence of continua approaching the boundary can be. The bound is given in the terms of the diameter of the continua involved. In order to prove this result, we need the following lemmas. This result is presented in [17, pp.638–639]. Lemma 6.1. Let w > 0 and t ∈ (0, min{w2 , 1/w}). Then

1 1 w 1 log < log < 2 log . 2 t t t √ Proof. Since t < w2 , we have 1/ t < w/t. On the other hand, t < 1/w, or w < 1/t, and hence w/t < 1/t2 . By taking logarithm the claim follows. Lemma 6.2. Let C ⊂ Bn be connected and 0 < d(C) ≤ 1. Then m ≡ d(0, C)/d(C) < ∞ and if m > 0, then 1 n M(Γ) ≥ τ (4m2 + 4m) ≥ 2−n τ (m); Γ = ∆(B (1/2), C; Bn ). 2

Proof. The second inequality holds by Lemma 4.25. To prove the first inequality, n we note that if C ∩ B (1/2) 6= ∅, then M(Γ) = ∞ and there is nothing to prove. n In what follows we may assume that C ∩ B (1/2) = ∅. Now the result follows from the symmetry property of the modulus Lemma 4.11 and Lemma 4.30. Theorem 6.3. Let f : Bn → Rn be a quasiconformal mapping or constant, α ∈ Rn and Cj a sequence of nondegenerate continua such that Cj → ∂Bn and |f (x)− α| < Mj when x ∈ Cj , where Mj → 0 as j → ∞. If n−1 1 1 lim sup τ log = ∞, d(Cj ) Mj j→∞ then f ≡ α. In particular, if lim sup log j→∞

1 d(Cj )

1−n

1 log Mj

n−1

= ∞,

then f ≡ α. Proof. Suppose that f is not constant. Let Γj = ∆(Bn (1/2), Cj ; Bn ). Then by Lemma 6.2 1 d(0, Cj ) −n ≥ 2−n τ . M(Γj ) ≥ 2 τ d(Cj ) d(Cj ) n

Let w = d(f B (1/2), α) > 0. Now by Lemma 4.14 1−n 1 1 w 1−n ≤ ωn−1 log , M(f Γj ) ≤ ωn−1 log Mj 2 Mj


259

whenever Mj < min{w2 , 1/w} by Lemma 6.1. Because M(Γj ) ≤ K(M(f Γj )), the estimates above yield n−1 1 1 τ ≤ 22n−1 Kωn−1 , log d(Cj ) Mj proving the first part of the claim. The estimate (4.26) yields 1−n

τ (t) ≥ 2

ωn−1 log

λ

n

2

√ 1−n √ 1+t+ t

where t = 1/d(Cj ). It follows that λ √ λ √ 1−n √ 1−n n n log ≥ log 1+t+ t (1 + 2 t) 2 2 λ 1−n 2 n = log . 1+ p 2 d(Cj ) We note that

log

λ

2 1+ p 2 d(Cj ) n

1−n

1−n λn ≥ 2 log p d(Cj )

whenever j is large enough. Let v = λn . Now by Lemma 6.1 1−n 1−n λ 1 n , ≥ 2 log 2 log p d(Cj ) d(Cj ) p for d(Cj ) < min{v 2 , 1/v}. Hence n−1 n−1 1−n 1 1 1 1 2−2n τ ≤2 ωn−1 log , log log d(Cj ) Mj d(Cj ) Mj which gives the second part of the claim.

References 1. L. V. Ahlfors: Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10 D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. 2. C. Andreian–Cazacu: Foundations of quasiconformal mappings, Handbook of complex analysis: geometric function theory, Vol. 2, edited by R. K¨ uhnau, 687–753, Elsevier, Amsterdam, 2005. 3. G. D. Anderson, M. K. Vamanamurty and M. Vuorinen: Conformal invariants, inequalities and quasiconformal mappings, Wiley-Interscience, 1997. 4. P. Caraman: n-dimensional quasiconformal (QCf) mappings, Editura Academiei Române, Bucharest; Abacus Press, Newfoundland, N.J., 1974. 5. F. W. Gehring: Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961) 499–519. 6. F. W. Gehring: The Carathéodory convergence theorem for quasiconformal mappings in the space, Ann. Acad. Sci. Fenn. Ser. A I 336/11, 1-21, 1963. 7. F. W. Gehring: Quasiconformal mappings, Complex analysis and its applications (Lectures, Internat. Sem., Trieste, 1975), Vol. II, 213–268, IAEA, Vienna, 1976.

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8. F. W. Gehring: Quasiconformal mappings in Euclidean spaces, Handbook of complex analysis: geometric function theory, Vol. 2, edited by R. K¨ uhnau, 1–29, Elsevier, Amsterdam, 2005. 9. V. Heikkala: Inequalities for conformal capacity, modulus and conformal invariants, Ann. Acad. Sci. Fenn. Math. Dissertationes 132 (2002), 1–62. 10. J. Heinonen: Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. 11. J. Heinonen and P. Koskela: Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. 12. J. Heinonen and P. Koskela: Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. 13. T. Iwaniec and G. Martin: Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. ¨hnau (ed.): Handbook of complex analysis: geometric function theory, Vol. 1–2, 14. R. Ku Amsterdam : North Holland/Elsevier, 2002, 2005. 15. O. Lehto: Univalent functions and Teichmller spaces, Graduate Texts in Mathematics, 109. Springer-Verlag, New York, 1987. 16. O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane, Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer-Verlag, New York-Heidelberg, 1973. 17. A. Rasila: Multiplicity and boundary behavior of quasiregular maps, Math. Z. 250 (2005), 611–640. 18. Yu. G. Reshetnyak: Space mappings with bounded distortion, Translations of Mathematical Monographs, 73, American Mathematical Society, Providence, RI, 1989. 19. S. Rickman: Quasiregular Mappings, Ergeb. Math. Grenzgeb. (3), Vol. 26, SpringerVerlag, Berlin, 1993. ïsa ¨ la ¨: On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I 298 20. J. Va (1961), 1–36. ïsa ¨ la ¨: Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in 21. J. Va Math., Vol. 229, Springer-Verlag, Berlin, 1971. 22. M. Vuorinen: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., Vol. 1319, Springer-Verlag, Berlin, 1988. 23. M. Vuorinen (ed.): Quasiconformal space mappings. A collection of surveys 1960–1990, Lecture Notes in Mathematics, 1508, Springer-Verlag, Berlin, 1992. Antti Rasila E-mail: [email protected] Address: Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, FIN-02015 HUT, Finland


The universal Teichm¨ uller space and related topics Toshiyuki Sugawa Abstract. In this survey, we give an expository account of the universal Teichm¨ uller space with emphasis on the connection with univalent functions. In the theory, the Schwarzian derivative plays an important role. We also introduce many interesting results involving Schwarzian derivatives and pre-Schwarzian derivatives, as well. Keywords. universal Teichm¨ uller space, univalent function, Schwarzian derivative, preSchwarzian derivative. 2000 MSC. Primary: 30F60, Secondary: 30C55, 30C62.

Contents 1. Preliminary

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1.1. Quasiconformal mappings

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1.2. Hyperbolic Riemann surfaces

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1.3. Quadratic differentials

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1.4. Univalent functions

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1.5. Grunsky inequality

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1.6. Schwarzian derivative

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2. The universal Teichm¨ uller space

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2.1. Definition 1: the quotient space of quasiconformal maps

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2.2. Definition 2: quasisymmetric functions

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2.3. Definition 3: marked quasidisks

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2.4. Definition 4: Bers embedding

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2.5. Equivalence of T1 through T4

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3. Analytic properties of the Bers embedding

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3.1. The Teichm¨ uller space of a Riemann surface

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3.2. Relationship with quasi-Teichm¨ uller spaces

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3.3. The Bers projection

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3.4. Convexity

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3.5. Teichm¨ uller distance and other natural distances (metrics)

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4. Pre-Schwarzian models 4.1. The models Tˆ(D) and Tˆ(H)

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4.2. The model Tˆ(D∗ )

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4.3. Loci of typical subclasses of S

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5. Univalence criteria 5.1. Univalence criteria due to Nehari and Ahlfors-Weill 5.2. Inner radius and outer radius 5.3. Pre-Schwarzian counterpart 5.4. Directions of further investigation

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References

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1. Preliminary In this section, we prepare basic tools to understand the universal Teichm¨ uller space. The material is more or less standard, but for convenience, an expository account will be given without proofs. The most convenient reference for overall topics is perhaps the recently published handbook [61]. 1.1. Quasiconformal mappings. A homeomorhism f of a plane domain D onto another domain D′ is called a quasiconformal map if f has locally square integrable partial derivatives (in the sense of distribution) and satisfies the inequality |fz¯| ≤ k|fz | almost everywhere in D, where k is a constant with 0 ≤ k < 1, fz = 21 (fx − ify ),

fz¯ = 21 (fx + ify )

and fx =

∂f , ∂x

fy =

∂f . ∂y

It turns out that f preserves sets of (2-dimensional) Lebesgue measure zero and, in particular, fz 6= 0 a.e. Thus the quotient µ = fz¯/fz is well defined as a Borel measurable function on D and satisfies kµk∞ ≤ k < 1. This function is sometimes called the complex dilatation of f and denoted by µf . More specifically, f is also called a K-quasiconformal map, where K = (1 + k)/(1 − k). The minimal K = (1 + kµk∞ )/(1 − kµk∞ ) is called the maximal dilatation of f and denoted by K(f ). It is known that a 1-quasiconformal map is conformal (i.e., biholomorphic) and vice versa. The composition of a K1 -quasiconformal map and a K2 -quasiconformal map is K1 K2 -quasiconformal map and the inverse map of a K-quasiconformal map is also K-quasiconformal. In particular, K-quasiconformality is

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preserved under composition with conformal maps. Therefore, K-quasiconformality and, hence, quasiconformality can be defined for homeomorphisms between Riemann surfaces. In particular, we can argue quasiconformality of a homeomorphism of the Riemann sphere b = C ∪ {∞}. C

More precise information about compositions of quasiconformal maps will be needed later. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then the complex dilatation of g ◦ f −1 is given by (1.1.1)

(µg◦f −1 ◦ f )

µg − µf fz = . fz 1 − µf · µg

In particular, we obtain the following lemma. Lemma 1.1.2. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then g ◦ f −1 is conformal on Ω′ if and only if µf = µg a.e. in Ω. Fundamental in the theory of quasiconformal maps is the following existence and uniqueness theorem. Theorem 1.1.3 (The measurable Riemann mapping theorem). For any measurable function µ on C with kµk∞ < 1, there exists a unique quasiconformal map f : C → C such that f (0) = 0, f (1) = 1 and fz¯ = µfz a.e. in C. For the proof of the theorem and for more information about quasiconformal maps, the reader should consult the books [3] and [69] as well as the paper [4] by Ahlfors and Bers. See also the article “Beltrami Equation”, by Srebro and Yakubov, in [61, vol. 2] for the recent development. We denote by Belt(D) the open unit ball of the space L∞ (D) for a domain (or, more generally, a measurable set) D. An element µ of Belt(D) is called a Beltrami coefficient on D. For a Beltrami coefficient µ on C, the function f given in the measurable Riemann mapping theorem will be denoted by f µ throughout the present survey. Let µ be a Beltrami coefficient on the outside D∗ of the unit disk. We extend µ to µ∗ ∈ Belt(C) by setting µ∗ (z) = µ(1/¯ z ) for z ∈ D. Let f be a quasiconformal autob fixing 1, −1, −i with µf = µ∗ . Since f (z) and 1/f (1/¯ z ) both have the morphism of C ∗ same complex dilatation µ and satisfy the same normalization condition, they must be equal by uniqueness part of the measurable Riemann mapping theorem. In particular, b→C b |f (z)|2 = 1 for z ∈ ∂D, and consequently, f maps D∗ onto itself. We define wµ : C by wµ = f. Recall that wµ fixes 1, −1 and −i.

The following fact was observed by Ahlfors and Bers [4].

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Theorem 1.1.4. Let µt be a family of Beltrami coefficients on C holomorphically parameterized over a complex manifold X. Then the map t 7→ f µt (z) is holomorphic on X for a fixed z ∈ C. We do not explain the meaning of “holomorphically parameterized” here. It is, however, sufficient practically to know that (tµ + ν)/(1 + t¯ ν µ) is a family of Beltrami coefficients holomorphically parameterized over the unit disk |t| < 1, where µ, ν ∈ Belt(C). 1.2. Hyperbolic Riemann surfaces. A connected complex manifold of complex dimension one is called a Riemann surface. The Poincaré-Koebe uniformization theorem tells us that every Riemann surface R admits an analytic universal covering projection p of the unit disk D = {z ∈ C : |z| < 1} onto R except for the case when R is conforb the complex plane C, the punctured complex mally equivalent to the Riemann sphere C,

plane C∗ = C \ {0} or a complex torus (a smooth elliptic curve). Those non-exceptional Riemann surfaces are called hyperbolic.

The group of analytic automorphisms of R is denoted by Aut(R). The group of disk automorphisms Aut(D) is identified with PSU(1, 1) and isomorphic to PSL(2, R) through the Möbius transformation M : H = {z : Im z > 0} → D defined by M (z) = (z−i)/(z+i). Thus Aut(D) inherits a structure of real Lie group. A subgroup Γ of Aut(D) is called Fuchsian if Γ is discrete. It is known that Γ is discrete if and only if Γ acts on D properly discontinuously. Note also that Γ is torsion-free if and only if Γ acts on D without fixed points. The covering transformation group Γ = {γ ∈ Aut(D) : p ◦ γ = p} is a torsionfree Fuchsian group and will be called the Fuchsian model of R. Conversely, for a given torsion-free Fuchsian group Γ the quotient space D/Γ has natural complex structure so that the projection D → D/Γ becomes an analytic universal covering. In this way, the theory of hyperbolic Riemann surfaces can be translated into that of torsion-free Fuchsian groups. Since the Poincaré metric ρD (z)|dz| = |dz|/(1−|z|2 ) is invariant under the pull-back by analytic automorphisms of D, it projects to a smooth metric, denoted by ρR = ρR (w)|dw|, on the hyperbolic Riemann surface R via p. The metric ρR is called the hyperbolic metric of R. Thus ρR is characterized by the relation ρD = p∗ (ρR ) = ρR (p(z))|p′ (z)||dz|. Note that ρR has constant Gaussian curvature −4, in other words, ∆ log ρR = 4ρ2R .

The Schwarz-Pick lemma implies the contraction property f ∗ ρS ≤ ρR for a holomorphic map f : R → S between hyperbolic Riemann surfaces R and S, where equality holds at some (hence every) point in R iff f is a covering projection of R onto S. 1.3. Quadratic differentials. Let H(D) be the set of analytic functions on the unit disk D and let n be a non-negative integer. For a Fuchsian group Γ, a function ϕ ∈ H(D) is said to be automorphic for Γ (with weight −2n) if ϕ satisfies the functional equation


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(ϕ ◦ γ)(γ ′ )n = ϕ for every γ ∈ Γ, that is to say, ϕ(z)dz n is an invariant n-form for Γ. The set of automorphic functions for Γ with weight −2n will be denoted by Hn (D, Γ). An element ϕ of Hn (D, Γ) for a torsion-free Fuchsian group Γ projects to a holomorphic n-form q = q(w)dwn on R = D/Γ so that p∗n q = ϕ(z)dz n , where p∗n q means the pull-back q(p(w))(p′ (w))n of the n-form q by the canonical projection p : D → D/Γ. We now define two norms for ϕ ∈ Hn (D, Γ) by ZZ kϕkAn (D,Γ) = |ϕ(z)|(1 − |z|2 )n−2 dxdy, ω

kϕkBn (D,Γ) = sup |ϕ(z)|(1 − |z|2 )n , z∈D

where ω is a fundamental domain for Γ, that is, a subdomain of D such that ω ∩ γ(ω) = ∅ S for every γ ∈ Γ with γ 6= id, γ∈Γ γ(¯ ω ) = D and ∂ω is of zero area. We denote by An (D, Γ) and Bn (D, Γ) the subsets of Hn (D, Γ) consisting of ϕ with finite norm kϕkAn (D,Γ) and kϕkBn (D,Γ) , respectively. It is easy to see that these become complex Banach spaces. When Γ is the trivial group 1, we write An (D) and Bn (D) for An (D, 1) and Bn (D, 1), respectively.

The definition of the spaces An (D) and Bn (D) can be extended for hyperbolic Riemann surfaces R. Let Hn (R) denote the set of holomorphic n-forms on R and set ZZ kϕkAn (R) = |ϕ(w)|ρR (w)2−n dxdy, R

kϕkBn (R) = sup |ϕ(w)|ρR (w)−n w∈R

for ϕ = ϕ(w)dwn in Hn (R). Here, we should note that |ϕ(w)|ρR (w)−n does not depend on the choice of the local coordinate w, in other words, |ϕ|ρ−n R can be regarded as a function on R. The Banach spaces An (D, Γ) and An (D/Γ) (resp. Bn (D, Γ) and Bn (D/Γ)) are isometrically isomorphic through the pull-back p∗n by the projection p : D → D/Γ. Also, the following invariance property is convenient to note. Lemma 1.3.1. Let R and S be hyperbolic Riemann surfaces and let p : R → S be a conformal homeomorphism. Then the pullback operator p∗n : Bn (S) → Bn (R) is a linear isometry, in other words, kp∗n ϕkBn (R) = kϕkBn (S) ,

ϕ ∈ Bn (S).

In the theory of Teichm¨ uller spaces, it is important to consider the spaces A2 and B2 as we shall see later. A 2-form q(w)dw2 is traditionally called a quadratic differential.

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1.4. Univalent functions. In connection with the universal Teichm¨ uller space, the theory of univalent functions is of particular importance. The best textbook in this direction is [67] by O. Lehto. We denote by S the set of analytic univalent functions f on the unit disk so normalized that f (0) = 0 and f ′ (0) = 1. An analytic function f around the origin is said to be strongly normalized if f (0) = f ′ (0) − 1 = f ′′ (0) = 0. Let S0 be the subset of S consisting of strongly normalized functions. For f ∈ S , the function g = f /(1 + af ), where a = f ′′ (0)/2, is strongly normalized but not necessarily analytic in D. It is thus natural to consider the wider class S˜0 = {f : meromorphic and univalent in D and strongly normalized} than S0 . The following meromorphic counterpart is also useful in the theory of univalent functions. Let Σ be the set of meromorphic univalent functions F on the exterior D∗ = {ζ ∈ b : |ζ| > 1} of the unit disk so normalized that C

(1.4.1)

F (ζ) = ζ + b0 +

b1 b2 + 2 + ... ζ ζ

in |ζ| > 1. For f ∈ S , the function F (ζ) = 1/f (1/ζ) belongs to Σ and satisfies the condition 0∈ / F (D∗ ), and vice versa. Let Σ′ denote the set of those functions F ∈ Σ that satisfy 0∈ / F (D∗ ). Moreover, b0 = 0 for a function F (ζ) = ζ + b0 + b1 /ζ + . . . in Σ if and only if f ∈ S˜0 , where f (z) = 1/F (1/z). Hence, if we set Σ0 = {F ∈ Σ : F (ζ) − ζ → 0 as ζ → ∞}, the correspondence f (z) 7→ F (ζ) = 1/f (1/ζ) gives bijections of S˜0 onto Σ0 and of S0 onto Σ′0 , where we define Σ′0 = Σ0 ∩ Σ′ . 1.5. Grunsky inequality. For a meromorphic function F near the point at infinity with an expansion of the form (1.4.1), we take a single-valued branch of log((F (ζ)−F (ω))/(ζ − ω)) in |ζ| > R and |ω| > R for sufficiently large R > 0 and expand it in the form ∞ X ∞ X F (ζ) − F (ω) bj,k =− log ζ −ω ζ j ωk j=1 k=1

there. The coefficients bj,k are called the Grunsky coefficients of F. It is easy to see that bj,k = bk,j and b1,k = bk for j, k ≥ 1, where bk is the coefficient in (1.4.1). The last relation is deduced in the following way. If we write F (ζ) = ζ + b0 + G(ζ), then G(ζ) = O(|ζ|−1 )


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as ζ → ∞. Fix ω for a moment. Since F (ζ) − F (ω) G(ζ) − G(ω) G(ζ) − G(ω) log = log 1 + + O(|ζ|−2 ), = ζ −ω ζ −ω ζ −ω we obtain ∞ X b1,k k=1

ωk

= − lim ζ log ζ→∞

= − lim ζ ζ→∞

= G(ω),

F (ζ) − F (ω) ζ −ω

G(ζ) − G(ω) ζ −ω

from which the required relation follows. The following theorem is greatly useful in the theory of Teichm¨ uller spaces as well as the theory of univalent functions. See [42], [29] or [86] for the proof and applications. Theorem 1.5.1 (Grunsky). A meromorphic function F (ζ) with expansion of the form (1.4.1) around ζ = ∞ is analytically continued to a univalent meromorphic function in |ζ| > 1 if and only if the inequality 2 ∞ ∞ ∞ X X X |xj |2 bj,k xj ≤ (1.5.2) k j j=1 j=1 k=1 holds for an arbitrary sequence of complex numbers x1 , x2 , . . . .

The inequality in (1.5.2) is known as the strong Grunsky inequality. Noting b1,k = bk , we take (x1 , x2 , x3 , . . . ) = (1, 0, 0, . . . ) to obtain (1.5.3)

∞ X k=1

k|bk |2 ≤ 1.

This inequality is known as Gronwall’s area theorem. It is also known that inequality (1.5.2) can be replaced in the above theorem by the (classical) Grunsky inequality: ∞ ∞ ∞ X X X |xj |2 bj,k xj xk ≤ (1.5.4) . j j=1 k=1

j=1

√ The symmetric matrix ( jkbj,k ) defines a linear operator on ℓ2 , where bj,k are the Grunsky coefficient of a meromorphic function F (ζ) around ζ = ∞. This is sometimes called the Grunsky operator and will be denoted by G[F ] in the following. The strong

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Grunsky inequality says that F ∈ Σ if and only if G[F ] is a bounded linear operator on ℓ2 with operator norm ≤ 1. Here, the operator norm kG[F ]k of G[F ] is defined by ∞ 2 ∞ X X p 2 kG[F ]k = sup jkbj,k yj , kyk2 =1 k=1

j=1

P where kyk2 = ( k |yk |2 )1/2 for y = (y1 , y2 , . . . ). Thus, F ∈ Σ ⇔ kG[F ]k ≤ 1. It is b if and only if kG[F ]k < 1. known (cf. [86]) that F has a quasiconformal extension to C See also the article “Univalent holomorphic functions with quasiconformal extensions”, by Krushkal, in [61, vol. 2]. 1.6. Schwarzian derivative. For a non-constant meromorphic function f on a domain, we define Tf and Sf by Tf =

f ′′ = (log f ′ )′ , f′

f ′′′ 3 1 Sf = (Tf ) − (Tf )2 = ′ − 2 f 2 ′

f′ f

2

.

These are called the pre-Schwarzian derivative and the Schwarzian derivative of f, respectively. Note that Tf is analytic at a finite point z0 if and only if f is analytic and injective around z0 . Similarly, Sf is analytic at z0 if and only if f is meromorphic and injective around z0 . The following two lemmas show usefulness of these operations. Lemma 1.6.1. Let f be a non-constant meromorphic function on a domain D. The pre-Schwarzian derivative of f vanishes on D if and only if f is (the restriction of ) a similarity. The Schwarzian derivative of f vanishes on D if and only if f is (the restriction of ) a Möbius transformation.

Lemma 1.6.2. Let f and g be non-constant meromorphic functions for which the composition f ◦ g is defined. Then Tf ◦g = (Tf ) ◦ g · g ′ + Tg = g1∗ (Tf ) + Tg , Sf ◦g = (Sf ) ◦ g · (g ′ )2 + Sg = g2∗ (Sf ) + Sg .

Combining these lemmas, we observe that SL◦f ◦M = M2∗ (Sf ) for Möbius transformations L and M. Thus the Schwarzian derivative behaves like a quadratic differential.


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2. The universal Teichm¨ uller space We have two choices to develop the theory of the (universal) Teichm¨ uller space; the unit disk model or the upper half-plane model. Although they can be translated into each other, in principle, via the Möbius transformation z 7→ (z − i)/(z + i), both models have their own advantage and thus can be chosen at will according to the purpose. In the present survey, we will take the unit disk model to connect with the theory of univalent functions in a direct way. 2.1. Definition 1: the quotient space of quasiconformal maps. We denote by QC(D) the set of quasiconformal automorphisms of the unit disk D. As we will observe later, every function in QC(D) extends to a unique homeomorphism of the closed unit disk D. Thus, we may think that every f ∈ QC(D) is a self-homeomorphism of the closed

uller equivalent and unit disk D. Two functions f and g in QC(D) are said to be Teichm¨ T

denoted by f ∼g if there exists a disk automorphism L ∈ Aut(D) such that g = L ◦ f on T

∂D. The quotient space QC(D)/∼ is a model of the universal Teichm¨ uller space and will be denoted by T1 for a moment. The equivalence class represented by f ∈ QC(D) will be denoted by [f ] below. Let f, g ∈ QC(D). The Teichm¨ uller distance between p = [f ] and q = [g] is defined by d1 (p, q) =

1 log K(g1 ◦ f1−1 ). f1 ∼f,g1 ∼g 2 inf

T

T

Recall here that K(f ) denotes the maximal dilatation of f. By a compactness property of quasiconformal maps, one can check that d1 (p, q) is indeed a distance on T1 . In this way, T1 becomes a metric space. It can also be shown that T1 is a complete metric space with metric d1 by a normality property of the set of normalized K-quasiconformal automorphisms of C (see [69]). 2.2. Definition 2: quasisymmetric functions. The notion of quasisymmetric functions was created by Beurling and Ahlfors [18] for functions on the real line. We give here a corresponding definition of quasisymmetric functions on the unit circle. A sensepreserving homeomorphism h of the unit circle ∂D is called quasisymmetric if |h(ei(s+t) ) − h(eis )| 1 ≤ ≤ M, M |h(eis ) − h(ei(s−t) )|

s ∈ R, 0 < t
0}. Therefore, we may restrict ourselves on the two cases ∆ = D and H. First let f ∈ S . By the well-known inequality (cf. [29]) (1 − |z|2 )Tf (z) − 2¯ z ≤ 4, (4.1.1)

ˆ we obtain kTf kB1 (D) ≤ 6. In particular, S(D) ⊂ B1 (D). Note also that the constant 6 is 2 ˆ sharp as the Koebe function K(z) = z/(1 − z) shows. It is easy to see that S(D) is closed

in B1 (D).

Let L(z) = (z − i)/(z + i). Note that kTL kB1 (H) = 4 and hence TL ∈ Tˆ(H). Since L∗1 : B1 (D) → B1 (H) is a linear isometry and Tf ◦L = L∗1 (Tf ) + TL , the space Tˆ(H) is

contained in B1 (H) and it is isometrically equivalent to Tˆ(D). In this sense, it is enough to consider only Tˆ(D).

ˆ We define the map π : B1 (D) → B2 (D) by π(ψ) = ψ ′ − ψ 2 /2. By definition, π(S(D)) = S(D) and π(Tˆ(D)) = T (D). Duren, Shapiro and Shields [30] noticed that this map is continuous (see also §5.3). Astala and Gehring [6] proved an analogous result to the case of Schwarzian derivative. ˆ Theorem 4.1.2. The interior of S(D) in B1 (D) is equal to Tˆ(D), while the closure of ˆ Tˆ(D) in B1 (D) is not equal to S(D). Moreover, ∂T (D) \ π(∂ Tˆ(D)) is not empty. Zhuravlev [116] revealed the following remarkable property of Tˆ(D). Theorem 4.1.3 (Zhuravlev). The space Tˆ(D) decomposes into the uncountably many connected components Tˆ0 and Tˆω , ω ∈ ∂D, where Tˆ0 = {Tf ∈ Tˆ(D) : f (D) is bounded }

and Tˆω = {Tf ∈ Tˆ(D) : f (z) → ∞ as z → ω}.

Moreover, {ψ ∈ B1 (D) : kψ − ψω kB1 (D) < 1} ⊂ Tˆω holds for each ω ∈ ∂D, where ψω (z) = 2¯ ω /(1 − ω ¯ z) is the pre-Schwarzian derivative of the function z/(1 − ω ¯ z). Note that the map π is not injective even in each connected component of Tˆ(D). Therefore, we should note that this model of the universal Teichm¨ uller space has some redundancy. 4.2. The model Tˆ(D∗ ). There is some subtlety in consideration of the pre-Schwarzian model of the universal Teichm¨ uller space Tˆ(D∗ ) on the exterior D∗ of the unit circle. The


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first thing to note is the fact that the Banach space B1 (D∗ ) is not the right space on which Tˆ(D∗ ) is modeled. We define ˆ ∗ ) = {TF : F ∈ Σ} S(D and b Tˆ(D∗ ) = {TF : F ∈ Σ extends to a quasiconformal map of C}.

If F (ζ) = ζ + b0 + b1 /ζ + b2 /ζ 2 + . . . , then TF (ζ) = 2b1 /ζ 3 + · · · = O(ζ −3 ) as ζ → ∞. Therefore, the norm B(ψ) = sup (|ζ|2 − 1)|ζψ(ζ)|

(4.2.1)

ζ∈D∗

is more natural. Indeed, Becker’s univalence criterion [12] and Avhadiev’s inequality [8] ′′ ζF (ζ) 2 ≤6 (4.2.2) (|ζ| − 1) ′ F (ζ)

imply the following result.

Theorem 4.2.3. If a meromorphic function F (ζ) = ζ + b0 + b1 /ζ + . . . in |ζ| > 1 satisfies B(TF ) ≤ 1, then F ∈ Σ. Conversely, every function F in Σ satisfies B(TF ) ≤ 6. We set B1′ (D∗ ) = {ψ ∈ B1 (D∗ ) : lim ζ 2 ψ(ζ) = 0}. ζ→∞

Then, it is easy to see that

B1′ (D∗ )

∗

= {ψ : D → C holomorphic and B(ψ) < ∞}. The

ˆ ∗ ) is a bounded subset of B1′ (D∗ ). above theorem now yields that S(D

We define π : B1′ (D∗ ) → B2 (D∗ ) as before by π(ψ) = ψ ′ − ψ 2 /2. Then π is continuous ˆ ∗ )) = S(D∗ ) and π(Tˆ(D∗ )) = T (D∗ ). Since T (D∗ ) [13, Lemma 6.1]. By definition, π(S(D is an open set and Tˆ(D∗ ) = π −1 (T (D∗ )), the set Tˆ(D∗ ) is also open in B ′ (D∗ ). In this 1

way, we see that the space Tˆ(D ) is a complex Banach manifold modeled on B1′ (D∗ ). We remark that π does not map B1 (D∗ ) into B2 (D∗ ). ∗

The set Tˆ(D∗ ) seems to be less investigated, but could be more useful. For instance, ˆ ∗ ) bijectively. Recall that the mapping F 7→ SF the mapping F 7→ TF sends Σ0 to S(D ˆ ∗ ) to S(D∗ ) bijectively. sends Σ0 to S(D∗ ) bijectively. Therefore, the mapping π sends S(D

4.3. Loci of typical subclasses of S . Since the differential operator Tf is closely related with geometric function theory, many classical subclasses of univalent functions ˆ correspond to sets with nice properties in S(D). We recall several fundamental classes in univalent function theory. We denote by A the set of analytic functions f in the unit disk D so normalized that f (0) = 0 and f ′ (0) = 1. A function f ∈ A is called starlike (convex) if f is univalent and if f (D) is starlike with

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respect to the origin (convex). We denote by S ∗ and K the sets of starlike and convex functions in A , respectively. A function f ∈ A is called close-to-convex if eiα f ′ /g ′ has positive real part in D for a convex function g and for a real constant α. Denote by C the set of close-to-convex functions in A . It is known that C ⊂ S (cf. [29]). It is interesting to see how pre-Schwarzians of those functions are located in the space ˆ S(D). The following result gives an answer to this question. Theorem 4.3.1 ([27], [51]). {Tf : f ∈ K } and {Tf : f ∈ C } are both convex subsets of ˆ S(D). It may be natural to conjecture the following. ˆ Conjecture 4.3.2 ([48]). The subset {Tf : f ∈ S ∗ } of S(D) is starlike with respect to the origin. Note that the vector operations in B1 (D) is translated to the Hornich operations in the space of uniformly locally univalent functions (see, for example, [48]). Also, see Casey [22] for relations between subclasses of S and (the closure) of Tˆ(D).

5. Univalence criteria As is well developed in Lehto’s textbook [67], univalence criteria are closely connected with the universal Teichm¨ uller space. The present section will be devoted to this topic. 5.1. Univalence criteria due to Nehari and Ahlfors-Weill. Nehari [78] proved the following result, which is fundamental in the Teichm¨ uller spaces. Theorem 5.1.1. Every meromorphic univalent function f on the unit disk satisfies the inequality kSf kB2 (D) ≤ 6. Conversely, if a meromorphic function f on the unit disk satisfies the inequality kSf kB2 (D) ≤ 2, then f must be univalent. The constants 6 and 2 are sharp since the Koebe function K(z) = z/(1 − z)2 satisfies kSK kB2 (D) = 6 and since the function f (z) = ((1 + z)/(1 − z))iǫ , ǫ > 0, is never univalent but kSf kB2 (D) = 2(1 + ε2 ) can approach 2 (Hille [44]). The former assertion was first proved by Kraus [56] and reproved by Nehari. Therefore, it is called nowadays the KrausNehari theorem. The Kraus-Nehari theorem is a consequence of the Bieberbach theorem. By the Möbius invariance of (1 − |z|2 )2 |Sf (z)|, it is enough to show the inequality only at the origin, namely, |Sf (0)| ≤ 6 for f ∈ S . A straightforward computation gives Sf (0) = 6(a3 − a22 ) for f (z) = z + a2 z 2 + a3 z 3 + . . . . If we set F (ζ) = 1/f (1/ζ) = ζ + b0 + b1 /ζ + . . . , then b1 = a22 − a3 , and thus the inequality |b1 | ≤ 1 (see (1.5.3)) implies the required one.


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The class N = {f ∈ A : kSf kB2 (D) ≤ 2} is sometimes called the Nehari class. Gehring and Pommerenke [41] showed that f ∈ N maps the unit disk conformally onto a Jordan domain unless f (D) is Möbius equivalent to the parallel strip {z : |Im z| < π/4}. For further development, see [24], [25] and [26]. In connection with Nehari’s theorem, Ahlfors and Weill established the following quasiconformal extension criterion. For ϕ ∈ B2 (D∗ ) with kϕkB2 (D∗ ) < 2, we set α[ϕ] ∈ Belt(D) by α[ϕ](z) = −ρD (z)−2 ϕ(1/¯ z )¯ z −4 /2. Note that the map α is the restriction of a bounded linear operator which maps B2 (D∗ , Γ)2 = {ϕ ∈ B2 (D∗ , Γ) : kϕkB2 (D) < 2} into Belt(D∗ , Γ) for every Fuchsian group Γ. Theorem 5.1.2 (Ahlfors-Weill). The map α : B2 (D∗ )2 → Belt(D) is the local inverse of the Bers projection Φ : Belt(D) → T (D∗ ), in other words, Φ(α[ϕ]) = ϕ for ϕ ∈ B2 (D∗ ) with kϕkB2 (D∗ ) < 2. Corollary 5.1.3. The universal Teichm¨ uller space T(D∗ ) contains the open ball centered at the origin with radius 2 in B2 (D∗ ). The map α : B2 (D∗ )2 → Belt(D) is sometimes called the Ahlfors-Weill section. b The inner 5.2. Inner radius and outer radius. Let D be a hyperbolic domain in C. radius σI (D) and the outer radius σO (D) of univalence is defined respectively by σI (D) = sup{σ ≥ 0 : kSf kB2 (D) ≤ σ ⇒ f is univalent in D},

b is univalent}. σO (D) = sup{kSf kB2 (D) : f : D → C

We also define the number τ (D) ∈ [0, +∞] as kSp kB2 (D) , where p is a holomorphic universal covering projection of D onto D. The quantity τ (D) is independent of the choice of p and thus well defined. Note that τ (D) < ∞ if and only if ∂D is uniformly perfect (cf. [87] or [103]). Summarizing theorems of Ahlfors [2], Gehring [38], Nehari [78], we obtain the following. Theorem 5.2.1. σI (∆) = 2, σO (∆) = 6, τ (∆) = 0 hold for a circle domain ∆. Let D be a simply connected hyperbolic domain. Then σO (∆) ≤ 12 and τ (D) ≤ 6. Moreover, D is a quasidisk if and only if σI (D) > 0. b be univalent and set The inequality σO (∆) ≤ 12 is shown as follows. Let f : D → C Ω = f (D). Take a conformal map g : D∗ → D and set h = f ◦ g. Then, by Lemmas 1.3.1, 1.6.2 and the Kraus-Nehari theorem, we obtain kSf kB2 (D) = kg2∗ (Sf )kB2 (D∗ ) = kSh − Sg kB2 (D∗ ) ≤ kSh kB2 (D∗ ) + kSg kB2 (D∗ ) ≤ 12.

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It is a remarkable fact due to Beardon and Gehring [10] that σO (D) ≤ 12 holds even for an arbitrary hyperbolic domain D. The inner and outer radii of univalence are better understood in the context of (quasi-) Teichm¨ uller space. Lemma 5.2.2. Let g : D∗ → D be a conformal homeomorphism of D∗ onto a simply connected hyperbolic domain D. Then {ϕ ∈ S(D∗ ) : kϕ−Sg k < σI (D)} is the maximal open ball centered at Sg contained in T (D∗ ). On the other hand, σO (D) = max{kϕ − Sg kB2 (D∗ ) : ϕ ∈ S(D∗ )}. Lehto [65] proved the following relations. Theorem 5.2.3. The relation σO (D) = τ (D) + 6 holds for a simply connected hyperbolic domain D. Furthermore, 2 − τ (D) ≤ σI (R) ≤ min{2, 6 − τ (D)}. As for the quantity τ (D), the following are known. For a convex domain D, we have τ (D) ≤ 2. This result is repeatedly re-discovered by many mathematicians; [85], [90], [112], [79], [65]. Suita [106] refined this result by showing the sharp inequality ( 2, 0 ≤ α ≤ 1/2, τ (f (D)) ≤ 8α(1 − α), 1/2 ≤ α ≤ 1 for a convex function f ∈ K of order α, namely, when Re (1 + zf ′′ (z)/f ′ (z)) > α. It is known that τ (D) ≤ 6(K 2 − 1)/(K 2 + 1) for a K-quasidisk D (see [67]). See also [68], [53], [23], [72], [9] for other classes of domains. It is not easy to determine, or even to estimate from below, the value of σI (D), in general. Known examples are sectors [62], triangles [64], the interiors and the exteriors of regular polygons [21], [64], some other polygonal domains [73], [74], the exteriors of hyperbolas [63]. For a general method of estimating σI (D) from below, see [66], [67] and [104]. See also [105]. 5.3. Pre-Schwarzian counterpart. One can define quantities similarly as in the previous section with respect to pre-Schwarzian derivative. We add the symbol ˆ to indicate it. For instance, σ Î (D) = sup{σ ≥ 0 : kTf kB1 (D) ≤ σ ⇒ f is univalent in D} for a hyperbolic domain D in C. In the case when D = D∗ , we adopt the norm B(ψ) : σ Î (D∗ ) = sup{σ ≥ 0 : B(TF ) ≤ σ ⇒ f is univalent in D∗ }.


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√ Duren, Shapiro and Shields [30] proved that σ Î (D) ≥ 2( 5−2) = 0.472 · · · by observing that kψ ′ kB2 (D) ≤ 4kψkB1 (D) and thus π(ψ) = ψ ′ − ψ 2 /2 is a continuous map of B1 (D) into √ √ B2 (D). Note that Wirths [114] found the sharp constant C = (13 3 + 55 11)/64 = 3.20204 . . . for the estimate kψ ′ kB2 (D) ≤ CkψkB1 (D) . Nowadays, the best value for this univalence criterion is known. Theorem 5.3.1. σ Î (∆) = 1 and σ Ô (∆) = 6 for ∆ = D, H and D∗ . Becker [11], [12] showed that σ Î (D) ≥ 1 and σ Î (D∗ ) ≥ 1 and Becker-Pommerenke [14] showed that equality hold for ∆ = D and that σ Î (H) = 1. Pommerenke [88] showed the ∗ sharpness for ∆ = D . By (4.1.1) and the fact that the Koebe function K satisfies kTK kB1 (D) = 6, we see that σ Ô (D) = 6. σ Ô (H) = 6 can be seen by noting the relation kψkB1 (H) = lim kψkB1 (∆r ) r→1−

for ψ ∈ B1 (H), where ∆r = {z : |z − i(1 + r2 )/(1 − r2 )| < 2r/(1 − r2 )}. The formula σ Ô (D∗ ) = 6 follows from the fact that the inequality in (4.2.2) is sharp for each ζ. For concrete estimates of τˆ(D) for several geometric classes of domains, see [115], [101], [81], [49], [50]. In spite of relative simplicity of the operation Tf , very little is known for quantities σ Î (D) and σ Ô (D). Stowe [97] gave non-trivial examples of domains D for which σ Î (D) ≥ 1. 5.4. Directions of further investigation. The Bers embedding of Teichm¨ uller spaces is still mysterious. We know very little about the shape of it. Pictures of one-dimensional Teichm¨ uller spaces were recently given in [54] and [55]. Note that the first attempt towards it was done by Porter [89] as early as in 1970’s. It is an interesting and important problem to describe the intersection of T (∆) or Tˆ(∆) with a (complex) one-dimensional vector subspace of B2 (∆) or B1 (∆) for a circle domain. Completely known examples are essentially, as far as the author knows, the linear hull of 1/(1 − z) in B1 (∆) [92] and the linear hull of z −2 in B2 (H) in [47], only. The results presented above could be generalized to various directions. We end this survey with remarks on possible ways to study furthermore.

In this section, we considered mainly the case when the domain is simply connected. When the domain is multiply connected, the problem will become much more difficult. See [83] and [84] for fundamental information. We were concerned here with only pre-Schwarzian and Schwarzian derivatives. On the other hand, several definitions of higher-order Schwarzian derivatives have been proposed (e.g., [109], [93]). Thus, we may develop the theory for those higher-order Schwarzian derivatives.

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Of course, we may consider domains in Cn or Rn but with great difficulty caused by the lack of canonical metrics such as hyperbolic metric, the lack of Riemann mapping theorem and so on. Note that Martio and Sarvas [70] gave some injectivity conditions even in higher dimensions.

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95. H. Shiga, Characterization of quasi-disks and Teichm¨ uller spaces, Tôhoku Math. J. 37 (1985), 541– 552. 96. H. Shiga and H. Tanigawa, Grunsky’s inequality and its applications to Teichm¨ uller spaces, Kodai Math. J. 16 (1993), 361–378. 97. D. Stowe, Injectivity and the pre-Schwarzian derivative, Michigan Math. J. 45 (1998), 537–546. 98. T. Sugawa, The Bers projection and the λ-lemma, J. Math. Kyoto Univ. 32 (1992), 701–713. 99. , On the Bers conjecture for Fuchsian groups of the second kind, J. Math. Kyoto Univ. 32 (1992), 45–52. 100. , On the space of schlicht projective structures on compact Riemann surfaces with boundary, J. Math. Kyoto Univ. 35 (1995), 697–732. , On the norm of the pre-Schwarzian derivatives of strongly starlike functions, Ann. Univ. 101. Mariae Curie-Sklodowska, Sectio A 52 (1998), 149–157. , Holomorphic motions and quasiconformal extensions, Ann. Univ. Mariae Curie-Sklodowska, 102. Sectio A 53 (1999), 239–252. 103. , Uniformly perfect sets: analytic and geometric aspects (Japanese), Sugaku 53 (2001), 387– 402, English translation in Sugaku Expo. 16 (2003), 225–242. 104. , A remark on the Ahlfors-Lehto univalence criterion, Ann. Acad. Sci. Fenn. A I Math. 27 (2002), 151–161. 105. , Inner radius of univalence for a strongly starlike domain, Monatsh. Math. (2003), 61–68. 106. N. Suita, Schwarzian derivatives of convex functions, J. Hokkaido Univ. Edu. (Sec. IIA) 46 (1996), 113–117. 107. L. A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003), 183–240. , Weil-Petersson geometry of the universal Teichm¨ uller space, Infinite dimensional algebras 108. and quantum integrable systems (Basel), Progr. Math., vol. 237, Birkhäuser, Basel, 2005, pp. 225– 233. 109. H. Tamanoi, Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1995), 127–181. 110. W. P. Thurston, Zippers and schlicht functions, The Bieberbach conjecture, Proceedings of the symposium on the occasion of the proof of the Bieberbach conjecture held at Purdue University, West Lafayette, Ind., March 11–14, 1985 (Providence, RI) (D. Drasin, P. Duren, and A. Marden, eds.), Mathematical Surveys and Monographs, vol. 21, American Mathematical Society, 1986, pp. 185–187. 111. M. Toki, On nonstarlikeness of Teichm¨ uller spaces, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 58–60. 112. S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266–267. 113. A. J. Tromba, Teichm¨ uller Theory in Riemannian Geometry, Birkhäuser, Basel, 1992. ¨ 114. K.-J. Wirths, Uber holomorphe Funktionen, die einer Wachstumsbeschr¨ ankung unterliegen, Arch. Math. 30 (1978), 606–612. 115. S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J. 28 (1999), 217–230. 116. I. V. Zhuravlev, Model of the universal Teichm¨ uller space, Siberian Math. J. 27 (1986), 691–697. ˇ 117. I. V. Zuravlev, Univalent functions and Teichm¨ uller spaces, Soviet Math. Dokl. 21 (1980), 252–255.


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Toshiyuki Sugawa Address: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan E-mail: [email protected]


Metrics and quasiregular mappings Matti Vuorinen Abstract. This series of lectures intends to provide a gateway to some selected topics of quasiconformal and quasiregular maps, in particular to the main themes of [Vu1] and [Vu3]. Some of the basic notions and tools are briefly reviewed. Several problems, exercises and open problems are given throughout the text. At the end of the paper a short list of some generic open problems is presented for metric spaces, which allow a great number of variations in specific cases. Keywords. quasiregular mappings, metric spaces. 2000 MSC. 30C65.

Contents 1. Introduction 2. Möbius transformations 3. Hyperbolic geometry 4. Quasihyperbolic geometry 5. Modulus and capacity 6. Conformal invariants 7. Distortion theory 8. Open problems References

291 295 300 305 306 313 319 322 323

1. Introduction The goal of these lectures is to provide an introduction to some of the main properties of quasiconformal and quasiregular mappings. One of the central themes here will be to study how these mappings deform distances and metrics and therefore it is natural to study our mappings between metric spaces. In most cases, the metrics will have some useful invariance or quasi-invariance properties under a set Γ of transformations, called rigid motions. An important example is the unit ball of Rn equipped with the hyperbolic metric in which case we may take the set Γ to be the group of the Möbius self-mappings of the ball. Version October 19, 2006.

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The material is largely drawn from [Vu3] and [AVV2]. In order to give the reader a chance to enter gradually this territory of mathematical research, problems of varying level are given, from easy exercises to research problems. Many more can be found in [Vu3] and [AVV2] (the exercises in [AVV2] come with solutions). Some research problems are collected at the end of the paper. Because of limitations of space, most of the details/proofs are omitted with the general reference to [V1] and [Vu3]. The idea of using invariance with respect to rigid motion to study function theory is very old. In fact, it can be traced back to nineteenth century, in particular, to the work of F. Klein. Perhaps the most natural notion of invariance is conformal invariance under the group of conformal self-maps of a given simplyconnected domain. Several conformal invariants emerged from the studies of H. Grötzsch, L. Ahlfors, and A. Beurling. A pair (X, d) is called a metric space if X 6= ∅ and d : X ×X → [0, ∞) satisfies the following four conditions  (M1) d(x, y) ≥ 0 for all x, y ∈ X ,    (M2) d(x, y) = 0 iff x = y , (1.1) (M3) d(x, y) = d(y, x) for all x, y ∈ X ,    (M4) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X .

Let (X, d1 ) and (Y, d2 ) be metric spaces and let f : X → Y be a continuous mapping. Then we say that f is uniformly continuous if there exists an increasing continuous function ω : [0, ∞) → [0, ∞) with ω(0) = 0 and d2 (f (x), f (y)) ≤ ω(d1 (x, y)) for all x, y ∈ X . We call the function ω the modulus of continuity of f . If there exist C, α > 0 such that ω(t) ≤ Ctα for all t > 0 , we say that f is Hölder-continuous with Hölder exponent α . If α = 1 , we say that f is Lipschitz with the Lipschitz constant C or simply C-Lipschitz. If f is a homeomorphism and both f and f −1 are C-Lipschitz, then f is C-bilipschitz or C-quasiisometry and if C = 1 we say that f is an isometry. These conditions are said to hold locally, if they hold for each compact subset of X . 1.2. Exercise. If h : [0, ∞) → [0, ∞) is a function and h(t)/t is decreasing, show that h(x + y) ≤ h(x) + h(y) for all x, y > 0. In particular, show that if (X, d) is a metric space, then also (X, dα ), α ∈ (0, 1), is. 1.3. Exercise. Let f : [0, ∞) → [0, ∞) be Hölder-continuous with exponent β > 1. Show that f (x) = f (0) for all x > 0 . 1.4. Example. Let f : Rn → Rn , f (x) = |x|α−1 x, f (0) = 0. Then f is Höldercontinuous with exponent α . In most examples below, the metric spaces will have some additional structure. The metrics will often have some quasiinvariance properties. For instance, we say that a pair of metric spaces (Xj , dj ), j = 1, 2, is quasiinvariant under a set Γ of mappings f : (X1 , d1 ) → (X2 , d2 ) if there exists C ≥ 1 such that 1/C ≤ d2 (f (x), f (y))/d1 (x, y) ≤ C for all x, y ∈ X1 , x 6= y and all f ∈ Γ . In particular,

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we will study metric spaces (X, d) where the group Γ of automorphisms of X acts transitively (i.e. given x, y ∈ X there exists h ∈ Γ such that hx = y .) If C = 1, then we say that d is invariant. 1.5. Examples. (1) The euclidean space Rn equipped with the usual metric Pn |x − y| = ( j=1 (xj − yj )2 )1/2 , Γ is the group of translation in Rn . (2) The unit sphere S n = {z ∈ Rn+1 : |z| = 1} equipped with the metric of R and Γ is the set of rotations of S n . n+1

(3) Let G ⊂ Rn , G 6= Rn , for x, y ∈ G set |x − y| jG (x, y) = log 1 + . min{d(x, ∂G), d(y, ∂G)}

Then one can prove that jG is a metric (this is a folklore result, see e.g. [S]). In fact, there exists a constant C > 1 such that for the unit ball Bn of Rn 1/C ≤ ρBn (x, y)/jBn (x, y) ≤ C

for all x, y ∈ Bn , x 6= y . Here ρBn is the hyperbolic metric of Bn and it is invariant under the group of Möbius self-mappings of Bn . For the definition of ρBn see below or [Vu3, Section2]. A basic geometric object of a metric space (X, d) is the ball BX (z, r) = {x ∈ X : d(x, z) < r} . In order to study how balls and their boundary spheres are deformed under homeomorphisms, we introduce a deformation measure Hf (x0 , r) of a ball under a homeomorphism f : (X1 , d1 ) → (X2 , d2 ) at a point x ∈ X1 d2 (f (x), f (y)) : d1 (z, x) = d1 (y, x) = r . Hf (x, r) = sup d2 (f (x), f (z))

f r x

Lr lr

f(x)

Figure 1. Hf (x, r) . If f maps spheres centered at x onto spheres centered at f (x) , then Hf (x, r) = 1. For instance the above radial mapping x 7→ |x|α−1 x has the property Hf (0, r) = 1 for all r > 0 . Recall from Complex Analysis that a conformal map f : D1 → D2 , Dj ⊂ C, j = 1, 2, satisfies limr→0 Hf (x, r) = 1 for all x ∈ D1 . We say that

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a homeomorphism f : (X1 , d1 ) → (X2 , d2 ) is quasiconformal (with respect to (d1 , d2 )), if there exists C ∈ [1, ∞) such that for all x ∈ X1 Hf (x) = lim sup Hf (x, r) ≤ C . r→0

If f is L-bilipschitz, then f satisfies the above condition with C = L2 . Let Gj ⊂ Rn , j = 1, 2, be domains and let f : G1 → G2 be a homeomorphism. Suppose now that there exists a constant C ≥ 1 such that for all subdomains D ⊂ G1 the mapping f |D : (D, jD ) → (f (D), jf (D) ) is C-Lipschitz. Fix x0 ∈ G1 and r ∈ (0, d(x0 , ∂G1 )/2) . If |x − x0 | = |y − x0 | = r and G = B n (x0 , 2r) \ {x0 } , then jG (x, y) ≤ log 3 and we obtain by the above C-Lipschitz-property log |f (x) − f (x0 )| ≤ jf G (f (x), f (y)) ≤ CjG (x, y) ≤ C log 3 , |f (y) − f (x0 )|

and hence Hf (x0 ) ≤ 3C , where we used the triangle inequality (Lemma 3.21 (3) below) and the fact that x0 ∈ ∂G . Now d1 and d2 are the usual metrics. Thus we see that our map is quasiconformal. In this argument the fact that x0 ∈ ∂G played a key role. For most of the metrics that we will consider here, even one single boundary point will be important. Most of the metrics will also be monotone with respect to the domain. Thus, for instance, if G1 ⊂ G2 ⊂ Rn are domains, then jG1 (x, y) ≥ jG2 (x, y) for all x, y ∈ G1 and for a fixed x0 ∈ G1 , jG1 (x0 , x) → ∞ as x → x1 ∈ ∂G . In the above argument, it was assumed that f |D is Lipschitz continuous for all subdomains D of G1 but we only used this property for subdomains of the form B n (x, r) \ {x} , x ∈ G1 . In order to motivate this condition let us recall that a conformal map is conformal also in every subdomain. Here we have studied the metric j , mainly because it is very easy to define and because it well represents the metrics we study here. There are now several questions: (a) Can we characterize the class of quasiisometries or isometries in the above sense? (b) Can we prove similar results for other metrics (and what are these metrics)? (c) Can we say more for the case when the domains are ”nice” (for instance quasidisks)? Conformal invariants and conformally invariant metrics have been an important topic in geometric function theory during the past century. One of the first promoters of these ideas was F. Klein. In the context of quasiconformal mappings these ideas emerged as a result of the pioneering works of H. Grötzsch, O. Teichm¨ uller, L. Ahlfors and A. Beurling on quasiconformal maps in plane domains [LV], [K]. Extension to higher dimensions is due to F.W. Gehring and J. Väisälä [G] , [V1]. The case of metric measure spaces has been studied recently by J. Heinonen, P. Koskela and many other people [H].


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In the setup presented here, the aforementioned questions (a)-(c) were studied already in [Vu1] and [Vu3]. But only very few answers are known, see [H1], [H2], [HI]. These questions could also be investigated for some particular classes of domains, which would bring a very wide spectrum of new questions into play. Some examples of such classes of domains would be uniform domains and quasiconformal balls. As we will see, there still are numerous open problems in this area. It is assumed that the reader is familiar with some basic facts and definitions of the theory of quasiconformal and quasiregular maps [V1], [Vu3].

2. M¨ obius transformations For x ∈ Rn and r > 0 let

B n (x, r) = { z ∈ Rn : |x − z| < r }, S n−1 (x, r) = { z ∈ Rn : |x − z| = r }

denote the ball and sphere, respectively, centered at x with radius r. The abbreviations B n (r) = B n (0, r), S n−1 (r) = S n−1 (0, r), Bn = B n (1), S n−1 = S n−1 (1) will be used frequently. For t ∈ R and a ∈ Rn \ {0} we denote P (a, t) = { x ∈ Rn : x · a = t } ∪ {∞}. n

Then P (a, t) is a hyperplane in R = Rn ∪ {∞} perpendicular to the vector a, at distance t/|a| from the origin. 2.1. Definition. Let D and D′ be domains in Rn and let f : D → D′ be a homeomorphism. We call f conformal if (1) f ∈ C 1 , (2) Jf (x) 6= 0 for all x ∈ D, and (3) |f ′ (x)h| = |f ′ (x)||h| for all x ∈ D and all h ∈ Rn . If D and n D′ are domains in R , we call a homeomorphism f : D → D′ conformal if the restriction of f to D \ {∞, f −1 (∞)} is conformal. 2.2. Examples. Some basic examples of conformal mappings are the following elementary transformations. (1) A reflection in P (a, t): f1 (x) = x − 2(x · a − t)

a , f1 (∞) = ∞ . |a|2

(2) An inversion (reflection) in S n−1 (a, r): f2 (x) = a +

r2 (x − a) , f2 (a) = ∞ , f2 (∞) = a . |x − a|2

(3) A translation f3 (x) = x + a , a ∈ Rn , f3 (∞) = ∞.

(4) A stretching by a factor k > 0: f4 (x) = kx , f4 (∞) = ∞. (5) An orthogonal mapping, i.e. a linear map f5 with |f5 (x)| = |x| , f5 (∞) = ∞ .

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2.3. Remarks. (1) The translation x 7→ x + a can be written as a composition of reflections in P (a, 0) and P (a, 21 |a|2 ). The stretching x 7→ kx, k > 0, can be √ written as a composition of inversions in S n−1 (0, 1) and S n−1 (0, k ). It can be proved, that an orthogonal mapping can be composed of at most n+1 reflections in planes (see [BE, p. 23, Theorem 3.1.3]). (2) It is easy to show that f1 (f1 (x)) = x and f2 (f2 (x)) = x for all x ∈ Rn , i.e. f1 and f2 are involutions. (3) It can also be shown that we have the difference formula r2 |x − y| |f2 (x) − f2 (y)| = |x − a||y − a|

for all x, y ∈ Rn \ {a}. (4) If an = 0, then one can show that f2 (Hn ) = Hn and that for all x, y ∈ Hn

|f2 (x) − f2 (y)|2 |x − y|2 = . (f2 (x))n (f2 (y))n (x)n (y)n (5) Reflections and inversions are sense-reversing. The composition of two sensereversing maps is sense-preserving. n

n

2.4. Definition. A homeomorphism f : R → R is called a M¨ obius transformation if f = g1 ◦ · · · ◦ gp where each gj is one of the elementary transformations in 2.2(1)–(5) and p is a positive integer. Equivalently (see 2.3) f is a Möbius transformation if f = h1 ◦ · · · ◦ hm where each hj is a reflection in a sphere or in a n hyperplane and m is a positive integer. If G ⊂ R the set of all (sense-preserving) Möbius transformations mapping G onto itself is denoted by GM(G) (M(G)). n

It will be convenient to identify R with the subset { x ∈ Rn : xn+1 = 0 }∪{∞} n+1 of R . The identification is given by the embedding (2.5) x 7→ x˜˜ = (x1 , . . . , xn , 0) ; x = (x1 , . . . , xn ) ∈ Rn .

We are now going to describe a natural two–step way of extending a Möbius n transformation of Rn to a Möbius transformation of Rn+1 . First, if f in GM(R ) ˜˜, t) or S n (a ˜˜, r), is a reflection in P (a, t) or in S n−1 (a, r), let f˜˜ be a reflection in P (a n respectively. Then if x ∈ R and y = f (x), by 2.2(1)–(2) we get g (2.6) f˜˜(x1 , . . . , xn , 0) = (y1 , . . . , yn , 0) = fg (x) . By (2.6) we may regard f˜˜ as an extension of f . Note that f˜˜ preserves the plane

xn+1 = 0 and each of the half–spaces xn+1 > 0 and xn+1 < 0. These facts follow n from the formulae 2.2(1)–(2). Second, if f is an arbitrary mapping in GM(R ) it has a representation f = f1 ◦ · · · ◦ fm where each fj is a reflection in a plane or a sphere. Then f˜˜ = f˜˜1 ◦ · · · ◦ f˜˜m is the extension of f , and it preserves the half–spaces xn+1 > 0, xn+1 < 0, and the plane xn+1 = 0. In conclusion, every n n+1 f in GM(R ) has an extension f˜˜ in GM(R ). It follows from [BE, p. 31,


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Theorem 3.2.4] that such an extension f˜˜ of f is unique. The mapping f˜˜ is called the Poincaré extension of f . In the sequel we shall write x, f instead of x˜˜, f˜˜, respectively. Many properties of plane Möbius transformations hold for n–dimensional n Möbius transformations as well. The fundamental property that spheres of R (which are spheres or planes in Rn , see [Vu1, Exercise 1.26, p.8]) are preserved under Möbius transformations is proved in [BE, p. 28, Theorem 3.2.1]. 2.7. Stereographic projection. The stereographic projection S n ( 21 en+1 , 21 ) is defined by (2.8)

π(x) = en+1 +

n

π: R

→

x − en+1 , x ∈ Rn ; π(∞) = en+1 2 |x − en+1 | n

Then π is the restriction to R of the inversion in S n (en+1 , 1). In fact, we can identify π with this inversion. Because f −1 = f for every inversion f , it follows n that π maps the “Riemann sphere” S n ( 21 en+1 , 21 ) onto R . n

The spherical (chordal) metric q in R is defined by (2.9)

n

q(x, y) = |π(x) − π(y)| ; x, y ∈ R ,

where π is the stereographic projection (2.8). From the definition (2.8) by calculating we obtain  |x − y|   q(x, y) = p p ; x 6= ∞ 6= y,    1 + |x|2 1 + |y|2 (2.10)   1   .  q(x, ∞) = p 1 + |x|2

For x ∈ Rn \ {0} the antipodal (diametrically opposite) point x˜, is defined by x (2.11) x˜ = − 2 |x|

and we set ∞ ˜ = 0, 0˜ = ∞ . Then, by (2.10), q(x, x˜ ) = 1 and hence π(x),π(˜ x) are indeed diametrically opposite points on the Riemann sphere. n

2.12. Balls in the spherical metric. For x ∈ R and r ∈ (0, 1) we define the spherical ball (2.13)

n

Q(x, r) = { z ∈ R : q(x, z) < r } .

Its boundary sphere is denoted by ∂Q(x, r). From the Pythagorean theorem it follows that (cf. (2.11)) √ n Q(x, r) = R \ Q x˜, 1 − r2 . (2.14)

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en+1 π(x) π(y)

y

0 x

_

n

Figure 2. Formulae (2.8) and (2.9) visualized.

Figure 3. A cross-section of the Riemann sphere. To gain insight into the geometry of spherical balls Q(x, r) it is convenient to study the image πQ(x, r) under the stereographic projection π (see figure 3). Indeed, by definition (2.9) we see that πQ(x, r) = B n+1 π(x), r ∩ S n ( 21 en+1 , 21 ). (2.15) Either by this formula or more directly by the definition of the spherical metric (plus the fact that Möbius transformations preserve spheres) we see that in the euclidean geometry, Q(x, r) is a point set of one of the following three kinds (a) an open ball B n (u, s), n (b) the complement of B (v, t) in Rn , (c) a half–space of Rn .


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n Clearly, ∂Q(x, r) is either a sphere √ or a hyperplane of R . Formula (2.14) shows, in particular, that πQ(x, 1/ 2 ) is a half–sphere of the Riemann sphere S n ( 21 en+1 , 21 ).

2.16. Absolute ratio. For an ordered quadruple a, b, c, d of distinct points in n R we define the absolute (cross) ratio by q(a, c) q(b, d) | a, b, c, d | = (2.17) . q(a, b) q(c, d) It follows from (2.10) that for distinct a, b, c, d in Rn |a − c| |b − d| . | a, b, c, d | = |a − b| |c − d| One of the most important properties of Möbius transformations is that they preserve absolute ratios, i.e. if f ∈ GM, then (2.18)

| f (a), f (b), f (c), f (d) | = | a, b, c, d | n

for all distinct a, b, c, d in R . As a matter of fact, the preservation of absolute ratios is a characteristic property of Möbius transformations. It is proved in [BE, n n p. 72, Theorem 3.2.7] that a mapping f : R → R is a Möbius transformation if and only if f preserves all absolute ratios. 2.19. Automorphisms in Bn . We shall give a canonical representation for the maps in M(Bn ). Assume that f is in M(Bn ) and that f (a) = 0 for some a ∈ Bn . We denote a a∗ = 2 , a ∈ Rn \ {0} (2.20) |a| ∗ ∗ and 0 = ∞, ∞ = 0. Fix a ∈ Bn \ {0}. Let (2.21)

σa (x) = a∗ + r2 (x − a∗ )∗ , r2 = |a|−2 − 1

be an inversion in the sphere S n−1 (a∗ , r) orthogonal to S n−1 . Then σa (a) = 0, σa (a∗ ) = ∞, σa (Bn ) = Bn . Let pa denote the reflection in the (n − 1)–dimensional plane P (a, 0) through the origin and orthogonal to a and define a sense–preserving Möbius transformation by Ta = pa ◦ σa . Then, by (2.21), Ta Bn = Bn , Ta (a) = 0, and with ea = a/|a| we have Ta (ea ) = ea , Ta (−ea ) = −ea . For a = 0 we set T0 = id, where id stands for the identity map. The proof of the following fundamental fact can be found in [A, p. 21], [BE, p. 40, Theorem 3.5.1]. We now define a spherical isometry tz in M(Rn ) which maps a given point z ∈ Rn to 0 as follows. For z = 0 let tz = id and for z = ∞ let tz = p ◦ f , where f is inversion in S n−1 and p is reflection in the (n − 1)–dimensional plane p x1 = 0. n n−1 2 For z ∈ R \ {0} let sz be inversion in S (−z/|z| , r), where r = 1 + |z|−2 . According to [Vu1, (1.45)], the inversion sz is a spherical isometry and it is easy to show that sz (z) = 0. Let pz be reflection in the plane P (z, 0). Defining (2.22)

tz = pz ◦ sz ,

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s S

r

1

0

a

b

Figure 4. Inversion in S n−1 , b = a∗ . we see that tz ∈ M(Rn ) is a spherical isometry with tz (z) = 0. Hence √ tz (Q(z, r)) = Q(0, r) = B n r/ 1 − r2 , (2.23)

|tz (x)|2 = for all x, z ∈ Rn , r ∈ (0, 1).

q(x, z)2 1 − q(x, z)2

2.24. Lemma. Let a ∈ Rn , r > 0, and let b ∈ Rn , u > 0, be such that B n (a, r) = Q(b, u). If f is the inversion in S n−1 (a, r), then f = t−1 b ◦ f1 ◦ tb ,

where tb √is the spherical isometry defined in (2.22) and f1 is the inversion in S n−1 (u/ 1 − u2 ) = ∂Q(0, u).

3. Hyperbolic geometry Hyperbolic geometry can be developed in the context of two spaces or, as they are sometimes called, models. These two models of the hyperbolic space are the unit ball Bn and the Poincaré half–space Hn = Rn+ = { (x1 , . . . , xn ) ∈ Rn : xn > 0 } .

These two models can be equipped with a hyperbolic metric ρ that is unique up to a multiplicative constant in either model. In either model the metric is normalized (by giving the element of length of the metric) in such a way that for all x, y ∈ Bn ρHn h(x), h(y) = ρBn (x, y) whenever h ∈ GM and hBn = Hn . Therefore both models are conformally compatible in the sense that the two metric spaces (Bn , ρ) and (Hn , ρ) can be identified. This compatibility is very convenient in computations because we may


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do a computation in that model in which it is easier, without loss of generality. In what follows we shall use the symbols Rn+ and Hn interchangeably. For A ⊂ Rn let A+ = { x ∈ A : xn > 0 }. We define a weight function w : Rn+ → R+ = { x ∈ R : x > 0 } by (3.1)

w(x) =

1 , x = (x1 , . . . , xn ) ∈ Rn+ . xn

If γ : [0, 1) → Rn+ is a continuous mapping such that γ[0, 1) is a rectifiable curve with length s = ℓ(γ), then γ has a normal representation γ 0 : [0, s) → Rn+ parametrized by arc length (see J. Väisälä [V, p. 5]). The hyperbolic length of γ[0, 1) is defined by Z s Z |dx| 0 ′ 0 ℓh (γ[0, 1)) = |(γ ) (t)| w(γ (t))dt = (3.2) . 0 γ xn If A ⊂ Rn+ is a (Lebesgue) measurable set we define the hyperbolic volume of A by Z (3.3) mh (A) = w(x)n dm(x) , A

where m stands for the n–dimensional Lebesgue measure and w is as in (3.1). If a, b ∈ Rn+ , then the hyperbolic distance between a and b is defined by Z |dx| (3.4) , ρ(a, b) = inf ℓh (α) = inf α∈Γab α∈Γab α xn where Γab stands for the collection of all rectifiable curves in Rn+ joining a and b. Sometimes the more complete notation ρRn+ (a, b) or ρHn (a, b) will be employed.

Figure 5. Some geodesics of Hn = Rn+ . The infimum in (3.4) is in fact attained: for given a, b ∈ Rn+ there exists a circular arc L perpendicular to ∂Rn+ such that the closed subarc J[a, b] of L with

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end points a and b satisfies (3.5)

ρ(a, b) = ℓh (J[a, b]) =

Z

|dx| . xn

J[a,b]

If a and b are located on a normal of ∂Rn+ , then J[a, b] = [a, b] = { (1 − t)a + tb : 0 ≤ t ≤ 1 } (cf. [BE, p. 134]). Because of the (hyperbolic) length–minimizing property (3.5), the arc J[a, b] will be called the geodesic segment joining a and b. Knowing the geodesics, we calculate the hyperbolic distance in two special cases. First, for r, s > 0 we obtain Z r dt r ρ(ren , sen ) = (3.6) = log . t s s Second, if ϕ ∈ (0, 12 π) we denote uϕ = (cos ϕ)e1 + (sin ϕ)en and calculate (3.7)

ρ(en , uϕ ) =

Z

J[uϕ ,en ]

dα = sin α

Zπ/2

dα = log cot 12 ϕ . sin α

ϕ

Figure 6. The points uϕ and en . We shall often make use of the hyperbolic functions sh x = sinh x, ch x = cosh x, th x = tanh x, cth x = coth x and their inverse functions. The above formulae (3.6) and (3.7) are special cases of the general formula (see [BE, p. 35]) |x − y|2 , x, y ∈ Hn = Rn+ . 2xn yn Note that by this formula the hyperbolic distance ρ(x, y) is completely determined once the euclidean distances xn = d(x, ∂Hn ), yn = d(y, ∂Hn ), and |x − y| are known. In passing we note that if f2 ∈ GM(Hn ) is as defined in Remark 2.3(4), then ρ(x, y) = ρ(f2 (x), f2 (y)) for all x, y ∈ Hn . For another formulation of (3.8) let z, w ∈ Hn , let L be an arc of a circle perpendicular to ∂Hn with z, w ∈ L and let {z∗ , w∗ } = L ∩ ∂Hn , the points being labelled so that z∗ , z, w, w∗ occur in this order on L. Then (cf. [BE, p. 133, (7.26)]) (3.8)

(3.9)

ch ρ(x, y) = 1 +

ρ(z, w) = log | z∗ , z, w, w∗ | .


303

Figure 7. The quadruple z∗ , z, w, w∗ . Note that (3.6) is a special case of (3.9) when z∗ = 0 and w∗ = ∞ because | 0, z, w, ∞ | = |w|/|z| for z, w ∈ Hn . The invariance of ρ is apparent by (3.9) and (2.18): If f in GM(Hn ), then for all x, y ∈ Hn

(3.10)

ρ(x, y) = ρ(f (x), f (y)) .

For a ∈ Hn and M > 0 the hyperbolic ball { x ∈ Hn : ρ(a, x) < M } is denoted by D(a, M ). It is well known that D(a, M ) = B n (z, r) for some z and r (this also follows from (3.10)! ). This fact together with the observation that λten , (t/λ)en ∈ ∂D(ten , M ), λ = eM (cf. (3.6)), yields   D(ten , M ) = B n (t ch M )en , t sh M , B n (ten , rt) ⊂ D(ten , M ) ⊂ B n (ten , Rt) , (3.11)  r = 1 − e−M , R = eM − 1 .

Figure 8. The hyperbolic ball D(ten , M ) as a euclidean ball. A counterpart of (3.8) for Bn is |x − y|2 (3.12) , x, y ∈ Bn , sh2 21 ρ(x, y) = 2 2 (1 − |x| )(1 − |y| )

(cf. [BE, p. 40]). As in the case of Hn , we see by (3.12) that the hyperbolic distance ρ(x, y) between x and y is completely determined by the euclidean

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quantities |x − y|, d(x, ∂Bn ), d(y, ∂Bn ). Finally, we have also (3.13)

ρ(x, y) = log | x∗ , x, y, y∗ | ,

where x∗ , y∗ are defined as in (3.9): If L is the circle orthogonal to S n−1 with x, y ∈ L, then {x∗ , y∗ } = L ∩ S n−1 , the points being labelled so that x∗ , x, y, y∗ occur in this order on L. It follows from (3.13) and (2.18) that (3.14)

ρ(x, y) = ρ(h(x), h(y))

for all x, y ∈ Bn whenever h is in GM(Bn ). Finally, in view of (2.18), (3.9), and (3.13) we have (3.15)

ρBn (x, y) = ρHn (g(x), g(y)) , x, y ∈ Bn ,

whenever g is a Möbius transformation with gBn = Hn . It is well known that the balls D(z, M ) of (Bn , ρ) are balls in the euclidean geometry as well, i.e. D(z, M ) = B n (y, r) for some y ∈ Bn and r > 0. Making use of this fact, we shall find y and r. Let Lz be a euclidean line through 0 and z and {z1 , z2 } = Lz ∩ ∂D(z, M ), |z1 | ≤ |z2 |. We may assume that z 6= 0 since with obvious changes the following argument works for z = 0 as well. Let e = z/|z| and z1 = se, z2 = ue, u ∈ (0, 1), s ∈ (−u, u). Then it follows that 1 + |z| 1 − s · =M , ρ(z1 , z) = log 1 − |z| 1 + s 1 + u 1 − |z| =M . · ρ(z2 , z) = log 1 − u 1 + |z| Solving these for s and u and using the fact that D(z, M ) = B n

1 (z 2 1

+ z2 ), 21 |u − s|

one obtains the following formulae:  D(x, M ) = B n (y, r)    (3.16) x(1 − t2 ) (1 − |x|2 )t   , r = , t = th 21 M ,  y= 2 2 2 2 1 − |x| t 1 − |x| t

and

(3.17)

 B n x, a(1 − |x|) ⊂ D(x, M ) ⊂ B n x, A(1 − |x|) ,    t(1 + |x|) t(1 + |x|)   , A= , t = th 12 M .  a= 1 + |x|t 1 − |x|t

We shall often need a special case of (3.16): (3.18)

1 D(0, M ) = B n (th M ) . 2


305

A standard application of formula (3.18) is the following observation. Let Tx be in M(Bn ) as defined in 2.19 with Tx (x) = 0. Fix x, y ∈ Bn and z ∈ J[x, y] with ρ(z, x) = ρ(z, y) = 12 ρ(x, y). Then Tz (x) = −Tz (y) and (3.18) yields |Tx (y)| = th 12 ρ(x, y) , (3.19) |Tz (x)| = th 14 ρ(x, y) . For an open set D in Rn , D 6= Rn , define d(z) = d(z, ∂D) for z ∈ D and |x − y| (3.20) jD (x, y) = log 1 + min{d(x), d(y)} for x, y ∈ D. Then it is well-known that jD is a metric (see, e.g. [S]).

3.21. Lemma. The following inequalities d(x) (1) jD (x, y) ≥ log , d(y) |x − y| d(x) + log 1 + ≤ 2 jD (x, y) (2) jD (x, y) ≤ log d(y) d(x) |x − z| (3) jD (x, y) ≥ log |y − z| hold for all x, y ∈ D, z ∈ ∂D . In the next lemma we show that jD yields simple two–sided estimates for ρD both when D = Bn and when D = Hn . 3.22. Lemma. (1) jBn (x, y) ≤ ρBn (x, y) ≤ 4 jBn (x, y) for x, y ∈ Bn . (2) jHn (x, y) ≤ ρHn (x, y) ≤ 2 jHn (x, y) for x, y ∈ Hn .

4. Quasihyperbolic geometry In an arbitrary proper subdomain D of Rn one can define a metric, the quasihyperbolic metric of D, which shares some properties of the hyperbolic metric of Bn or Hn . We shall now give the definition of the quasihyperbolic metric and state without proof some of its basic properties which we require later on. The quasihyperbolic metric has been systematically developed and applied by F. W. Gehring and his collaborators. Throughout this section D will denote a proper subdomain of Rn . In D we define a weight function w : D → R+ by 1 (4.1) ; x∈D. w(x) = d(x, ∂D) Using this weight function one defines the quasihyperbolic length ℓq (γ) = ℓD q (γ) of a rectifiable curve γ by a formula similar to (3.2). The quasihyperbolic distance between x and y in D is defined by Z D kD (x, y) = inf ℓq (α) = inf (4.2) w(x)|dx| , α∈Γxy

α∈Γxy

α

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where Γxy is as in (3.4). It is clear that kD is a metric on D. It follows from (4.2) that kD is invariant under translations, stretchings, and orthogonal mappings. (As in (3.3) one can define the quasihyperbolic volume of a (Lebesgue) measurable set A ⊂ D, but we shall not make use of this notion.) Given x, y ∈ D there exists a geodesic segment JD [x, y] of the metric kD joining x and y (cf. [GO]). However, very little is known about the structure of such geodesic segments JD [x, y] when D is given. For some elementary domains, the geodesics were recently studied by H. Lindén [L]. 4.3. Remarks. Clearly, kHn = ρHn , and we see easily that ρBn ≤ 2 kBn ≤ 2 ρBn (cf. (4.1)). Hence, the geodesics of (Hn , kHn ) are those of (Hn , ρHn ), but it is a difficult task to find the geodesics of kD when D is given. The following monotone property of kD is clear: if D and D′ are domains with D′ ⊂ D and x, y ∈ D′ , then kD′ (x, y) ≥ kD (x, y). In order to find some estimates for kD (x, y) we shall employ, as in the case of Hn and Bn , the metric jD defined in (3.20). The metric jD is indeed a natural choice for such a comparison function since both kD and jD are invariant under translations, stretchings and orthogonal mappings. A useful inequality is ([GP, Lemma 2.1]) (4.4)

kD (x, y) ≥ jD (x, y) ; x, y ∈ D .

In combination with 3.22, (4.4) yields d(x) kD (x, y) ≥ log (4.5) , d(z) = d(z, ∂D) . d(y) For easy reference we record Bernoulli’s inequality

(4.6)

log(1 + as) ≤ a log(1 + s) ; a ≥ 1 , s > 0 .

4.7. Lemma. (1) If x ∈ D, y ∈ Bx = B n (x, d(x)), then |x − y| . kD (x, y) ≤ log 1 + d(x) − |x − y| (2) If s ∈ (0, 1) and |x − y| ≤ s d(x), then 1 jD (x, y) . kD (x, y) ≤ 1−s

5. Modulus and capacity For the definition and basic properties of the modulus we refer the reader to [V1]. The main sources for this section are [V1], [Vu3], [AVV2]. One of the main reasons why the modulus of a curve family is studied is that we have a simple rule of transformation for the modulus of a curve family under quasiconformal mappings. Further, we would like to use modulus as an instrument so as to gain insight about ”the geometry”. Roughly speaking we can say that the modulus of the family of all curves joining two connected nonintersecting continua E, F ⊂ Rn behaves like min{d(E), d(F )}/d(E, F ) , where


307

d stands for the euclidean diameter. A long series of estimates is needed to reach this conclusion and its variants and some of these estimates are given in this and the following section. Some of these variants involve hyperbolic or quasihyperbolic metric. In this fashion we step by step approach our goal, the study of how quasiconformal mappings between metric spaces deform distances. 5.1. Lemma. The p–modulus Mp is an outer measure in the space of all curve families in Rn . That is, (1) (2) (3)

Mp (∅) = 0 , Γ1 ⊂ Γ2 implies Mp (Γ1 ) ≤ Mp (Γ2 ) , ∞ ∞ X [ Mp (Γi ) . Γi ≤ Mp i=1

i=1

Let Γ1 and Γ2 be curve families in Rn . We say that Γ2 is minorized by Γ1 and write Γ2 > Γ1 if every γ ∈ Γ2 has a subcurve belonging to Γ1 . 5.2. Lemma. Γ1 < Γ2 implies Mp (Γ1 ) ≥ Mp (Γ2 ). 5.3. Remark. The family of all paths joining E and F in G is denoted by n ∆(E, F ; G) see [Vu3, p.51]. If G = Rn or R we often denote ∆(E, F ; G) by ∆(E, F ). Curve families of this form are the most important S∞ for what follows. The following subadditivity property is useful. If E = j=1 Ej and cE (F ) = P Mp ∆(E, F ) = cF (E), then cF (E) ≤ cF (Ej ), see 5.1(3). More precisely if n G ⊂ R is a domain and F ⊂ G is fixed, then cG F (E) = Mp ∆(E, F ; G) is an outer measure defined for E ⊂ G. In a sense which will be made precise later on, cE (F ) describes the mutual size and location of E and F . Assume now that n D is an open set in R and that F ⊂ D. It follows from 5.1(2) that Mp ∆(F, ∂D; D \ F ) ≤ Mp ∆(F, ∂D; D) ≤ Mp ∆(F, ∂D) . On the other hand, because ∆(F, ∂D; D) < ∆(F, ∂D) and ∆(F, ∂D; D \ F ) < ∆(F, ∂D; D) , 5.2 yields (5.4) Mp ∆(F, ∂D) = Mp ∆(F, ∂D; D) = Mp ∆(F, ∂D; D \ F ) . n

5.5. Lemma. Let D and D′ be domains in R and let f : D → D′ be a conformal mapping. Then M(f Γ) = M(Γ) for each curve family Γ in D where f Γ = { f ◦ γ : γ ∈ Γ }. 5.6. Lemma. Let p > 1 and let E and F be subsets of Rn+ . Then 1 Mp ∆(E, F ; Rn+ ) ≥ Mp ∆(E, F ) . 2 n

5.7. Corollary. Let E and F be sets in R with q( E, F ) ≥ a > 0. Then M ∆(E, F ) ≤ c(n, a) < ∞. 5.8. Lemma. (1) Let 0 < a < b and let E, F be sets in Rn with E ∩ S n−1 (t) 6= ∅ 6= F ∩ S n−1 (t)

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for t ∈ (a, b). Then

IWQCMA05

b M ∆( E, F ; B n (b) \ B n (a) ) ≥ cn log . a

Equality holds if E = (ae1 , be1 ), F = (−be1 , −ae1 ). Here cn > 0 depends only on n (see [V1, (10.11),(10.4)]). (2) Let 0 < a < b . Then M ∆( S n−1 (a), S n−1 (a); B n (b) \ B n (a) ) = ωn−1 (log(b/a))1−n ,

where ωn−1 is the (n − 1)-dimensional surface area of S n−1 .

5.9. Corollary. If E and F are non–degenerate continua with 0 ∈ E ∩ F then M ∆(E, F ) = ∞.

Proof. Apply 5.8 with a fixed b such that S n−1 (b) ∩ E 6= ∅ 6= S n−1 (b) ∩ F and let a → 0. 5.10. Canonical ring domains. A domain (open, connected set) D in Rn is called a ring domain or, briefly, a ring, if Rn \ D consists of two components C0 and C1 . Sometimes we denote such a ring by R(C0 , C1 ). In our study two canonical ring domains will be of particular importance. These are the Grötzsch ring RG,n (s), s > 1, and the Teichm¨ uller ring RT,n (t), t > 0, defined by RG,n (s) = R(Bn , [se1 , ∞]), s > 1, (5.11) RT,n (t) = R([−e1 , 0], [te1 , ∞]), t > 0. Sometimes we also use the bounded Grötzsch ring R(Rn \ Bn , [0, re1 ]) . An important conformal invariant associated with a ring is the modulus of the family of curves joining its complementary components. In the case of Grötzsch ring RG,n (s) and Teichm¨ uller ring RT,n (t) the modulus is denoted by γn (s) and τn (t) respectively. It is a well-known basic fact that γn : (1, ∞) → (0, ∞) is a decreasing homeomorphism and that for all s > 1 γn (s) = 2n−1 τn (s2 − 1) .

(5.12)

t

s

Bn

se1 cap R

G,n

(s) = M( s ) = (s) n

∞

-e1

0

∞

t e1

cap RT,n (t) = M(

Figure 9. The Grötzsch and Teichm¨ uller rings.

t

) = n(t)


309

w

z f

1

t

-t

-1

z

-1

-t

- r

-1

1

r

w

i K '/2

t

-K

1

w = r sn( ,r),

K -1 - r 2 i K Log z + K = π t

r

1

Figure 10. Conformal map of an annulus onto a disk minus a symmetric slit. z

w g

f

-1

-t

t

1

-1

- r

r

1

-1

0

a

1

g f

Figure 11. Conformal map of an annulus onto a bounded Grötzsch ring. 5.13. Elliptic integrals and γ2 (s). In the plane every ring domain can be conformally mapped onto an annulus {z ∈ C : 1 < |z| < M } for some M . For the Grötzsch ring this conformal mapping is given by the elliptic sn-function [AVV2]. For more information on the involved special functions see [QV]. As shown in [LV, II.2]

(5.14)

γ2 (s) = 2π/µ(1/s)

for s > 1 where

√ Z 1 π K( 1 − r2 ) µ(r) = , K(r) = [(1 − x2 )(1 − r2 x2 )]−1/2 dx 2 K(r) 0

for 0 < r < 1. The function K(r) is called a complete elliptic integral of the first kind and its values can be found in tables.

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5.5

5.5

5

5

4.5

4.5

4

4

3.5

3.5

3

3

K’

2.5

K

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

0.5

0

1

µ

0

0.5

1

Figure 12. The functions K(r) and µ(r) . The modulus µ(r) satisfies the following three functional identities  √ 2 = 1 π2 , r µ(r)µ 1 −   4  1 2 1−r µ(r)µ 1+r = 2 π , (5.15) √    µ(r) = 2µ 2 r . 1+r

From (5.15) one can derive several estimates for µ(r) [LV, p. 62]. By [LV, p. 62] the following inequalities hold √ 1 1 + 3 1 − r2 4 (5.16) log < log < µ(r) < log r r r for 0 < r < 1. From (5.16) it follows that limr→0+ µ(r) = ∞ whence, by virtue of the functional identities (5.15), limr→1− µ(r) = 0. Therefore, µ : (0, 1) → (0, ∞) is a decreasing homeomorphism. For the sake of completeness we set µ(0) = ∞ and µ(1) = 0. By (5.14) and (5.15) we obtain 4 s − 1 , s>1. (5.17) γ2 (s) = µ π s+1

5.18. Exercise. In the study of distortion theory of quasiconformal mappings in Section 7 below the following special function will be useful 1 ϕK,n (r) = −1 γn (Kγn (1/r))


311

for 0 < r < 1, K > 0. (Note: [Vu1, Lemma 7.20] shows that γn is strictly decreasing and hence that γn−1 exists.) Show that ϕAB,n (r) = ϕA,n (ϕB,n (r)) and ϕ−1 A,n (r) = ϕ1/A,n (r) and that ϕK,2 (r) = ϕK (r) = µ−1 K1 µ(r) . Verify also that

√

ϕ2 (r) = 21+rr √ 2 (2) ϕK (r)2 + ϕ1/K 1 − r2 = 1 . (1)

Exploiting (1) and (2) find ϕ1/2 (r). Show also that 1−ϕK (r) 1−r = 1+ϕ , (3) ϕ1/K 1+r K (r) √ √ 2 ϕ (r) (4) ϕK 21+rr = 1+ϕKK(r) . Lemma 7.22 in [Vu3] yields the inequalities

ωn−1 (log λn s)1−n ≤ γn (s) ≤ ωn−1 (log s)1−n ,

(5.19)

1−n

ωn−1 (log(λ2n s))

(5.20)

≤ τn (s − 1) ≤ ωn−1 (log s)1−n ,

for s > 1. 5.21. Theorem. The function gn (t) = (ωn−1 /γn (t))1/(n−1) −log t is an increasing function on (1, ∞) with limt→∞ gn (t) = log λn where λn ∈ [4, 2en−1 ), λ2 = 4 , is so-called Grötzsch ring constant. 5.22. Theorem. For s ∈ (1, ∞) and n ≥ 2

√ 1−n , (1) γn (s) ≤ ωn−1 µ(1/s)1−n < ωn−1 log(s + 3 s2 − 1 ) n−1

(2) 2

cn log

s + 1 s−1

n−1

≤ γn (s) ≤ 2

cn µ

Moreover, if s ∈ (0, ∞) and a = 1 + 2(1 +

s − 1

√

s+1

n−1

1 and K > 0, where α = K 1/(1−n) . From (5.26) it follows immediately that (5.27)

rα λn −α ≤ ϕK,n (r) ≤ λn rα

holds for all K > 0 and r ∈ (0, 1). For K ≥ 1 this inequality can be refined if we use Theorem 5.22 (1).


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5.28. Theorem. For n ≥ 2, K ≥ 1, and 0 ≤ r ≤ 1

(1) ϕK (r) ≤ λ1−α rα , α = K 1/(1−n) , n (2) ϕ1/K (r) ≥ λ1−β rβ , β = K 1/(n−1) . n

A compact set E ⊂ Bn is said to be of capacity zero, denoted cap E = 0 , if n M(∆(E, S n−1 (2))) = 0 . A compact set E ⊂ R is said to be of capacity zero, if it can be mapped by a Möbius transformation onto a set E1 ⊂ Bn of capacity zero. Sets of capacity zero are very small: they have zero Hausdorff dimension, see [Vu3, p.86]. For many purposes they are negligible. The next theorem provides a convenient tool for estimating moduli of curve families in terms is geometric quantities and a set function. 5.29. Theorem. For n ≥ 2 there exist positive numbers d1 , . . . , d4 and a set n function c(·) in R such that n n n and E ⊂ R . (1) c(E) = c(hE) whenever h : R → R is a spherical isometry S n (2) c(∅) = 0, A ⊂ B ⊂ R implies c(A) ≤ c(B) and c ∞ j=1 Ej P∞ n ≤ d1 j=1 c(Ej ) if Ej ⊂ R . n (3) If E ⊂ R is compact, then c(E) > 0 if and only if cap E > 0. Moreover n c(R ) ≤ d2 < ∞. n (4) c(E) ≥ d3 q(E) if E ⊂ R is connected and E 6= ∅. n (5) M ∆(E, F ) ≥ d4 min{ c(E), c(F ) }, if E, F ⊂ R . Furthermore, for n ≥ 2 and t ∈ (0, 1) there exists a positive number d5 such that n (6) M ∆(E, F ) ≤ d5 min{c(E), c(F )} whenever E, F ⊂ R and q(E, F ) ≥ t.

6. Conformal invariants

In the preceding sections we have studied some properties of the conformal invariant M ∆(E, F ; G) . In this section we shall introduce two other conformal invariants, the modulus metric µG (x, y) and its ”dual” quantity λG (x, y), where n G is a domain in R and x, y ∈ G. The modulus metric µG is functionally related to the hyperbolic metric ρG if G = Bn , while in the general case µG reflects the “capacitary geometry” of G in a delicate fashion. The dual quantity λG (x, y) is also functionally related to ρG if G = Bn . As shown in [Vu3] for a wide class of domains in Rn , the so–called QED–domains[GM], two–sided estimates for λG (x, y) in terms of rG (x, y) =

|x − y| . min{ d(x, ∂G), d(y, ∂G) }

6.1. The conformal invariants λG and µG . If G is a proper subdomain of n R , then for x, y ∈ G with x 6= y we define λG (x, y) = inf M ∆(Cx , Cy ; G) (6.2) Cx ,Cy

where Cz = γz [0, 1) and γz : [0, 1) → G is a curve such that γz (0) = z and γz (t) → ∂G when t → 1, z = x, y. It follows from 5.5 that λG is invariant under

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conformal mappings of G. That is, λf G (f (x), f (y)) = λG (x, y), if f : G → f G is conformal and x, y ∈ G are distinct.

Cy G

G

• y

Cxy •

Cx

x

•

• y

x G(x,y)

G (x,y)

Figure 14. The conformal invariants λG and µG . n

6.3. Remark. If card(R \ G) = 1, then λG (x, y) ≡ ∞ by 5.9. Therefore λG is n n of interest only in case card(R \ G) ≥ 2. For card(R \ G) ≥ 2 and x, y ∈ G, x 6= y, there are with C x ∩ C y = ∅ and thus continua Cx and Cy as in (6.2) n M ∆(Cx , Cy ; G) < ∞ by 5.7. Thus, if card(R \ G) ≥ 2, we may assume that the infimum in (6.2) is taken over continua Cx and Cy with C x ∩ C y = ∅. 6.4. The extremal problems of Gr¨ otzsch and Teichm¨ uller. The Grötzsch and Teichm¨ uller rings arise from extremal problems of the following type, which were first posed for the case of the plane: Among all ring domains which separate two given closed sets E1 and E2 , E1 ∩ E2 = ∅, find one whose module has the greatest value. Let E1 be a continuum and E2 consist of two points not separated by E1 . By the conformal invariance of the modulus one may then suppose that E1 = S 1 and E2 = {0, r}, 0 < r < 1. Then the extremal problem is solved by the bounded Grötzsch ring R(R2 \ B 2 , [0, r]). In other words, cap(B 2 , E) ≥ γ2 (1/r), where E ⊂ B 2 is any continuum joining the points 0 and r ∈ R. For details we refer the reader to [LV, Ch. II]. The following function is the solution of the generalization of the Teichm¨ uller problem to Rn . For x ∈ Rn \ {0, e1 }, n ≥ 2, define (6.5)

p(x) = inf M(△(E, F )), E,F

n

where the infimum is taken over all pairs of continua E and F in R with 0, e1 ∈ E, x, ∞ ∈ F . Teichm¨ uller applied a symmetrization method to prove that for n = 2, p(x) ≥ p((1 + |x − e1 |)e1 ) with equality for x = (1 + |x − e1 |)e1 . For more details, see [HV] and [SoV]. n

For a proper subdomain G of R and for all x, y ∈ G define µG (x, y) = inf M ∆(Cxy , ∂G; G) (6.6) Cxy


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3 2.5 2

∞

F

1.5 1

x 0.5 0 −0.5

0

1 E

−1 −1

0

1

2

3

4

Figure 15. The extremal problem of Teichm¨ uller. where the infimum is taken over all continua Cxy such that Cxy = γ[0, 1] and γ is a curve with γ(0) = x and γ(1) = y. It is clear that µG is also a conformal invariant in the same sense as λG . It is left as an easy exercise for the reader to verify that µG is a metric if cap∂G > 0. [Hint: Apply 5.3 and 5.29.] If cap∂G > 0, we call µG the modulus metric or conformal metric of G. 6.7. Remark. Let D be a subdomain of G. It follows from 5.3 and (5.4) that µG (a, b) ≤ µD (a, b) for all a, b ∈ D and λG (a, b) ≥ λD (a, b) for all distinct a, b ∈ D. In what follows we are interested only in the non–trivial case n card(R \ G) ≥ 2. Moreover, by performing an auxiliary Möbius transformation, n we may and shall assume that ∞ ∈ R \ G throughout this section. Hence G will have at least one finite boundary point. In a general domain G, the values of λG (x, y) and µG (x, y) cannot be expressed in terms of well–known simple functions. For G = Bn they can be given in terms of ρ(x, y) and the capacity of the Teichm¨ uller condenser. 6.8. Theorem. The following identities hold for all distinct x, y ∈ Bn : ! 1 1 n−1 =γ , (1) µBn (x, y) = 2 τ th 21 ρ(x, y) sh2 21 ρ(x, y) 1 2 1 (2) λBn (x, y) = τ sh ρ(x, y) . 2 2

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6.9. Remark. (1) In [Vu3, p. 193] it was stated as an open problem, whether λD (x, y)1/(1−n) is a metric when D = Rn \ {0} and n = 2 . Subsequently the problem was solved by A. Solynin [So] and J. Jenkins [J] for n = 2 . J. Ferrand [F] proved that λD (x, y)1/(1−n) is a metric for all D ⊂ Rn , n ≥ 2. (2) From 5.22(3) we obtain the following inequality for x, y ∈ Bn (exercise) 1 1 1 τ sh2 ρ(x, y) ≥ −cn log th ρ(x, y) 2 2 4 1

1

= 2cn arth e− 2 ρ(x,y) ≥ 2cn e− 2 ρ(x,y) .

Here the identities 2 ch2 A = 1 + ch 2A , sh 2A = 2 ch A sh A , and log th s = −2 arth e−2s were applied. Recall that |x − y|2 1 sh2 ρ(x, y) = 2 (1 − |x|2 )(1 − |y|2 )

by (3.12). Similarly, by 5.22(3) we obtain also 1 1 1 4 1 1 τ sh2 ρ(x, y) ≤ cn µ th2 ( ρ(x, y)) < cn log 2 1 2 2 2 4 2 th 4 ρ(x, y) 2 . = cn log 1 th 4 ρ(x, y) 6.10. Lemma. Let G be a proper subdomain of Rn , x ∈ G, d(x) = d(x, ∂G), Bx = B n (x, d(x)), let y ∈ Bx with y 6= x, and let r = |x − y|/d(x). Then the following two inequalities hold: 1 r2 1 (1) λG (x, y) ≥ λBx (x, y) = τ > cn log 2 2 1−r r 1 1−n 1 (2) µG (x, y) ≤ µBx (x, y) = γ ≤ ωn−1 log . r r 6.11. Lemma. The inequality p(x) ≥ max τ (|x|) , τ (|x − e1 |) holds for all x ∈ Rn \ {0, e1 }. Equality holds if x = se1 and s < 0 or s > 1. The following theorem summarizes some properties of p(x). 6.12. Theorem. For |x − e1 | ≤ |x|, x ∈ Rn \ {0, e1 } (1) (2) (3)

p(x) ≤ 2 τ (|x − e1 |) when |x + e1 | ≥ 2, p(x) ≤ 4 τ (|x − e1 |) when |x| ≥ 1, p(x) ≤ 2n+1 τ (|x − e1 |).

This result was improved by D. Betsakos [B] who proved the next theorem. The sharp constant in Theorem 6.13 is not known for n > 2 , for n = 2, see [BV]. 6.13. Theorem. For x ∈ Rn \ {0, e1 } (6.14)

p(x) ≤ 4τ (min{|x|, |x − e1 |}) .


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For x ∈ Rn \{0} we denote by rx a similarity map with rx (0) = 0 and rx (x) = e1 . Then it is easy to see that |rx (y) − e1 | = |x − y|/|x|. It follows immediately from the definitions (6.2) and (6.5) that (6.15)

λRn \{0} (x, y) = min{ p(rx (y)) , p(ry (x)) } .

Next we deduce the following two–sided inequality for λRn \{0} (x, y). 6.16. Theorem. For distinct x, y ∈ Rn \ {0} the following inequality holds 1 ≤ λRn \{0} (x, y) τ |x − y|/ min{|x|, |y|} ≤ 4 .

6.17. Corollary. Let G be a proper subdomain of Rn , x and y distinct points in G and m(x, y) = min{d(x), d(y)}. Then λG (x, y) ≤ inf λRn \{z} (x, y) ≤ 4 τ |x − y|/m(x, y) . z∈∂G

Proof. The first inequality follows from 6.7. For the second one fix z0 ∈ ∂G with m(x, y) = d({x, y}, {z0 }). Applying 6.16 to Rn \ {z0 } yields the desired result.

We next show that 6.17 fails to be sharp for a Jordan domain G in Rn . For t ∈ (0, 15 ) consider the family Gt = B n (−e1 , 1) ∪ B n (e1 , 1) ∪ B n (t) of Jordan domains. Then by 6.17 λGt (−e1 , e1 ) ≤ 4 τ (2) for all t ∈ (0, 51 ). But this is far from sharp because in fact

λGt (−e1 , e1 ) ≤ M ∆( [−2e1 , −e1 ] , [e1 , 2e1 ] ; Gt ) 1 1−n −→ 0 ≤ ωn−1 log t as t → 0. However, for a wide class of domains, which we shall now consider, the upper bound in 6.17 is essentially best possible.

−e1

e1

Figure 16. The family ∆( [−2e1 , −e1 ] , [e1 , 2e1 ] ; Gt ) .

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6.18. QED domains. A closed set E in R is called a c–quasiextremal distance set or c-QED exceptional set or c-QED set, c ∈ (0, 1], if for each pair of disjoint n continua F1 , F2 ⊂ R \ E n M ∆(F1 , F2 ; R \ E) ≥ c M ∆(F1 , F2 ) . (6.19) n

n

If G is a domain in R such that R \ G is a c-QED set, then we call G a c-QED domain.

6.20. Examples. (1) The unit ball Bn is a 12 –QED set by [GM1] or by the above Lemma 5.6. (2) If E is a compact set of capacity zero, then E is a 1–QED set. For instance all isolated sets are 1–QED sets. The class of all 1–QED sets contains all closed sets in Rn of vanishing (n − 1)–dimensional Hausdorff measure (see [V3], [GM1]). (3) B2 \ [0, e1 ) is not a c-QED set for any c > 0. 6.21. Theorem. Let G be a c-QED domain in Rn . Then λG (x, y) ≥ c τ (s2 + 2s) ≥ 21−n c τ (s)

where s = |x − y|/ min{d(x), d(y)}.

It should be noted that the lower bound of 6.21 is very close to that of 6.16; in fact it differs only by a multiplicative constant. In the next few theorems we shall give some estimates for the conformal metric µG . 6.22. Lemma. Let G be a proper subdomain of Rn , s ∈ (0, 1), x, y ∈ G. If kG (x, y) ≤ 2 log(1 + s), then 1 (1) µG (x, y) ≤ γ . th(kG (x, y)/(1 − s))

Moreover, there exist positive numbers b1 and b2 depending only on n such that (2) µG (x, y) ≤ b1 kG (x, y) + b2 for all x, y ∈ G.

It should be observed that (6.22(2)) is a generalization of the upper bound in (5.25) to the case of an arbitrary domain. The lower bound in (5.25) will next be generalized to the case of domains with connected boundary. 6.23. Lemma. Let G be a domain in Rn such that ∂G is connected. Then for all a, b ∈ G, a 6= b, (1) µG (a, b) ≥ τ (4m2 + 4m) ≥ cn jG (a, b)

where cn is the constant in 5.8 and m = min{d(a), d(b)}/|a − b|. If, in addition, G is uniform, then (2) µG (a, b) ≥ B kG (a, b) for all a, b ∈ G.


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7. Distortion theory For the basic properties and definitions of K-quasiconformal and K-quasiregular mappings we refer the reader to [Vu3] as well as to the other papers in this same volume. See, in particular, Rasila’s paper. One of the key ideas is that under a K-quasiconformal mapping, the modulus is changed at most by a constant c ∈ [1/K, K] . The notions introduced in the previous chapters enable us to formulate this basic property in a more concrete and geometric way, in terms of metrics. Theorem 7.1 and Corollary 7.2 are the key results of this paper, and the other results in this section are just consequences. One should carefully observe that although the transformation rule in Corollary 7.2 looks like a bilipschitz property; the mappings need not be bilipschitz in the euclidean metric. This is because the metric µG behaves in a non-linear fashion. In the euclidean metric quasiconformal mappings are Hölder-continuous as the results below show. 7.1. Theorem. If f : G → Rn is a non–constant qr mapping, then (1) µf G (f (a), f (b)) ≤ KI (f ) µG (a, b) ; a, b ∈ G .

In particular, f : (G, µG ) → (f G, µf G ) is Lipschitz continuous. If N (f, G) < ∞, then (2) λG (a, b) ≤ KO (f ) N (f, G) λf G (f (a), f (b))

for all a, b ∈ G with f (a) 6= f (b).

7.2. Corollary. If f : G → G′ = f G is a qc mapping, then (1) µG (a, b) KO (f ) ≤ µf G (f (a), f (b)) ≤ KI (f ) µG (a, b) , (2) λG (a, b) KO (f ) ≤ λf G (f (a), f (b)) ≤ KI (f ) λG (a, b) hold for all distinct a, b ∈ G.

7.3. Theorem. Let f : Bn → Rn be a non–constant K–qr mapping with f Bn ⊂ Bn and let α = KI (f )1/(1−n) . Then α 1 th ρ(x, y) , (1) th 21 ρ f (x), f (y) ≤ ϕK th 12 ρ(x, y) ≤ λ1−α n 2 (2) ρ f (x), f (y) ≤ KI (f ) ρ(x, y) + log 4 ,

hold for all x, y ∈ Bn , where λn is the Grötzsch ring constant.

7.4. Corollary. Let f : Bn → Bn be a K–qr mapping with f (0) = 0 and let α = KI (f )1/(1−n) . Then α 1−1/K K|x|1/K , (1) |f (x)| ≤ ϕK,n (|x|) ≤ λ1−α n |x| ≤ 2 1 + |x| KI (f ) a−1 , a= 4 , (2) |f (x)| ≤ a+1 1 − |x|

for all x ∈ Bn .

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7.5. Example. Let g : B2 → B2 \ {0} = gB2 be the exponential function z+1 ), z ∈ B2 . We shall show that g : (B2 , ρ) → (gB2 , kgB2 ) fails to g(z) = exp( z−1 be uniformly continuous. To this end, let xj = (ej − 1)/(ej + 1), j = 1, 2, . . . . Then it follows that ρ(0, xj ) = j and thus ρ(xj , xj+1 ) = 1. Let Y = B2 \ {0}. Since g(xj ) = exp(−ej ) we get by (4.4) and (3.20) kY g(xj ), g(xj+1 ) ≥ jY g(xj ), g(xj+1 ) = log 1 + exp ej+1 exp(−ej ) − exp(−ej+1 ) = log 1 + exp(ej+1 − ej ) − 1 = ej+1 − ej → ∞

as j → ∞. In conclusion, g : (B2 , ρ) → (Y, kY ) cannot be uniformly continuous, because ρ(xj , xj+1 ) = 1. 7.6. Theorem. Let f : Bn → Rn be a non–constant qr mapping, let E ⊂ Rn \ f Bn be a non–degenerate continuum such that ∞ ∈ E, and let G = Rn \ E be a domain. (1) Then f : (Bn , ρ) → (G, jG ) is uniformly continuous. (2) If G is uniform, then f : (Bn , ρ) → (G, kG ) is uniformly continuous. 7.7. Theorem. Suppose that f : G → Rn is a bounded qr mapping and that F α is a compact subset of G. Let α = KI (f )1/(1−n) and C = λ1−α n d(f G)/d(F, ∂G) where λn is the Grötzsch constant. Then f satisfies the H¨ older condition (7.8) for x ∈ F , y ∈ G.

|f (x) − f (y)| ≤ C |x − y|α

7.9. Theorem. Let f : Bn → Rn be a non–constant qr mapping. (1) If ϕ ∈ (0, 21 π) and f Bn ⊂ C(ϕ), then for all x ∈ Bn 1 + |x| aϕ |f (x)| ≤ |f (0)| 4aϕ 1 − |x|

where a depends only on n and KI (f ). (2) If f Bn ⊂ { x ∈ Rn : x21 + · · · + x2n−1 < 1 }, then for all x, y ∈ Bn |f (x)| ≤ |f (y)| + A KI (f ) (ρ(x, y) + log 4)

where A is a positive constant depending only on n.

7.10. Theorem. Let f : Bn → Bn be a qr mapping with N (f, Bn ) = N < ∞. Then β th 41 ρ f (x), f (y) ≤ 2 th 14 ρ(x, y) holds for all x, y ∈ Bn where β = 1/(N KO (f )). Furthermore, if f (0) = 0, then for all x ∈ Bn β |x| |f (x)| p p ≤2 . 1 + 1 − |f (x)|2 1 + 1 − |x|2


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7.11. Exercise. Assume that f : Bn → Bn is K–qc with f (0) = 0 and f Bn = Bn . Show that p |f (x)|2 ≤ min ϕ2K,n (|x|) , 1 − ϕ21/K,n 1 − |x|2 , p 1 − |x|2 . |f (x)|2 ≥ max ϕ21/K,n (|x|) , 1 − ϕ2K,n √ Note that in the case n = 2 we have ϕ2K,2 (r) = 1 − ϕ21/K,2 ( 1 − r2 ) for all K > 0 and 0 < r < 1, while the analogous relation fails to hold for n ≥ 3. The next result is a famous theorem of A. Mori from 1956 [LV]. The theorem has, however, one esthetic flaw: it is not sharp when K = 1 . It was conjectured in the 1960’s that the constant 16 in the theorem could be replaced by 161−1/K and also shown in [LV] that this would be sharp. In 1988 it was proven in [FeV] that we can replace 16 by M (K) → 1 as K → 1 . Perhaps the latest paper dealing with the problem of reducing the constant M (K) was written by S.-L. Qiu [Q], but as far as we know it is still an open problem whether the constant 161−1/K could be achieved. Settling this problem would be remarkable progress, since a lot of work has been done. For the spherical chordal metric this problem was recently discussed by P. Hästö [H3]. 7.12. Theorem. Let f : B2 → B2 be a K–qc mapping with f (0) = 0 and f B2 = B2 . Then |f (x) − f (y)| ≤ 16 |x − y|1/K for all x, y ∈ B2 . Furthermore, the number 16 cannot be replaced by any smaller absolute constant. 7.13. An open problem. For K ≥ 1, n ≥ 2, and r ∈ (0, 1) let

ϕ∗K,n (r) = ϕ∗K (r) = sup{ |f (x)| : f ∈ QC K (Bn ), f (0) = 0, |x| ≤ r }

where QC K (Bn ) = { f : Bn → f Bn | f is K–qc and f Bn ⊂ Bn }. As shown in [LV, p. 64] (7.14)

ϕ∗K,2 (r) = ϕK,2 (r) ≤ 41−1/K r1/K

for each r ∈ (0, 1) and K ≥ 1. By 7.4(1) (7.15)

ϕ∗K,n (r) ≤ ϕK,n (r) ≤ λ1−α rα , α = K 1/(1−n) , n

for n ≥ 2, K ≥ 1, r ∈ (0, 1). A. V. Sychev [SY, p. 89] has conjectured that (7.16)

ϕ∗K,n (r) ≤ 41−α rα

for all n ≥ 2 and K ≥ 1. Because λ2 = 4, (7.16) agrees with (7.14) for n = 2. In [AVV4] it is shown that ϕ∗K,n 6≡ ϕK,n for n ≥ 3. It follows from 7.10 and 7.11 that ( ϕ∗ (r) ≤ 4 r1/K , √ K,n 2 (7.17) ϕ∗K,n (r) ≤ 1 − ϕ21/K,n ( 1 − r2 ) .

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From (7.17) and (7.15) it follows, as shown in [AVV], that (7.18)

2

ϕ∗K,n (r) ≤ 41−1/K r1/K

holds for all n ≥ 2, K ≥ 1, r ∈ (0, 1). Note that the right hand side of (7.18) is bounded when K → ∞. Recall that λn → ∞ as n → ∞ and that λ1−α ≤ n 21−1/K K. Note that Sychev’s conjecture (7.16) still remains open. 7.19. Another open problem. In [Vu4], the following problem was stated. Let QC K (Rn ) = { f : Rn → Rn | f is K–qc and f (e1 ) = e1 }. Is it true that (7.20) sup{|g(x)| : |x| = r, g ∈ QC K (Rn )} = sup{|f (−re1 )| : f ∈ QC K (Rn )} , when r > 0? For n = 2 the answer is in the affirmative by [LVV].

8. Open problems Assume that G ⊂ Rn is a proper subdomain. For what follows, we will be interested mainly in the cases when the domain is a member of some well-known class of domains. Some examples are uniform domains, QED-domains, domains with uniformly perfect (in the sense of Pommerenke [Su]) boundaries and quasiballs, i.e. domains G of the form G = f Bn for quasiconformal f : Rn → Rn . We denote the class of domains with D . Let us consider collection of metrics 1/(1−n) M = {αG , hG , jG , kG , λG , µG , q, | · |} where hG refers to the hyperbolic metric when n = 2. Interesting categories of mappings, we denote them by C, are Hölder, Lipschitz, isometries, quasiisometries and identity mappings. The problems that we list below are just examples. There are a great many variations, by letting the domain, mapping and metric independently vary over the categories D , C , and M . 8.1. Convexity of balls and smoothness of spheres. Fix m ∈ M . Does there exist constant T0 > 0 such that Dm (x, T ) = {z ∈ G : m(x, z) < T }, is convex (in euclidean geometry) for all T ∈ (0, T0 )? Is ∂Gm (x, T ) smooth for T < T0 ? For instance, in the case m = kG both of these problems seem to be open. In passing, we remark that it follows from (4.4) and Theorem 4.7 (2) that when the radius tends to 0, quasihyperbolic balls become more and more round. The quasihyperbolic metric is used as a tool for many applications, but very little about the metric itself is known. See the theses [MA] and [L] and also Lindén’s paper in this volume. 8.2. Lipschitz-constant of identity mapping. For x, y ∈ Bn , x 6= y , the following inequality holds [Vu3, (2.27)] ρBn (x, y) ρBn (x, y) < . |x − y| ≤ 2 th 4 2 We may now regard this result as an inequality for the modulus of continuity of id : (Bn , ρBn ) → (Bn , | · |) . Instead of considering the identity mapping we could now take any mapping in our category of mappings and consider the problem


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of estimating the modulus of continuity between any two metric spaces in our category of metric spaces, see [Vu1], [S]. We list several particular cases of our problem. For G = Rn \ {0} does there exist constants A or B such that for all x, y ∈ G q(x, y) ≤ AkG (x, y), and 1/(1−n) q(x, y) ≤ BλG (x, y) ? For G = C \ {0, 1} does there exist constant C such that for all x, y ∈ G q(x, y) ≤ ChG (x, y) ,

For G = Rn \ {0} does there exist a constant E such that for all x, y ∈ G 1/(1−n)

λG

(x, y) ≤ EjG (x, y) ?

8.3. Characterization of isometries and quasiisometries. Given two metric spaces in our category of spaces, does there exist a quasiisometry, mapping the one space onto the other space? Again, we could consider, in place of quasiisometries, any other map in our category of maps. 1/(1−n)

What is the modulus of continuity of id : (G, µG ) → (G, λG 1/(1−n) (G, λG )

)?

1/(1−n) (f G, λf G )

Is a quasiisometry f : → quasiconformal? J. LelongFerrand raised this question in [LF] and the question was answered in the negative in [FMV] . There it was also shown that the answer is affirmative under the 1/(1−n) 1/(1−n) stronger requirement that f : (D, λD ) → (f D, λf D ) be uniformly continuous for all subdomains D of G . However, it is not known what the isometries are. Are isometries f : (G, αG ) → (f G, αf G ) Möbius transformations? (see Beardon [BE2], Hästö and Ibragimov [HI] and also Hästö’s paper in this volume). 8.4. Conformal invariants. The conformal invariant p(x) is relatively wellknown. See [HV] for further information. However, much less is known about the invariants µG and λG . For domains whose boundaries are uniformly perfect (in the sense of Pommerenke), there are some inequalities for µG in terms of jG , see [Vu2] and [JV]. Some results for λG when G = Bn \ {0} , were proved in [H] and [BV]. But even the basic question of finding a formula for λB2 \{0} (x, y) is open. Some of these problems may be hard, some are very easy. Because of the very general setup, it would require some effort even to single out the interesting combinations of domains in D , mappings in C , and metrics in M .

References [A]

L. V. Ahlfors: Collected papers. Vol.1 and 2. Edited with the assistance of Rae Michael Shortt. Contemporary Mathematicians. Birkhäuser, Boston, Mass., 1982. xix+515 pp., xix+520 pp., ISBN: 3-7643-3076-7, ISBN: 3-7643-3075-9

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[AVV1] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen: Sharp distortion theorems for quasiconformal mappings. Trans. Amer. Math. Soc. 305 (1988), 95–111. [AVV2] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Conformal invariants, inequalities and quasiconformal mappings. J. Wiley, 1997, 505 pp. [BE1] A. F. Beardon: The geometry of discrete groups. Graduate Texts in Math. Vol. 91, Springer-Verlag, Berlin–Heidelberg–New York, 1982. [BE2] A. F. Beardon: The Apollonian metric of a domain in Rn . Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 91–108, Springer, New York, 1998. [B] D. Betsakos: On conformal capacity and Teichm¨ uller’s modulus problem in space. J. Anal. Math. 79 (1999), 201–214. [BV] D. Betsakos and M. Vuorinen: Estimates for conformal capacity, Constr. Approx. 16 (2000), 589–602. [FeV] R. Fehlmann and M. Vuorinen: Mori’s theorem for n-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I 13 (1988), 111–124. [F] J. Ferrand: Conformal capacities and extremal metrics, Pacific J. Math. 180 (1997), 41–49. [FMV] J. Ferrand, G. Martin, and M. Vuorinen: Lipschitz conditions in conformally invariant metrics. J. Anal. Math. 56 (1991), 187–210. [G] F.W. Gehring: Quasiconformal mappings in Euclidean spaces. Handbook of complex analysis: geometric function theory. Vol. 2, ed. by R. K¨ uhnau, 1–29, Elsevier, Amsterdam, 2005. [GM] F. W. Gehring and O. Martio: Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math. 45 (1985), 181–206. [GO] F. W. Gehring and B. G. Osgood: Uniform domains and the quasi–hyperbolic metric. J. Anal. Math. 36 (1979), 50–74. [GP] F. W. Gehring and B. P. Palka: Quasiconformally homogeneous domains. J. Anal. Math. 30 (1976), 172–199. ¨sto ¨ : The Apollonian metric: uniformity and quasiconvexity. Ann. Acad. [H1] P. A. Ha Sci. Fenn. Math. 28 (2003), no. 2, 385–414. ¨sto ¨ : The Apollonian metric: quasi-isotropy and Seittenranta’s metric. Com[H2] P. A. Ha put. Methods Funct. Theory 4 (2004), no. 2, 249–273. ¨sto ¨ : Distortion in the spherical metric under quasiconformal mappings. [H3] P. A. Ha Conform. Geom. Dyn. 7 (2003), 1–10 (electronic). ¨sto ¨ and Z. Ibragimov: Apollonian isometries of planar domains are [HI] P. A. Ha Möbius mappings. J. Geom. Anal. 15 (2005), no. 2, 229–237. [He] V. Heikkala: Inequalities for conformal capacity, modulus, and conformal invariants. Ann. Acad. Sci. Fenn. Math. Diss. No. 132 (2002), 62 pp. [HV] V. Heikkala and M. Vuorinen: Teichm¨ uller’s extremal ring problem.- Math. Z. (to appear) and Preprint 352, April 2003, University of Helsinki, 20 pp. [H] J. Heinonen: Lectures on Analysis on Metric Spaces. Springer, 2001. [HB] D. A. Herron and S. M. Buckley: Uniform domains and capacity. Manuscript, 2005, 20pp. ¨rvi and M. Vuorinen: Uniformly perfect sets and quasiregular mappings. J. [JV] P. Ja London Math. Soc. (2) (1996), 515–529. [J] J. A. Jenkins: On metrics defined by modules, Pacific J. Math. 167 (1995), 289–292. ¨hnau, ed.: Handbook of complex analysis: geometric function theory. Vol. 1. [K] R. Ku and Vol. 2, Elsevier Science B.V., Amsterdam, 2002. xii+536 pp, ISBN 0-444-82845-1 and 2005, xiv+861 pp. ISBN 0-444-51547-X . [LV] O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane. Die Grundlehren der math. Wissenschaften Vol. 126, Second ed., Springer-Verlag, Berlin– Heidelberg–New York, 1973.


[LVV] [LF] [L] [MA] [Q]

[QV]

[S] [So]

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[SY] [T1] [T2] [V1] [V2] [Vu1] [Vu2] [Vu3] [Vu4] [Vu5]

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ïsa ¨ la ¨: Contributions to the distortion theory O. Lehto, K. I. Virtanen and J. Va of quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I No. 273 (1959) 14 pp. J. Lelong-Ferrand: Invariants conformes globaux sur les varietes riemanniennes, J. Differential Geom. 8 (1973), 487–510. ń: Quasihyperbolic geodesics and uniformity in elementary domains. Ann. H. Linde Acad.Sci. Fenn. Math. Diss. No 146, (2005), 50 pp. G. Martin: Quasiconformal and bilipschitz mappings, uniform domains and the hyperbolic metric. Trans. Amer. Math. Soc. 292 (1985), 169–192. S.-L. Qiu: On Mori’s theorem in quasiconformal theory. A Chinese summary appears in Acta Math. Sinica 40 (1997), no. 2, 319. Acta Math. Sinica (N.S.) 13 (1997), no. 1, 35–44. S.-L. Qiu and M. Vuorinen: Special functions in geometric function theory. Handbook of complex analysis: geometric function theory. Vol. 2, ed. by R. K¨ uhnau, 621– 659, Elsevier, Amsterdam, 2005. P. Seittenranta: Möbius-invariant metrics. Math. Proc. Cambridge Philos. Soc. 125, 1999, 511–533. A. Yu. Solynin: Moduli of doubly-connected domains and conformally invariant metrics (in Russian), Zap. Nautsh. Semin. LOMI, tom 196 (1991), 122–131, Sankt Peterburg “Nauka,” 1991. A. Yu. Solynin and M. Vuorinen: Extremal problems and symmetrization for plane ring domains, Trans. Amer. Math. Soc. 348 (1996), 4095–4112. T. Sugawa: Uniformly perfect sets: analytic and geometric aspects [translation of S¯ ugaku 53 (2001), no. 4, 387–402]. Sugaku Expositions. Sugaku Expositions 16 (2003), no. 2, 225–242. A. V. Sychev: Moduli and n–dimensional quasiconformal mappings. (Russian). Izdat. “Nauka”, Sibirsk. Otdelenie, Novosibirsk, 1983. ¨ller: Untersuchungen u O. Teichmu ¨ber konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621–678. ¨ller: Gesammelte Abhandlungen, ed. by L. V. Ahlfors and F. W. O. Teichmu Gehring, Springer-Verlag, Berlin, 1982. ïsa ¨ la ¨: Lectures on n–dimensional quasiconformal mappings. Lecture Notes in J. Va Math. Vol. 229, Springer-Verlag, Berlin–Heidelberg–New York, 1971. ïsa ¨ la ¨: Domains and maps. Quasiconformal space mappings, 119–131, Lecture J. Va Notes in Math., 1508, Springer, Berlin, 1992. M. Vuorinen: Conformal invariants and quasiregular mappings. J. Anal. Math. 45 (1985), 69–115. M. Vuorinen: On quasiregular mappings and domains with a complete conformal metric. Math. Z. 194 (1987) 459–470. M. Vuorinen: Conformal geometry and quasiregular mappings. (Monograph, 208 pp.). Lecture Notes in Math. Vol. 1319, Springer-Verlag, 1988. M. Vuorinen: Quadruples and spatial quasiconformal mappings. Math. Z. 205 (1990), no. 4, 617–628. M. Vuorinen: Quasiconformal images of spheres. Mini-Conference on Quasiconformal Mappings, Sobolev Spaces and Special Functions, Kurashiki, Japan, 2003-01-08, available at http://www.cajpn.org/complex/conf02/kurashiki/vuorinen.pdf

Matti Vuorinen Address: University of Turku

E-mail: [email protected]


Circle Packings, Quasiconformal Mappings, and Applications G. Brock Williams Abstract. We provide an overview of the connections between circle packings and quasiconformal mappings, with particular attention to applications to string theory and image recognition. Keywords. Circle Packing, Quasiconformal Maps. 2000 MSC. 52C26, 30F60.


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2. Quasiconformal Maps

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2.1. Analytic Definition of Quasiconformality

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2.2. Geometric Definition of Quasiconformality

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2.3. An Important Example

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3. Conformal Welding

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3.1. Quasisymmetries and Quasicircles

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3.2. Conformal Welding Theorem

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4. Circle Packing

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4.1. Definitions and Examples

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4.2. Packings and Maps

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4.3. The Rodin-Sullivan Theorem

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5. Applications

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5.1. Image Recognition

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5.2. Radnell-Schippers Quantum Field Theory

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5.3. Circle Packing Measurable Riemann Mapping Theorem

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References Version June 10, 2006. Supported by NSF Grant #0536665.

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1. Introduction The deep connections between the combinatorial and geometric properties of circle packings and the analytic properties of the maps they induce have been the subject of intense study in recent years. In 1985, William Thurston conjectured, and Burt Rodin and Dennis Sullivan proved, that maps between circle packings were nearly analytic [Thu85, RS87]. Since then the study of circle packings has exploded to impact a great many other fields including conformal mapping [HS93, HS96,Ste05], complex analysis [BS91,DS95a,Ste97,Ste02,Ste03] Teichm¨ uller theory [BS90, Bro96, Wil01b, BW02, Wil03, BS04b], brain mapping [Bea99, Kra99], random walks [Ste96, Dub97, HS95, McC98, DW05], tilings [BS97, Rep98], minimal surfaces and integrable systems [BS04a], numerical analysis [Moh93, CS99], metric measure spaces [BK02] and much more. The fundamental folk theorem of circle packing is that “packings desperately want to be conformal.” They react to combinatorial or geometric changes in precisely the same way as conformal maps. Maps between packings seem determined to approximate conformal maps. There is, however, much to be said about the relationships between circle packings and quasiconformal maps. It is principally with these connections and the applications arising from them that we will concern ourselves in this paper. After some initial background on quasiconformal mappings in Section 2, we describe the crucial concept of conformal welding in Section 3. We review the fundamental concepts of circle packing in Section 4, and then describe three applications of circle packings and quasiconformal maps in Section 5. Namely, we discuss the use of packings in image recognition, in implementing RadnellSchippers quantum field theory, and in constructing quasiconformal maps.

2. Quasiconformal Maps 2.1. Analytic Definition of Quasiconformality. Quasiconformal mappings form the heart of Teichm¨ uller theory as developed in the 1950’s and 1960’s. They are the natural generalization of analytic functions. For more detailed explanations, a number of excellent resources are available, including [Ahl66, LV73, Leh87, Nag88, IT92, GL00]. Definition 2.1. A homeomorphism f ∈ L2 is quasiconformal if (2.1)

∂z f = µ∂z f

for some µ ∈ L∞ , ||µ||∞ < 1. Recall the complex partial derivatives are defined by 1 ∂z f = (∂x f + ∂y f ) 2 1 ∂z f = (∂x f − ∂y f ) . 2

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f

Figure 1. A geometric measure of quasiconformality. The quotient of the length of the dashed lines measures how close the curve on the right is to being a circle. Equation 2.1 is called the Beltrami equation and µ, a Beltrami differential. Notice that when µ ≡ 0, the Beltrami equation becomes ∂z f ≡ 0, which when separated into real and imaginary parts is precisely the familiar Cauchy-Riemann equations. Thus µ determines how “quasi” a quasiconformal map really is. This measure of the “quasi-ness,” or distortion of a map is most often expressed in terms of the dilatation 1 + ||µ||∞ K= ≥1 1 − ||µ||∞ of the map. A quasiconformal map f with dilatation K is called a K-quasiconformal map; a 1-quasiconformal map is thus conformal. The Beltrami differential µ corresponding to a quasiconformal map is often called its complex dilatation. Notice, however, that the complex dilatation is a complex function and actually measures the distortion of f at every point in its domain. The dilatation, on the other hand, is a single real number and provides a global bound on the distortion of f over the entire domain. 2.2. Geometric Definition of Quasiconformality. An equivalent measure of the distortion of a quasiconformal map is provided by the dilatation quotient supθ |f (z + reiθ ) − f (z)| . iθ r→0+ inf θ |f (z + re ) − f (z)| The dilatation quotient has a simple geometric interpretation. If we consider a small circle of radius r about z in the domain, it will be mapped to some curve about f (z) in the range. The dilatation quotient is then the ratio of the maximal to the minimal distance from f (z) to this curve. See Figure 1. Df (z) = lim sup

Recall that conformal maps preserve angles; moreover, if f ′ (z) = reiθ 6= 0, then df = f ′ (z) dz = reiθ dz. Thus infinitesimally, f acts geometrically like z 7→ reiθ z + C

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for some number C; that is, f acts like the composition of a scaling, rotation, and translation. This not only explains the reason analytic maps with non-vanishing derivative preserve angles, but also implies that they must map infinitesimal circles to infinitesimal circles. Consequently, the dilatation quotient of a conformal map is identically 1. It turns out that the dilatation of a quasiconformal map is nothing more than the supremum of the dilatation quotient over the domain. Thus we have the following equivalent definition of quasiconformality. Definition 2.2. A homeomorphism f is K-quasiconformal if it is absolutely continuous on lines and Df (z) ≤ K for all z in its domain. 2.3. An Important Example. If we think of the complex plane as R2 and x x + iy as , then it is natural to consider the effect of linear and affine y transformations. Suppose e x a b , + (2.2) f (x + iy) = f y c d

where ad − bc 6= 0. A moment’s linear algebra shows f can be re-written as ax + by + e (2.3) f (x + iy) = . cx + dy + f Then (2.4)

1 ∂z f = 2

1 a+d −d a = − b c 2 c−b

1 ∂z f = 2

1 a−d a −d + . = c b 2 c+b

Notice that ∂z f = 0 ifand only if a = d and c = −b, in which case, mula b tiplication by the matrix is equivalent to multiplication by the complex c d number a + ib. In general, however, we will have (a − d) + i(c + b) ∂z f = , µ= ∂z f (a + d) + i(c − b) and f will be quasiconformal. Geometrically, f will map the basis vectors 1 and i to a + ic and b + id, respectively, and then translate by e + if . It is easy to check that µ = 0 if and


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only if the new basis vectors a + ic and b + id are perpendicular, and |µ| increases toward 1 as the angle decreases toward 0. Notice that affine maps have constant complex dilatation; conversely, if µ is constant, it is a simple exercise to solve for the affine map whose dilatation is µ. The importance of this example becomes apparent when we consider the infinitesimal behavior of any quasiconformal map. Just as we observed that conformal maps act infinitesimally by rotation, scaling, and translation, quasiconformal maps act infinitesimally as affine maps.

3. Conformal Welding 3.1. Quasisymmetries and Quasicircles. We continue our exploration of quasiconformal maps with an investigation of their boundary values [BA56,LV73, DE86, LP88, GL00]. Note that when maps extend continuously or smoothly to the boundary, we will use same notation for the extended maps. Definition 3.1. A homeomorphism ϕ : ∂D → ∂D is quasisymmetric or a quasisymmetry if it is the boundary function of some quasiconformal map of D onto itself. As might be expected, quasisymmetries have a beautiful geometric characterization as well [BA56, LV73, Leh87, Krz87]. Definition 3.2. An orientation preserving homeomorphism ϕ : ∂D → ∂D is a k - quasisymmetry if |ϕ(I)|D 1 ≤k ≤ k |ϕ(J)|D for any two adjacent intervals (subarcs) I and J of ∂D having equal length |I|D = |J|D . Essentially, this definition says quasisymmetries can’t map adjacent symmetric intervals to extremely non-symmetric intervals. Next, we temporarily leave quasisymmetries to consider the effect of quasiconformal maps on circles. However, as we will see, these quasicircles are intimately connected to quasisymmetries. Definition 3.3. A Jordan curve Γ is a K-quasicircle if it is the image of the unit circle under a K-quasiconformal map of C onto itself. As might be expected by now, quasicircles have both analytic and geometric definitions [Ahl63, Ahl66]. Definition 3.4. A Jordan curve Γ is a quasicircle if there exists R > 1 so that for all points x, y ∈ Γ diam(Γx,y ) ≤ R|x − y|, where Γx,y is the sub-arc of Γ connecting x and y which has the smaller diameter.

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f

g Figure 2. If Γ is a Jordan curve, then the Riemann Mapping Theorem promises the existence of a conformal map f from the inside of Γ to the inside of the unit disc D. Similarly, there exists a conformal map g from the outside of Γ to the outside of the unit disc.

f −1

g Figure 3. Since f and g extend to the boundary, they induce a homeomorphism ϕ = g ◦ f −1 : ∂D → ∂D. Loosely speaking, this condition limits “pinching” – a quasicircle cannot visit a point x, wander far away, and then return to a point very near x. Fred Gehring’s monograph [Geh82] contains an extensive list of these and other characterizations of quasicircles. 3.2. Conformal Welding Theorem. The intimate connection between quasisymmetries and quasicircles is illustrated by the following two theorems [Pfl51, LV73, Leh87, GL00]. Theorem 3.5. Suppose Γ is Jordan curve dividing the plane into complementary components Ω and Ω∗ . Let f : Ω → D and g : Ω∗ → D∗ be conformal homeomorphisms, the existence of which are promised by the Riemann Mapping Theorem. Then f and g extend to homeomorphisms of the boundary and g ◦ f −1 : ∂D → ∂D

is a quasisymmetry if Γ is a quasicircle. See Figures 2 and 3.


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The converse is also true. Given a quasisymmetry ϕ : ∂D → ∂D, we can glue D ∗ iθ iθ and D together by attaching points e ∈ ∂D to their image points ϕ e ∈ ∂D∗ . The result is a topological sphere. As D and D∗ struggle to fit together after the welding, the “seam” between them will be pushed and pulled into a quasicircle. Conformal Welding Theorem. Let ϕ : ∂D → ∂D be a quasisymmetry. Then ϕ induces a conformal welding of D and D∗ . That is, there exist conformal maps f : Ω → D and g : Ω∗ → D∗ of complementary Jordan domains in C with boundary values satisfying g ◦ f −1 (eiθ ) = ϕ(eiθ ).

Moreover, the Jordan curve Γ = f −1 (∂D) = g −1 (∂D∗ ) is unique up to M¨ obius transformations. For quasisymmetries defined on ∂D, it is customary to normalize our welding maps so that f −1 (1) = g −1 (1) = ϕ(1) = 1, f (0) = 0, and g(∞) = ∞. With these normalizations, the maps f and g and the curve Γ are unique.

4. Circle Packing 4.1. Definitions and Examples. Since William Thurston’s work in the mid1980’s, the connections between circle packings and analytic functions have been widely studied. More detailed information is contained in the rapidly expanding literature, including several recent survey articles [DS95b,Ste97,Ste02,Ste03] and Ken Stephenson’s excellent new book [Ste05]. Definition 4.1. A CP-complex K is an abstract simplicial 2-complex such that 1. 2. 3. 4.

K is simplicially equivalent to a triangulation of an (orientable) surface. Every boundary vertex of K has an interior neighbor. The collection of interior vertices is nonempty and edge-connected. There is an upper bound on the degree of vertices in K.

The restrictions imposed by conditions 2 through 4 are extremely mild and are met by most any reasonable triangulation. Notice that a CP-complex is a purely combinatorial object. It possesses no geometric structure until it is embedded in a surface by a circle packing. To emphasize this fact, we will often refer to a CP-complex simply as an abstract triangulation. Definition 4.2. A circle packing is a configuration of circles with a specified pattern of tangencies. In particular, if K is a CP-complex, then a circle packing P for K is a configuration of circles such that 1. P contains a circle Cv for each vertex v in K, 2. Cv is externally tangent to Cu if [v, u] is an edge of K, 3. hCv , Cu , Cw i forms a positively oriented mutually tangent triple of circles if hv, u, wi is a positively oriented face of K.

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Figure 4. A finite circle packing (left). The underlying triangulation can be recovered by connecting centers of tangent circles with line segments (middle). The resulting collection of triangles forms the carrier of the packing (right).

Figure 5. A portion of the “regular hex” packing. Notice that every circle has the same radius. A packing is called univalent if none of its circles overlap, that is, if no pair of circles intersect in more than one point. A univalent circle packing produces a geometric realization of its underlying complex. Vertices can be embedded as centers of their corresponding circles, and edges can be realized as geodesic segments joining centers of circles. The collection of triangles embedded in this way is called the carrier of the packing, written carr P . See Figure 4. Example 4.3. William Thurston’s original interest in packings began with the infinite “regular hex packing” in which every circle touches exactly 6 others. He showed that the only univalent packing with this combinatorial pattern is the one in which every circle has the same radius. (It remains an open question to characterize the non-univalent ones.) See Figure 5.


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Figure 6. A portion of the “ball bearing” packing. The carrier has been drawn in to emphasize the lattice structure. Example 4.4. Another useful infinite packing is the “ball bearing packing” named by Tomasz Dubejko and Ken Stephenson [DS95b]. The underlying triangulation is created from a lattice, and the original lattice structure is still apparent in the resulting packing. Consequently, the carrier of the packing can be decomposed into small squares. Moreover, there is a natural refinement of the triangulation and carrier created by replacing each square with four copies of the original. See Figure 6. 4.2. Packings and Maps. The connection between circle packings and function theory arises from the investigation of maps between the carriers of two e are different packings for the same abstract complex. That is, suppose P and P both Euclidean circle packings for the same underlying complex K. Then every face in K is realized as both a Euclidean triangle T in carr P and a triangle Te e It is easy now to construct an affine map between triangles T and Te. in carr P. If we translate one vertex of each to the origin, then the two edges meeting at the origin form a basis for R2 and can be mapped one onto the other by a linear map. e by a piecewise Thus the entire carrier of P can be mapped onto the carrier of P affine map defined triangle by triangle. Notice that the individual triangle maps agree on adjacent edges, so the complete map is continuous. Circle packing maps constructed in this way are called discrete conformal maps. See Figure 7. 4.3. The Rodin-Sullivan Theorem. Recall from Section 2.3, that affine maps are quasiconformal. The dilation on each triangle will be constant and depend only by the difference between corresponding angles. If there are only finitely many circles in the packings, the dilatation of a discrete conformal map will be finite and depend only on the maximal difference in corresponding angles between triangles in the two carriers.

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Figure 7. Two circle packings with the same underlying triangulation. The carrier for each is indicated and one pair of corresponding triangles are shaded. Each triangle in the carrier on the left can be mapped via an affine map to its corresponding triangle in the carrier on the right. At this point in our story, we come to Burt Rodin and Dennis Sullivan’s Ring Lemma, the first connection between the analytic properties of discrete conformal maps and the combinatorial properties of packings [RS87]. Ring Lemma. In a univalent packing, there is a lower bound Cn on the ratio of the radius of any interior circle to the radius of any of its neighbors. This bound depends only on the degree n (the number of neighbors) of the circle. The sharp value of the bound Cn was determined by Dov Aharonov [Aha97]. Lemma 4.5. If {an } is the Fibonacci sequence, then 1 Cn = 2 . an−2 + a2n−1 − 1 Moreover,

Cn converges to the square of the golden ratio. Cn+1

The Ring Lemma thus connects a purely combinatorial property of the packing (the degree) with a geometric property of the packing (the ratio of the radii of adjacent circles). This geometric constraint on the circles implies angles in the carrier must be bounded away from 0 and π. Hence there is a uniform bound on the difference between corresponding angles in the carriers of two packings with the same underlying triangulation. Consequently, the associated discrete conformal map is quasiconformal with a bound on the dilatation determined only the degree. In this way, a combinatorial property of the triangulation leads directly to an analytic property of the associated discrete conformal maps. In 1985, William Thurston conjectured the relationships between the combinatorics, geometry, and mapping properties of packings run much deeper [Thu85].


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Figure 8. A cross-shaped packing (left) which has been re-packed in the unit disc (right). Since both packings share the same underlying triangulation, there is discrete conformal map between them which approximates the classical Riemann map. Suppose Ω ( C is a bounded simply connected region and p, q ∈ Ω, p 6= q. The Riemann Mapping Theorem implies there is a unique conformal map f : Ω → D with f (p) = 0 and f (q) > 0. Now suppose Pn is a sequence of packings in Ω with mesh (radius of the largest circle) decreasing to 0 and carr Pn → Ω as n → ∞. Let Kn be the underlying triangulation of Pn . Paul Koebe [Koe36], E. M. Andreev [And70a, And70b], and William Thurston [Thu] independently proved that any finite, simply connected CP-complex (such as Kn ) can be realized by a packing in D which is “maximal” in the sense that boundary circles are tangent to ∂D. This maximal, or Andreev, packing is unique up to disc automorphisms. en ⊂ D with the same Thus for each Pn ⊂ Ω, there is a maximal packing P en so that underlying triangulation Kn as Pn . Moreover, we can normalize P if Cp and Cq are the nearest circles in Pn to p and q, respectively, then the ep and C eq in P en are centered at 0 and on the positive real corresponding circles C axis, respectively. en share the same underlying triangulation, there is a discrete Since Pn and P conformal map en . fn : carr Pn → carr P William Thurston conjectured that fn → f locally uniformly on Ω as n → ∞ [Thu85]. This was quickly proven by Burt Rodin and Dennis Sullivan [RS87]. See Figure 8. Rodin-Sullivan Theorem. The discrete conformal maps described above converge locally uniformly to the conformal map f : Ω → D with f (p) = 0 and f (q) > 0. Recall that if the degree of Kn is uniformly bounded for all n (Thurston’s original conjecture was for packings with degree 6), then the Ring Lemma implies

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each fn will be K-quasiconformal, with K independent of n. It remains to show that the dilatation of fn must actually decrease to 1 as n → ∞. This follows from the uniqueness of infinite packings. Theorem 4.6. Every infinite, simply connected CP-complex has a packing in either C or D. This packing is unique up to conformal automorphisms. Various versions of Theorem 4.6 have been proven. Thurston’s original proof was only for the regular hex packing of Example 4.3 and relied on deep results from the theory of hyperbolic 3-manifolds [Thu]. Later improvements by Ken Stephenson [Ste96], Alan Beardon and Ken Stephenson [BS90], Yves Colin de Verdiére [dV89, dV91], Zheng-Xu He and Burt Rodin [HR93], and Zheng-Xu He and Oded Schramm [HS96, HS98] utilized probabilistic techniques, variational principles, the Perron method, or elementary topology. The effect of Theorem 4.6 is to force the dilation of fn to decrease to 1 as n → ∞. Consider a circle C “deep inside” Pn , that is, separated from ∂Ω by a great many generations of other circles. If C is far enough from the boundary, it can hardly tell if it is part of a finite packing, or the unique infinite one. The e in P en ⊂ D. Thus triangles in same must be true for the corresponding circle C en which are far from the boundary, must be nearly the same carr Pn and carr P (up to scaling, translation, and rotation). In particular, the corresponding angles must be nearly the same, and the resulting affine map must be nearly conformal. This is usually stated as the Packing Lemma [Ste96, Ste05]. Packing Lemma. Suppose Kn is a sequence of simply connected CP-complexes with uniformly bounded degree and having univalent packings Pn in a bounded en is any other sequence of univalent packings simply connected domain Ω. If P for Kn , then the maximum difference between corresponding angles in carr Pn en goes to 0 locally uniformly as n → ∞. and carr P

Finally, recall that we assumed the mesh of Pn decreased to 0 as n → ∞; thus on compact subsets of Ω, the number of generations of circles between the compact subset and the boundary must go uniformly to infinity as n → ∞. Consequently, the dilatation of fn will decrease to 1 uniformly on compact subsets of Ω.

5. Applications 5.1. Image Recognition. In work with Ken Stephenson, we have applied circle packing techniques to two-dimensional image recognition problems. David Mumford and Eitan Sharon have recently developed a technique for studying two-dimensional shapes (Jordan curves) by means of the Weil-Peterson metric on their associated welding homeomorphisms [MS04]. They restrict their attention to smooth curves which then produce diffeomorphisms of ∂D. The Weil-Peterson metric on these diffeomorphisms is invariant under Möbius transformations; thus shapes which differ only by scaling or rotation are recognized as being the same.


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(6.283,6.283)

(0.000,0.000)

Figure 9. A cross-shaped quasicircle (left) and the graph of the resulting quasisymmetry, parametrized as a map from [0, 2π] onto [0, 2π]. It is relatively easy to extend their program to shapes bounded by quasicircles and to quasisymmetric maps on ∂D. By packing both the inside Ω and outside Ω∗ of a quasicircle Γ, then repacking in D and D∗ , respectively, we can create discrete analytic functions

where Ωn → Ω and Ω∗n → Ω∗ .

fn :Ωn → D gn :Ω∗n → D∗ ,

It is much trickier to compare the boundary values of fn and gn since the packings in Ω and Ω∗ don’t necessarily match up on the boundary. However, it is possible with careful application of the geometry of quasicircles and a dash a topology to create a map ϕn : ∂D → ∂D −1 which is essentially given by gn ◦fn . We then have the following theorem [Wil01a]: Theorem 5.1. The mappings ϕn converge uniformly to the quasisymmetry ϕ induced by the quasicircle Γ. Moreover, fn and gn converge locally uniformly to the Riemann maps f : Ω → D and g : Ω → D∗ , respectively. For example, consider the cross-shaped curve in Figure 9. Creating discrete conformal maps as described above (Recall Figure 8), we can approximate the corresponding quasisymmetry. Repeating this procedure for a T-shaped curve and a hand-drawn cross in Figures 10 and 11, the similarities and differences with the straight-sided cross are easy to see. A more difficult problem is to recover the shape given the map ϕ : ∂D → ∂D. The Conformal Welding Theorem guarantees that this is possible, but is no help in actually computing the shape. Again, circle packing comes to the

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(6.283,6.283)

(0.000,0.000)

Figure 10. A T-shaped quasicircle (left) and the graph of the resulting quasisymmetry (right).

(6.283,6.283)

(0.000,0.000)

Figure 11. A hand-drawn cross (left) and the graph of the resulting quasisymmetry (right). Compare with Figures 9 and 10.

1 Figure 12. A discrete welding for the map ϕ(eiθ ) = ei(θ+ 3 sin(3θ)) . The circles corresponding to the “seam” in packing (left) are shaded. The packing provides a realization on S 2 of the welding triangulations (middle). The edges along the “seam” form the discrete welding curve (right).


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rescue. Instead of welding D to D∗ , we will weld triangulations of discs. For example, if ϕ is a homeomorphism from the boundary of an triangulation K to the boundary of K∗ , we use ϕ to glue the triangulations together. After a few minor refinements and adjustments, we attach every boundary edge e of K to its e of a sphere. image ϕ(e). This discrete welding then yields a triangulation K e The welded triangulation K can be realized by a unique circle packing on S 2 . The uniqueness of this packing is exactly analogous to the uniqueness of the conformal structure on S 2 . The circles must push and pull against each other to e in precisely settle in locations compatible with the global pattern provided by K the same way that two welded discs settle in locations compatible with the global conformal structure on S 2 . This circle packing provides a geometric realization of the formerly purely combinatorial welding. In particular, the “seam” between the original triangulations is realized as a polygonal Jordan curve, a discretized version of the conformal welding curve. e contains a copy of both K and K∗ . Thus we can define Notice also that K e This is, of discrete analytic functions from K and K∗ onto their copies in K. course, analogous to the existence of classical welding maps f and g onto come plementary regions of S 2 . Moreover, because of the way we used ϕ to weld K −1 together, a version of the welding condition g ◦ f = ϕ also holds. In fact, the discrete version is more than just analogous to the classical case – it converges to it as well. Welding finer and finer triangulations using the same quasisymmetric map produces discrete welding curves that converge uniformly to the classical conformal welding curve. Moreover, the discrete analytic functions converge locally uniformly to the classical conformal welding maps [Wil04]. Discrete Conformal Welding Theorem. Given a quasisymmetric map ϕ : ∂D → ∂D, our construction produces discrete analytic functions {fn } and {gn } converging locally uniformly to the conformal welding maps f and g induced by ϕ. Moreover, the discrete conformal welding curves Γn converge uniformly to the quasicircle Γ induced by ϕ. 5.2. Radnell-Schippers Quantum Field Theory. Recently David Radnell and Eric Schippers [RS05] have developed a two-dimensional quantum field theory based on conformal welding and rigged Teichm¨ uller spaces. Very briefly, one of fundamental ideas of string theory is that a one-dimensional closed string will sweep out a surface, called its world sheet, as it travels through time. As a string breaks apart and rejoins with itself, it alters the topology of the world sheet. See Figure 13. Dennis Sullivan and Moira Chas have in this manner described the topology of all world sheets in terms of the splitting and joining of strings [Sul01]. While the topology of the world sheet captures the splitting and re-joining of a string, it is the conformal structure of the world sheet that captures features such as the relative size of the string and length of time between splittings and joinings. Thus for many computations it is necessary to consider all possible conformal structures on all possible surfaces. The Universal Teichm¨ uller space contains

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Figure 13. A depiction of a string traveling through time. As the string breaks into two pieces and then rejoins, a topological handle is created. the Teichm¨ uller spaces of all Riemann surfaces and as such has recently gained the attention of physicists as a possible setting for string theory computations [Pek94, Pek95]. Two common models for the Universal Teichm¨ uller space are the space of normalized quasicircles and the space of normalized quasisymmetries. The process of conformal welding described in Section 3.2 provides the mechanism for switching between the two models [Leh87, Krz95]. Our method of discrete conformal welding described above provides the means for actually computing this correspondence as well [Wil01a, Wil04]. In the Radnell-Schippers model of quantum field theory, the ends of the world sheets are parametrized (“rigged”) by quasisymmetric maps. The interaction between two strings then corresponds to the welding of the two worldsheets via the rigging [RS05]. These operations can be carried out using circle packings to approximate the world sheets. The packable surfaces are dense [Bro86, Bro92, Bro96, BS92, BS93, Wil03] in the moduli space of all surfaces, so nothing is lost in this approach, while much is gained by the ability to actually compute the new welded surface. 5.3. Circle Packing Measurable Riemann Mapping Theorem. Recall that the distortion of a quasiconformal map f is described by its complex dilatation µ, defined by the Beltrami equation (5.1)

∂f = µ ∂f.

The classical Measurable Riemann Mapping Theorem asserts that given a Beltrami differential µ on a simply connected domain Ω ( C, there is a corresponding quasiconformal map f µ from Ω to the unit disc D having µ as its complex dilatation. If f µ is normalized to send two points p, q ∈ Ω, p 6= q, to 0 and the positive real axis, respectively, then f µ is unique [LV73, Leh87, GL00].. The original Riemann Mapping Theorem follows from the special case µ = 0. Circle packings have been used previously by Zheng-Xu He [He90] to solve Beltrami differential equations, but they appear indirectly. By applying our discrete conformal welding technique, however, we can create quasiconformal maps directly from their complex dilatation.


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Given a Beltrami differential µ on a bounded simply connected region Ω ( C, we pack Ω with a “ball bearing” packing. See Figure 6. The carrier divides Ω into small squares. We approximate µ by a constant function on each square. Recall from Section 2.3 that a map with constant dilatation is affine. In work with Roger Barnard, we showed that the conformal structure on any compact torus can be transformed into any other by cutting it open appropriately and welding it back together [BW02]. However, the conformal structures of compact tori can also be distorted by affine maps. Thus our work on welding tori provides the mechanism for creating the effect of affine maps. By refining our ball-bearing packing and performing a discrete conformal welding on each of the small squares in Ω, we can create a normalized discrete quasiconformal map fn whose dilatation is approximately equal to µ on each square [Wil]. Circle Packing Measureable Riemann Mapping Theorem. As the packings are refined, the discrete quasiconformal maps fn converge to the similarly normalized quasiconformal map f µ : Ω → D with dilation µ.

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[Wil01a]

, Approximating quasisymmetries using circle packings, Discrete and Comput. Geom. 25 (2001), no. 1, 103–124. , Earthquakes and circle packings, J. Anal. Math. (2001), no. 85, 371–396. [Wil01b] [Wil03] , Noncompact surfaces are packable, J. Anal. Math. 90 (2003). , Discrete conformal welding, Indiana Univ. Math. J. 53 (2004), no. 3, 765– [Wil04] 804. G. Brock Williams Address: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409 E-mail: [email protected] URL: http://www.math.ttu.edu/∼williams

List of registered participants for the workshop (excluding unregistered research scholars, Department of Mathematics, IIT Madras) Australia: T-W. Ma School of Mathematics and Statistics The University of Western Australia Nedlands, W.A., 6009 Australia [email protected]

Finland: Peter A. Hästö Department of Mathematical Sciences P.O. Box 3000, FI-90014 University of Oulu, Finland [email protected] Ilkka Holopainen Dept. of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected] Riku Klén Department of Mathematics University of Turku FI-20014 Turku, Finland [email protected]

Raimo Näkki Dept. of Mathematics and Statistics P.O. Box 35 (MaD) FIN-40014 University of Jyväskylä, Finland [email protected] Istvan Prause Dept. of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected] Antti Rasila Helsinki University of Technology Institute of Mathematics P.O. Box 1100, FIN-02015 HUT, Finland [email protected] Matti Vuorinen Department of Mathematics University of Turku FI-20014 Turku, Finland [email protected]

Henri Lindén Dept. of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected]

India: Javid Ali Department of Mathematics Aligarh Muslim University Aligarh–202 002, India [email protected]

Olli Martio Dept. of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland [email protected]

Meena S. Atak Maharashtra Academy of Engineering Alandi–Pune Pin–412 105 meena [email protected]

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A. Avudainayagam Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

Sukhjit Singh Dhaliwal Department of Mathematics Sant Longowal Inst. of Engg. & Tech. Longowal-148106 (Punjab) sukhjit [email protected]

R Balasubramanian Department of Mathematics The Institute of Mathematical Sciences C.I.T. Campus, Taramani Chennai – 600 113 [email protected]

Geetha Department of Mathematics Dheivanai Ammal College for Women Villupuram – 605 602 Tamil Nadu

S.S. Bhoosnurmath Department of Mathematics Karnatak University Pavate Nagar, Dharwad – 580 003 Karnataka State Bappaditya Bhowmik Department of Mathematics IIT Madras Chennai - 600 036 [email protected] Bhuvana Hindustan Engineering College Padur, Kanchipuran Dist. [email protected] K. Chandrasekhran R-3 Sivakami Apartments Somasundharam Street Muthuvel Nagar, East Tambaram Chennai-59 [email protected] S.A. Choudum Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

K.R. Karthikeyan 24, Dhanalakshmi Ammal Street Reddiar Garden, Kamaraj Nagar Avadi, Chennai–600 071 kr [email protected] S. Karthikeyan Department of Mathematics Sona College of Technology Sona Nagar Thiagarajar Polytechnic College Road Salem – 636 005, Tamil Nadu [email protected] S.M. Khairnar Department of Mathematics Anuradha Engineering College, Chikhli–443201, Dist. Buldana, (M.S) [email protected] Kokila Department of Mathematics Dheivanai Ammal College for Women Villupuram – 605 602 Tamil Nadu V. Lakshmi 52, Choolai high road choolai, chennai – 600 112 Tamil Nadu [email protected]


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J. Lourthu Mary Department of Mathematics Madras Institute Technology Anna University Chennai – 600 044

Tarakanta Nayak Department of Mathematics IIT Guwahati, North Guwahati Guwahati – 781039 [email protected]

N. Marikkannan Department of Applied Mathematics Sri Venkateswara College of Engg. Pennalur, Sripeumbudur – 602 105 [email protected]

Sanjay kumar Pant Deen Dayal Upadhyaya college(DU) Shivaji Marg, Karampura New Delhi – 110 015 [email protected]

A.K. Misra Department of Mathematics Berhampur University Bhanja Bihar Berhampur, Orissa [email protected]

R. Parvatham Ramanujan Institute (RIASM) University of Madras Chennai – 600 005 [email protected]

Debasisha Mishra Department of Mathematics Utkal University, Vani Vihar Bhubaneswar – 751004 kapa [email protected]

J. Patel Department of Mathematics Utkal University, Vani Vihar Bhubaneswar–751004 [email protected]

Sumit Mohanty C/O Jagannath Patel Department of Mathematics Utkal University, Vani Vihar Bhubaneswar – 751004 sumit [email protected]

Santosh Kumar Pattanayak Department of Mathematics Chennai Mathematical Institute Plot H1, SIPCOT IT Park Padur PO, Siruseri – 603 103 [email protected]

S. Nanda North Orissa University Sriram Chandra Vihar, Takatpur Mayurbhanj Baripada – 757003 [email protected]

S. Ponnusamy Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

U.H. Naik Department of Mathematics Willingdon College SANGLI – 416415 [email protected]

D.J. Prabhakaran Dept. of Appl. Sci. & Humanities M.I.T. Campus, Anna University Chennai – 600 044 [email protected]

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Z. Rebekal Dr. M.G.R. Deemed University No. 122, Saniyasipuram 2nd street, Medawalkam Kilpauk, Chennai – 600 010

S. Selvaganesh 1/D Jawaharlal Street, Madupet Gudiyattam – 632602 Vellore District, Tamil Nadu selva [email protected]

Rathiga Department of Mathematics Aarupadai Veedu Inst. of Tech. Paiyanoor [email protected]

C. S. Seshadri Chennai Mathematical Institute Plot H1, SIPCOT IT Park Padur PO, Siruseri – 603 103 [email protected]

Satyajit Roy Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

R. Sivakumar Department of Mathematics Sona College of Technology Sona Nagar Thiagarajar Polytechnic College Road Salem – 636 005, Tamil Nadu [email protected]

Pravati Sahoo Department of Mathematics Mahila Maha Vidyalaya (MMV) Banaras Hindu University Varanasi – 221005 [email protected] Swadesh Kumar Sahoo Department of Mathematics IIT Madras Chennai - 600 036 swadesh [email protected] N.D. Sangle Annasaheb Dange College of Engg. & Tech. Ashta, Dist. Sangli – 416 301 Maharastra navneet [email protected]

R. Sathya Priya R.M.K. Engineering College Kavarai Pattai Thiruvalluvar – 601 206

S. Sivaprasad Kumar Department of Applied Mathematics Delhi College of Engineering Bawana Road, New Delhi – 110 042 [email protected] C.M. Subalakshmi Dr. M.G.R. Deemed University No. 23, Bashyam Reddy 2nd street Ooteri, Chennai – 600 012 K. Subathra Department of Mathematics Rajalakshmi Engg.College Rajalakshmi Nagar, Thaudalam k [email protected]

M. Suganthi Dept. of Math. and Computer Appl. PSG College of Technology Coimbatore – 641004 msugan [email protected]


S. Sundar Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

P. Veeramani Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

Jagmohan Tanti Department of Mathematics Chennai Mathematical Institute Plot H1, SIPCOT IT Park Padur PO, Siruseri – 603 103 [email protected]

Murali K. Vemuri Department of Mathematics Chennai Mathematical Institute Plot H1, SIPCOT IT Park Padur PO, Siruseri – 603 103 [email protected]

G. Thirupathi Adhiyamaan College of Engineering Hosur–635109,Tamil Nadu gt [email protected]

V. Vetrivel Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

E. Umamaheswari Dr. M.G.D. Educational & Research Inst. No. 23, Venkatachala Mudali street Vepery – 600 007 Vanitha Department of Mathematics Dheivanai Ammal College for Women Villupuram – 605 602 Tamil Nadu Allu Vasudevarao Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

P. Vasundhra Department of Mathematics IIT Madras Chennai - 600 036 [email protected]

Iran: R. Aghalary Department of Mathematics University of Urmia Urmia, Iran [email protected] Saeid Shams Department of Mathematics University of Urmia Urmia, Iran [email protected]

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Japan: Toshiyuki Sugawa Department of Mathematics Graduate School of Science Hiroshima University 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 JAPAN [email protected]

C.M. Pokhrel Department of Mathematics Nepal Engineering College G.P.O. Box 10210 Kathmandu, Nepal [email protected]

Hiroshi Yanagihara Department of Applied Science Yamaguchi University Faculty of Engineering Tokiwadai Ube 755 – 8611 Japan [email protected]

USA: Om P. Ahuja Department of Mathematics Kent State University Ohio, U.S.A. [email protected]

Korea: Jae Ho Choi Department of Mathematics Education Daegu National University of Education 1797-6 Daemyong 2 dong, Namgu Daegu 705-715, Korea [email protected]

Mexico: R. Michael Porter K. Departamento de Matematicas CINVESTAV-I.P.N. Apdo. Postal 14-740 07000 Mexico, D.F. MEXICO [email protected]

Nepal: Ajaya Singh Central Department of Mathematics Tribhuvan University Kirtipur, Kathmandu Nepal ajayas [email protected]

Roger W. Barnard Dept. of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A [email protected] Phillip Brown Texas A&M University Galveston PO Box 1675, Galveston Texas 77553 -1675, U.S.A [email protected] J. Lee Bumpus Dept. of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A jlee [email protected] Atul Dixit Dept. of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A [email protected] D. Freeman Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, U.S.A [email protected]


David A. Herron Graduate Program Director Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025, U.S.A [email protected] Casey Hume Dept. of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A [email protected] William Ma Department of Mathematics Pennsylvania College of Technology 22 Hillview Avenue, Williamsport, PA 17701 U.S.A [email protected] S.S. Miller Department of Mathematics State University of New York Brockport, NY 14420, U.S.A [email protected] David Minda Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, U.S.A E-mail: [email protected] Eric Murphy U.S. Air Force 6611 Comet Circle Apt no. 302 Springfield, VA 22150, U.S.A [email protected] Kent Pearce Dept. of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, U.S.A [email protected]

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Len Ruth Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, U.S.A [email protected] Alex Williams Dept of Mathematics and Statistics Texas Tech University Lubbock, TX 79409 alejandros [email protected] Brock Williams Dept. of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409, U.S.A [email protected]

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List of unregistered participants (Research scholars, Department of Mathematics, IIT Madras) • • • • • • • • • • • • •

A. Anthony Eldred J. Anuradha A. Chandrashekaran R. Indhumathi Nachiketa Mishra V. Murugan Param Jeet H. Ramesh Jajati Keshori Sahoo V. Sankaraj E. Satyanarayana S. Suresh Kumar Manoj Yadav

Quasiconformal mappings and their applications. IWQCMA05

Lectures on quasiconformal mappings

Lectures on Quasiconformal Mappings

Quasiconformal Space Mappings

Quasiconformal mappings in the plane

Quasiconformal mappings in the plane

Quasiconformal Mappings in the Plane

Lectures on n-Dimensional Quasiconformal Mappings

Stable mappings and their singularities

Moduli of Families of Curves for Conformal and Quasiconformal Mappings

Moduli of Families of Curves for Conformal and Quasiconformal Mappings

Construction of Mappings for Hamiltonian Systems and Their Applications

Construction of mappings for Hamiltonian systems and their applications

Some applications of multivalued mappings

Orders and their Applications

Metagraphs and Their Applications

Wavelets and their applications

Mappings

Metagraphs and Their Applications

Wavelets and their applications

Cellulases and Their Applications

Metaclasses and Their Applications

Geosynthetics and their applications

Metaclasses and Their Applications

Wavelets and their Applications

Wavelets and Their Applications

Semigroups and their applications

Real submanifolds in complex space and their mappings

Universal Spaces and Mappings

Real Submanifolds in Complex Space and Their Mappings

Baecklund transformations and their applications

Quasiconformal mappings and their applications. IWQCMA05

Lectures on quasiconformal mappings

Lectures on Quasiconformal Mappings

Quasiconformal Space Mappings

Quasiconformal mappings in the plane

Quasiconformal mappings in the plane

Quasiconformal Mappings in the Plane

Lectures on n-Dimensional Quasiconformal Mappings

Stable mappings and their singularities

Moduli of Families of Curves for Conformal and Quasiconformal Mappings

Moduli of Families of Curves for Conformal and Quasiconformal Mappings

Construction of Mappings for Hamiltonian Systems and Their Applications

Construction of mappings for Hamiltonian systems and their applications

Some applications of multivalued mappings

Orders and their Applications

Metagraphs and Their Applications

Wavelets and their applications

Mappings

Metagraphs and Their Applications

Wavelets and their applications

Cellulases and Their Applications

Metaclasses and Their Applications

Geosynthetics and their applications

Metaclasses and Their Applications

Wavelets and their Applications

Wavelets and Their Applications

Semigroups and their applications

Real submanifolds in complex space and their mappings

Universal Spaces and Mappings

Real Submanifolds in Complex Space and Their Mappings

Baecklund transformations and their applications

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