Functionals of Finite Riemann Surfaces (Princeton mathematical series, no.16)

FUNCTIONALS OF FINITE RIEMANN SURFACES By M. Schiffer and D. C. Spencer FUNCTIONALS OF FINITE RIEMANN SURFACES FUNCT...

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FUNCTIONALS OF FINITE RIEMANN SURFACES By M. Schiffer and D. C. Spencer

FUNCTIONALS OF FINITE RIEMANN SURFACES

FUNCTIONALS OF FINITE RIEMANN SURFACES

Preface This monograph is an outgrowth of lectures given by the authors at Princeton University during the academic year 1949-1950, and it is concerned with finite Riemann surfaces - that is to say with Riemann surfaces of finite genus which have a finite number of non-degenerate boundary components. The main purpose of the monograph is the investigation of finite Riemann surfaces from the point of view of functional analysis, that is, the study of the various Abelian differentials of the surface in their dependence on the surface itself. Riemann surfaces with boundary are closed by the doubling process and their theory is thus reduced to that of closed surfaces. Attention is centered on the differentials of the third kind in terms of which the other differentials may be expressed. The relations between the functionals of two Riemann surfaces one of

which is imbedded in the other are studied, and series developments are given for the functionals of the smaller surface in terms of those of the larger. Conditions are found in order that a local holomorphic

imbedding can be extended to an imbedding in the large of one surface in the other. It may be remarked that the notion of imbedding is a natural generalization of the concept of schlicht functions in a

plane domain since these functions imbed the plane domain into the sphere. If a surface imbedded in another converges to the larger surface, asymptotic formulas are obtained which lead directly to the variational theory of Riemann surfaces. A systematic development of the variational calculus is then given in which topological or conformal

type may or may not be preserved. The variational calculus is applied to the study of relations between the various functionals of a given Riemann surface and to extremum problems in the imbedding of one surface into another. By specialization, applications to classical conformal mapping are obtained. iii

In a final chapter some aspects of the generalization of the theory to Kahler manifolds of higher dimension are discussed.

The first three chapters contain a development of the classical theory along historical lines, and these chapters may be omitted by the specialist. It was felt to be desirable to include these chapters as a means of providing a historical perspective of the field. A more modern treatment is included in Chapter 9 as the special case of a Kahler manifold of complex dimension 1 (a Riemann surface may

always be made into a Kahler manifold by the construction of a Kahler metric). The monograph is self-contained except for a few places where references to the literature are given. M. SCHIFFER and D. C. SPENCER,

Hebrew University, Jerusalem, and Princeton University December, 1951

}v

Acknowledgments The authors wish to thank the Mathematical Sciences Division, Office of Naval Research, and the Office of Ordnance Research, United States Army Ordnance, since this monograph was written while the authors were engaged in research projects sponsored by these agencies. They also thank Princeton University for funds supplied during the Summer of 1950, and Dr. G. F. D. Duff for his able assistance during this period. Finally, the authors wish to acknowledge their great indebtedness to Dr. Helen Nickerson, who read the manuscript in detail, made many critical suggestions, and in numerous instances recast the proofs to make them more intelligible,

and who has made most of the corrections in the galley and page proof. The authors thank the Princeton University Press and its

Director, Mr. Herbert S. Bailey Jr., for their patient cooperation during the final stages of preparation of this monograph.

June, 1953

Contents Page CHAPTER 1. 1.1

1.2 1.3

1.4 1.5

CHAPTER 2. 2.1 2.2 2.3

2.4

2.6 2.6

Geometrical and physical considerations . . . Conformal flatness. Beltrami's equation. . . . Exterior differential forms . . . . . . . . Differential forms on Riemann surfaces. . . . Elementary topology of surfaces . . . . . . Integration formulas . . . . . . . . . Existence theorems for finite Riemann surfaces . . . . . . Definition of a Riemann surface The double of a finite Riemann surface . . . Hilbert space . . . . . . . . . . . Orthogonal projection . . . . . . . . . The fundamental lemma. . . . . . . . . The existence of harmonic differentials with prescribed

20 25 25 29

periods

44

...

Existence of single-valued harmonic functions with . . . . . . . singularities . . . Boundary-value problems by the method of ortho2.8 . . . . . . . . gonal projection . . 2.9 Harmonic functions of a finite surface . . 2.10 The Uniformization Principle for finite surfaces .

2.7

...

2.11 CHAPTER

3.6

33

40 42

48 51

58 59

.

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62

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64 64

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. .

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. .

The order of a differential . . . . . . . The Riemann-Roch theorem for finite Riemann .

3.7

surfaces . . . . . . . . . . . . . Conformal mappings of a finite Riemaun surface onto . . . . . . . . . . . itself .

3.8

Reciprocal and quadratic differentials

.

.

.

.

Bilinear differentials .

.

.

.

.

CHAPTER 4.

11 15 17

Conformal mapping onto canonical domains of

higher genus . . . . . . Relations between differentials 3. . . . 3.1 Abelian differentials . 3.2 The period matrix . . . Normalized differentials . . 3.3 . . . 3.4 Period relations . . 3.5

1 1

.

.

.

.

.

Bilinear differentials and reproducing kernels . . 4.2 Definition of the Green's and Neumann's functions in terms of the 4.3 Differentials of the first kind ... defined . . . . . . . . Green's function 4.1

vi

71 72

74 76 78 83 85 88 88 93 101

4.4 Differentials of the first kind defined in terms of the

Neumann's function

. .

.

.

.

.

.

.

.

.

105

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.

107

4.5

Period matrices

4.6

Relations between the Green's and Neumann's func-

.

.

Page

tions . . . . . . . . . . . 4.7 Canonical mapping functions . . . . 4.8 Classes of differentials . . . 4.9 The bilinear differentials for the class F

.

.

.

.

. .

.

.

.

.

.

.

109

110 114 117

4.10 Construction of the bilinear differential for the class

M in terms of the Green's function.

.

.

.

.

121

4.11 Construction of the bilinear differential for the class F 126

4.12 Properties of the bilinear differentials . . . . 4.13 Approximation of differentials . . . . . . 4.14 A special complete orthonormal system . . . CHAPTER 5. Surfaces imbedded in a given surface . . . . 5.1 One surface imbedded in another . . . . . 5.2 Several surfaces imbedded in a given surface . . 5.3 Fundamental identities . . . . . . . 5.4 Inequalities for quadratic and Hermitian forms . 5.5 Extension of a local complex analytic imbedding of one surface in another .

.

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. . .

.

. .

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.

. . . 6.2 Scalar products of transforms . . . . 6.3 The iterated operators . . 6.4 Spaces of piecewise analytic differentials . . 8.5 Conditions for the vanishing of a differential . 8.6 Bounds for the operators T and T . . . . 6.7 Spectral theory of the t-operator. . . . . 6.8 Spectral theory of the t-operator . . . .

.

.

.

5.6 Applications to schlicht functions 5.7 5.8 CHAPTER 6. 6.1

6.9

Extremal mappings . . Non-schlicht mappings . Integral operators . .

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

Definition of the operators T, T and S

Spectral theory of the s-operator

.

.

.

. . . .

. .

6.10 Minimum-maximum properties of the eigen-differentials . . . . . . . . . . . . . .

129 137 138 143 143 147 148 153 158

168 173 178 181 181 185 189

198 199 208 212 219 223

230

6.11 The Hilbert space with Dirichlet metric . . . 233 241 6.12 Comparison with classical potential theory . . 6.13 Relation between the eigen-differentials of I)2 and -?M

244

. . 6.14 Extension to disconnected surfaces 6.15 Representation of domain functionais of D1 in terms of the domain functionals of lR . . . . . .

Vii

252

254

Page

6.16 The combination theorem . CHAPTER 7. 7.1

7.2

.

.

262

Variations of surfaces and of their functionals Boundary variations . . . . . . . .

. .

273 273

.

.

.

.

Variation of functionals as first terms of series developments

7.3 7.4 7.5 7.6 7.7 7.8 7.9

.

.

.

.

.

.

.

.

.

.

.

Variation by cutting a hole . . . . . . Variation by cutting a hole in a closed surface Attaching a handle to a closed surface . . . Attaching a handle to a surface with boundary . . . Attaching a cross-cap . . .

. .

. .

277 283 290 293 299 303

Interior deformation by attaching a cell. First method 310

Interior deformation by attaching a cell. Second method

.

.

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.

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314

. . 7.10 The variation kernel . . . . . . 316 7.11 Identities satisfied by the variation kernel. . . 323 7.12 Conditions for conformal equivalence under a defor. . mation. . . . . . . . 331 7.13 Construction of the variation which preserves con-

formal, type

.

334

7.14 Variational formulas for conformal mapping . . 7.15 Variations of boundary type . . . . . . . Applications of the variational method . . . . CHAPTER S. 8.1 Identities for functionals . . . . . . . . 8.2 The coefficient problem for schlicht functions . 8.3 Imbedding a circle in a given surface . . . . 8.4 Canonical cross-cuts on a surface R . . . . . 8.5 Extremum problems in the conformal mapping of plane domains . . . . . . . . . . . CHAPTER 9. Remarks on generalization to higher dimensional Kahler mani fold s . . . . . . . . . . 9.1 Kahler manifolds . . . . . . . . . .

347 354 357 357 364 376 385

.

.

.

.

.

.

.

.

.

.

408 408

9.2

Complex operators

.

.

.

.

.

.

.

.

.

.

415

9.3

Finite manifolds

.

.

.

.

.

.

.

.

.

.

420

9.4

Currents

.

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.

423

Hermitian metrics . . . . . . . . . . Dirichlet's principle for the real operators. . . Bounded manifolds . . . . . . . . . . Dirichlet's principle for the complex operators . Bounded Kahler manifolds . . . 9.10 Existence theorems on compact Kahler manifolds 9.11 The L-kernels on finite Kdhler manifolds . . . 9.12 Intrinsic definition of the operators . . . . .

425 425 433 438 439 440 443 445

.

448

.

.

.

9.5 9.6 9.7 9.8 9.9

INDEX

396

.

viii

1. Geometrical and Physical. Considerations 1.1. CONFORMAL FLATNESS. BELTRAMI'S EQUATION

In the preface to his book [7], F. Klein states that he is not sure that he would ever have reached a well-defined conception of Riemann's theory of functions had not Prym commented to him in the year 1874 that "Riemann surfaces originally are not necessarily many-sheeted surfaces over the plane, but that, on the contrary, complex analytic functions can be studied on arbitrarily given curved surfaces in exactly the same way as on the surfaces over the plane." Klein then goes on to say that he believes the starting point of Riemann's investigations was the physical conception of a steady flow of a fluid, say an electric fluid, over a surface. In this chapter we bring together various geometrical and physical concepts relating to surfaces which have motivated the development of the theory of Riemann surfaces and of functions of a complex variable. In recent years there has developed a general theory of harmonic tensors on Riemannian manifolds of dimension n. It is illuminating to observe how various features of this theory combine in the case of dimensionality 2 to produce an especially elegant and simple theory. In particular, although a neighborhood of each point of a two-dimensional Riemannian manifold 22 cannot generally be mapped isometrically onto a domain of the plane, it can be mapped conformally onto such a domain. In other words, a j8$ may be said

to be locally conformally flat (Euclidean); this property is not generally true for a 2", n > 2. Furthermore, it turns out that the equations defining harmonic functions and harmonic vectors on a Q32 are invariant to conformal transformations. These two properties, conformal flatness and the fact that harmonic functions and vectors

depend only on the conformal structure, enable us, in the study of C1]

2

GEOMETRICAL AND PHYSICAL CONSIDERATIONS

(CHAP. I

harmonic functions and vectors, to disregard the metric of the F82 entirely, retaining only the conformal structure. Thus we are led to the axiomatic definition of a Riemann surface as a surface every point of which has a neighborhood which can be mapped conformally

onto a region of the plane. In this chapter we shall discuss briefly the differential geometry V. Motivation

of a J82, and we define harmonic functions on a general

for this definition arises naturally if we consider the flow of an ideal incompressible fluid over the surface. Consider a two-dimensional Riemannian manifold 3z for which

the element of arc length ds is given by the formula ds2 =

(1.1.1)

where

Gdat2,

and 21 are curvilinear coordinates and E, F, and G are

functions of and ,7. Given an arbitrary point of the manifold, we wish to introduce new coordinates x(E, ii), rt) in some neighborhood of this point so that

this neighborhood is mapped conformally onto a region of the Euclidean plane. We consider the equations

G(a }E 2Fa ax ay

Gaeai

12-2Fa ay ay

f

a

(ax ay

ax ay\

art+E(arl)2

ax ay

These equations are equivalent to

VEG-F2 where

F' = H may have either choice of sign, or to

§ 1.1]

CONFORMAL FLATNESS. BELTRAMI'S EQUATION r

ax

E-Fay

aE

VEG-F2

(1.1.3)'

3

Fay-Ga ax a??

Thus x satisfies the partial differential equation

Fix-G 7

a

-G --F2 ) + E

a

FL-E VEn

G -- F2

1927

__

0,

and y satisfies the same equation. The equation (1. 1.4) was first introduced by Beltrami, and bears his name (see [2]). Let x and y satisfy (1.1.3), and therefore (1.1.2). Then a(x,y) (1.1. 5)

_

ax ay

ax ay

a($' n) - a a, - an k ax ax

hi{Glail$-2Fa

+E( )

0

unless ax/ai = ax/an = ay/a = ay/an = 0. For the existence of twice continuously differentiable solutions satisfying (1.1.4) such that at least one of these four quantities is different from zero at each point of a neighborhood of the given point it is sufficient that E, F, and G be twice continuously differentiable or, more generally, continuously differentiable with the first partial derivatives satisfying a Holder condition, in some neighborhood of the given point. Weaker

assumptions (E, F, and G continuous and satisfying a Holder condition) assure the existence of a continuously differentiable solution of (1.1.3) with non-vanishing Jacobian in a neighborhood of the given point (see [9a, b]). The Holder condition cannot be dropped

in either statement (see [4]). In fact, it is possible to take E, F, and G continuously differentiable in a neighborhood of the given point but such that the only twice continuously differentiable solution of (1.1.4) is z = constant. Similarly, it is possible to choose E, F, and


4

[CHAP. I

G continuous but such that the only continuously differentiable solution of (1.1.3) is x = constant, y = constant. In the exceptional cases there can be no non-constant harmonic or analytic functions, respectively, in the neighborhood. If E, F, and G have derivatives of all orders or are real analytic, the same will be true of the solutions x and y of (1.1.4). We remark that the Jacobian has the same sign

as H. If we introduce x and y as local coordinates in this neighborhood we have

d=

r7) /ay dx

a(x, y)

I

- an adyl1,

_ a

a

dx +

dy),

and by (1.1.2) (1.1.7)

where a(x,y)=Ha(.n)>0

a(x, y)

using (1.1.5).

Thus the function z = x + iy maps a neighborhood of the given point of the 62 one-one and conformally onto a neighborhood in the complex plane with the Euclidean metric I dz 12 = dx2 + dy2. The mapping cannot be chosen so as to be isometric, in general, since

a(x, y) is not constant. A continuously differentiable function of the type z = x + iy will be called a local uniformizer of the surface 82 in the neighborhood of

the given point. If z' = x' + iy' is another uniformizer in the same neighborhood, then x' and y' satisfy equations (1.1.2), which are necessary for a relationship of the form (1.1.7), and by (1.1.3), (1.1.3)'

and (1.1.6) we have ax' (1.1.8)

TX

_ ± ay', ay

ax'

ay

=

Tay' ax

according as z' and z correspond to the same or opposite choices of

the sign of H, so z' is an analytic function of z or of its complex

§ 1.11

CONFORMAL FLATNESS. BELTRAMI'S EQUATION

5

conjugate 2 in this neighborhood. Conversely, any analytic function z' of z with dz'fdz L 0 can be introduced as a uniformizer. In particular, 2

(1.1.9)

axe

a2 X1

+

=

aY2

0.

The Cauchy-Riemann differential equations (1.1.8) are the special case of (1.1.3), and (1.1.9) is the special case of (1.1.4), in which E = G, F = 0. A complex-valued function on the surface will be called an analytic function in a neighborhood of a point of the F82 if it is analytic in the ordinary sense as a function of the local uniformizer.

While it is always possible to map a Za locally on the plane with preservation of angles, this is in general not true in the large since, for topological reasons, it is not always possible to map a F82 "globally". For example, it is not possible to map a torus onto a subdomain of the plane. However, it follows from the general Uniformization Principle given in Chapter 2 that, if a global map is possible at all, then the §l can be mapped conformally in the large. Beltrami's equation is the analogue for the surface F8= of the Laplace equation (1.1.9). In order to understand this analogy even better, suppose that we consider an incompressible fluid contained between two planes parallel to the x, y plane. The exact nature of

this fluid is irrelevant but for many reasons it is convenient to identify it with an electric fluid. Suppose that there is a potential u = u(x, y) (in the case of an electric fluid the electrostatic potential) which gives rise to the streaming, and let it = v(x, y) be the stream-function. Then (1 1 10)

au

av

au

av

ax

ay'

ay

ax'

a2u

azu __

a2v 0,

axi+

ay=

asv

ax$ + ayQ =

0.

The curves u = constant are the equipotential curves, and the curves it = constant are the stream-lines. It is clear that u and it are only

determined up to additive constants. Moreover, the equations (1.1.10) remain unchanged if we replace u by it and it by - u. Corresponding to this we obtain a second system of streaming which Klein calls the conjugate streaming.


6

[CuAp. I

From the physical point of view, the Laplace equation z

z

(1.1.11)

axe -{-

=0

aYas

for the potential u expresses the property that as much fluid flows into an element of area per unit time as flows out. The Beltrami equation (1,1.4) has the same physical interpretation when the fluid is considered as streaming over a curved surface X32 in space. For

let , ,, be curvilinear coordinates on the surface, and let the arc length be given by (1.1.1). Let u be a function of position on the surface, and let the direction of fluid motion on the surface at every

point be perpendicular to the curve u = constant passing through that point. Let the velocity be an, where an is the element of arc

drawn on the surface, normal to the curve. At a point of the surface X32 the direction parameters of a E-curve

(that is, a curve alopg which i is constant) are Al = _l,E;L2 = 0 ,

and the orthogonal direction is defined by

1- -

F

1

VEG-F2

J,y2 _

-VIE

EG-F2

That is, EAlµi + F(Aiu2 + 12p') + GA21i2

(-F + F) = 0.

=

VEG - F2 The velocity of the flow at right angles to the E-curve therefore is ( 1.1.12)

au a

au

1

a + a l /2

VEF

1

EG - F2 `

au au Fad -}- E an).

By symmetry the velocity of flow across the ,-curve is given by 11 (..12)

1

1

VG EG - F2

au au Fa -}-Gay).

Now consider a small element of the surface J82 bounded by the coordinate curves corresponding to the parameter values $, $ + d$

§ 1.1]


7

and 27, n + dd. The flow across the coordinate curve extending from

i

the point (, -n) to the point ( + de, r!) is 1

(- F

1

-r-

F2 t

E} Ede F

1

EG - F2

.

ae

while the flow across the segment of the coordinate curve from + de, 27 + d1) is given by 17 + d17) to 1

- VEG - F2

(1.1.14)

(_F+E)ae a an F

E

a + a17J d 1/EG - F2 \ The difference in flow across these two boundary lines of the surface element is therefore equal to 1 a au + aI I

(1.1.15)

}d77

(_F+E)}dd17.

E

Adding the difference in flow across the other two boundary lines and setting the result equal to zero we obtain Beltrami's equation: (1.1.16)

a

Fa_-Ga

Fay- Ear

a

a VEG - F2 + N

EG - F2 -

0.

From the form of (1.1.16) it follows that for every u which satisfies (1.1.16) another function v can be found having a reciprocal relation to u. For by (1.1.16) the following equations hold simultaneously:

(1.1.17)

a'I

EG-F2

3


[CHAP. I

These equations define v up to an additive constant. From the geometrical meaning of the equations (1.1.17) we see that the systems of curves u = constant and v constant are in general orthogonal; in fact (1.1.18)

2Fd4dn + Gdt 2),

due -F dv2 = Ads2 =

so u + iv is a complex analytic function of position. If the surface Z2 is mapped conformally upon another surface with arc length dsi, then ds2 = ,uds1.

Hence by (1.1.18) due + dv2 = A,udsi = Aidsi,

so u + iv is transformed into a complex analytic function in Vi. This property is essentially a consequence of the fact that the equations (1.1.16) and (1.1.17) are homogeneous of degree zero in E, F and G. The stream-lines and equipotential curves on the one surface are mapped into stream-lines and equipotential curves on the other, but the velocity of the flow at corresponding points on the two surfaces is in general quite different. In particular, a harmonic function remains harmonic under a conformal transformation. This theorem has the following converse: If two complex analytic functions on two surfaces are given and if the surfaces can be mapped onto one another in such a way that at corresponding points of the surfaces the two complex analytic functions have the same values,

then the surfaces are mapped conformally onto one another. In fact, this criterion will later be used as the definition of conformal equivalence.

Another approach to Beltrami's differential equation is obtained by considering extremum problems on the W. Let q,($, n) be a real, twice continuously differentiable function on the surface. We seek the direction of steepest descent of c at a given point of the surface. For this purpose suppose that we proceed from the point along an

arbitrary curve and compute the derivative of T with respect to the are length s along this curve. We obtain (1.1.19)

dip

_

d

d7j ds=9'eds+9',

ds

§ 1.11


9

The fact that s is an arc length is expressed by the condition 2

(1.1.20)

= 1. ds l + G (j)

0 = E (h.) 2+ 2F (ds

We determine the extrema of (!)2under the side condition (1.1.20)

by means of the Lagrange multiplier rule. In this way we find (1.1.21)

max( )2=(EG-F2)-1CE(

)2-2F Lgo

`8 dijl

+G(8s)2]

In analogy with the terminology of plane geometry, we might call the expression on the right of (1.1.21) [grad 4,]2. By virtue of its definition, [grad rp]2 is independent of the choice of the coordinates and 77 of X32. It is called the first differentiator of Beltrami and can be expressed as follows: - 1 E F qof (1.1.22)

[grad (p]2 =

EG - F2

F G -p'

.

Let 97 and yr be two twice continuously differentiable functions on 22 and let p be a constant. We have [grad (p + ,uV)]2 = (grad y,)2 + 2p grad rp . grad V + ,u2 (grad u)s where

(1.1.22)'

-1

grad p grad V =EG

EF4'e FGg . ''$ V11 0

Clearly this expression is also invariant to changes of the ($, coordinate system. The operator

(1.1,28)

dq _

8

1

,BEG -

F2

a

E

G9 - F4 p,,

a

BT, - Fipj

+ai

F'J

is called the second differentiator of Beltrami [2, 5]. Its close relation

to the first differentiator is exhibited by the identity

G.LOMETRICAL AND PHYSICAL CONSIDERATIONS

IO

[Cxep. I

grad T - grad ip + VAT

(1.1.24)

VEG -

FT,,

a

1

F2

a

a

EG - F2 ) T a; \

VEG

-

l F2I

This identity is a generalization to curvilinear coordinates of the well-known vector identity grad T - grad' + yzlq = div (y, grad T). A more detailed study of vector analysis on Riemannian manifolds of higher dimensions will be given in Chapter 9. The reader can regard formula (1.1.24) as an illustration of the general theory. We wish to apply (1.1.24) as follows: Let y, vanish on a smooth 1-cycle a j which bounds a sub-domain of the 582. In view of (1.1.24) we have

$grad p - grad iV \/EG

- F2

J0.V/EG - F2 ded77, a

a

since the divergence integral is equal to an integral over the bounand this integral vanishes. dary The function q, is said to be harmonic in if AT = 0 in . Thus we have proved the following result, namely: If q, is harmonic in

j, and ' vanishes on aJ, then (1.1.25)

Jgrad q, grad y, V EG - F2 dddj7 = 0. a

Let now x be another twice continuously differentiable function in the closure of , which has the same boundary values as the harmonic function T. The difference x - ' is a function W which vanishes on the boundary. Hence, by (1.1.25)

f (grad x)2

E1" G

- F2

f(grad 'p)2i/EG - F2 d&d-j a

a

+ J(grad)2VEG_Pddn. a

Thus we have verified Dirichlet's Principle in a formal way for the region , namely, that any given harmonic function has a smaller

§ 1.2]

EXTERIOR DIFFERENTIAL FORMS

11

integral of the squared gradient than any other smooth function with the same boundary values. Dirichlet's Principle has served as a starting point for proving the existence of harmonic functions with prescribed boundary values. The corresponding existence proof in Section 2.8 represents a modifi-

cation of the original Dirichlet's Principle which is due to Kelvin, Riemann, and Hilbert. 1.2. EXTERIOR DIFFERENTIAL FORMS

So far we have been concerned mainly with the potential function of the fluid flow. We now examine briefly the vector field defined by the fluid velocity. First, however, let us make some remarks of general character. Let P, f2 be a pair of continuously differentiable functions of the rectangular coordinates x1, x2 in a domain as of the plane. We may consider these functions as defining a change of variables in _a2. If the Jacobian aft afl a(f1, fa) a(x1' xa)

-

ax1

axa

afa

a/2

ax,

axa

of t1' 12 (in this order) with respect to x1, x2 does not vanish over the

domain of integration we then have the formula (1.2.1)

fdIldl 2

=

f a (P, P) j a (x1. xa)

dx1 dxs,

as

the first integral being evaluated over the image of a8, regarding t1' 12 as independent variables. Even if t, fl do not map 1 in a one-one mariner, it is a matter of convenient notation to have the formula a (1.2.2)

d f 1d f $ =

a(x1, x2)

dxldx*.

Since

a(f1, fa) a (x1. Z2)

d U21 f1)

a (x1, x2) ,

12


[CHAP. 1

we should have (1.2.2)'

dfldf2 = - df2dfl.

We now proceed formally. A differential of degree 1 is an expression of the form a = aidxl + a2dx2. If ft = bidxl + b2dx2

is another such differential, we define addition of two differentials by the natural formula or + ,B = (al + b,)dxi + (a= + b2)dx2.

If f is a function (or a differential of degree zero), then we define fa = faxdxi + fa2dx2.

The operator d is defined as usual by the formula df =1 a fi dxl -I- at &2 s

when f has continuous first partial derivatives. A differential of degree 2 is an expression of the form adxidx2.

The sum of two such differentials, and multiplication by a function are defined in the obvious manner. We now define a multiplication for differentials, called exterior multiplication. The first rules (definitions) are given by the formulas (1.2.3)

dxldxs = -- dx2dx1; dxldxl = dx2dx2 = 0.

Furthermore, the multiplication shall be distributive, associative, and shall reduce to the ordinary scalar multiplication when defined for functions. Scalars (that is, functions or differentials of degree zero) shall commute witjl all differentials. Thus we have the formulas

a(fY)= (afl)Y; a(fj + Y) = afl + acv; 0%)# = q- (to) 14 where a, f, Y are differentials of any degree, and where f is a function. In other words, we may multiply in the natural way, being

EXTERIOR DIFFERENTIAL FORMS

y 1.23

1s

careful to use (1.2.3) for simplification purposes. Finally, if a = aldxi + a2dx2 is a differential of degree 1, we define (1.2.4) da = (dal)dxi + (da2)dx2. Thus doc is a differential of degree 2. Explicitly carrying out this calculation, we have da =

(axj

dx' + aal dx2) dx'

(1.2.4)'

(aa2

+

(axldx1 + axe dx2}

dx2

aall

8x1, 5x2) dx1 dx2,

by (1.2.3) and the rules of calculation. If /1 and /2 are two (continuously differentiable) functions, we have d f 1d f 2

= (f.1axi + axe dx2) (t_dx1 + ate dx2 )

_

afi aft

all aft1

ax-1 ax2

ax2 ax11

(fl,

f2)

a(xl, x2)

dxidx2

dxidx2 = -d12df1,

which is formula (1.2.2). The calculus of differential forms which satisfy the above laws was introduced by Grassmann and E. Cartan in a more general form and has been extensively used in the case of manifolds of higher dimension by Bochner, E. and H. Cartan, Hodge, Kodaira, de Rham and others.

In the case of 2-dimensional manifolds, it is almost trivial, but it still provides a formal simplification which is useful. We now return to the velocity field of the fluid flow, but we assume

that the flow is over a plane with rectangular coordinates xi, x2. Let al, a2 be the components of the fluid velocity in the directions xi, x2 respectively. Using the notation (1.2.4)' we can write the classical Stokes formula in the form

f da=J a, a3

2of

14


[CHAP. I

where a = aldxl + a$dx2 is a differential form of degree 1 and where aa$ denotes the positively oriented boundary of a2. The integral on the right in (1.2.5) is the circulation of the fluid around the boundary of a3. If this circulation is to be zero for every a2, we must have da = 0,

(1.2.6)

that is, (1.2.7)

aa1

aa2 _

ax,

8xl

0.

The equation (1.2.7) expresses the property that the vector field a, is irrotational, and it implies the existence (locally) of a velocity potential u(xl, x2) such that au (1.2.8)

du = adx1 + aedx$; a1 =

axl

a$ _ au

axt.

If the fluid is incompressible, we have seen in Section 1.1 that the potential function u satisfies (1.1.11); that is, by (1.2.8), aa1

aay

ail

axe

(1.2.9)

0.

The equation (1.2.9) is the so-called equation of continuity, and it expresses the property that the divergence of the vector field a, vanishes or that the vector field is solenoidal. A vector field a{ which is both irrotational and solenoidal is said to be harmonic. In the case of the flat plane the harmonic character of a vector field (a1, a.) is expressed by equations (1.2.7) and (1.2.9). The question arises as to the nature of these equations on a surface

which is not flat, and it turns out that the equations which characterize a harmonic vector field on a s,82 remain unchanged under a conformal transformation. Since a F82 is locally conformal to a flat surface, we see then that we may always express the equations characterizing a harmonic vector field in terms of the Euclidean coordinates of the flat plane. In other words, we may use the equa-

tions (1.2.7) and (1.2.9) to define a harmonic vector field on a general F8'.

§ 1.33

DIFFERENTIAL FORMS

15

The fact that the harmonic character of differential forms of degree 1 on a F82 depends only on the conformal structure is a special case of a more general result which has often been emphasized by H. Weyl,

namely that on Riemannian manifolds of dimension n = 2P the harmonic p-vectors (or harmonic p-forms) depend only on the conformal structure. The proof of this result is based on tensor calculus, and will therefore be omitted. 1.3. DIFFERENTIAL FORMS ON RIEMANN SURFACES

At the end of Section 1.2 it was pointed out that the conformal structure alone enters into the definition of harmonic functions and harmonic differentials on a :82. Since a neighborhood of each point can be mapped conformally onto a domain of the plane, p of a the metric may be disregarded entirely in so far as harmonic functions and 1-vectors are concerned. However, this is not true for 2-vectors; and for this reason we shall not consider them further. If x, y are the Euclidean coordinates of the plane which correspond to points

of 22 in the neighborhood of p, the function z = x + iy is a complex function of position in the neighborhood of P in the sense of Section, 1.1, and is called a (local) uniformizer at the point p. We are thus led to consider surfaces which have uniformizers at each of their points, a pair of uniformizers valid over a common neighborhood being related by a conformal mapping. Such a surface is called a Riemann surface, a concept which will be defined more precisely in Section 2.1. A differential of degree 1 on the Riemann surface is a linear expression of the form a = adx + bdy

where a and b are functions (not necessarily analytic) of the local

uniformizer z = x + iy. There is one such expression for each uniformizer, and the coefficients, are supposed to depend on the uniformizer in such a way that a is invariant. We suppose that a and b are continuous together with their first partial derivatives. A differential of degree 2 on the Riemann surface is an expression cdxdy

with a corresponding invariance property.


16

[CHAP I

Finally a function is a differential of degree 0 whose value at a point does not depend on the choice of uniformizer. An analytic function on a Riemann surface is a complex scalar which, expressed in terms of a local uniformizer, is a power series. In particular, a uniformizer is locally an analytic function. Let

z = x + iy be a uniformizer at the point p, and introduce the Wirtinger operators a a__1 - 1 is - _ i -)

ax

al

a

' az

2 (TX

+

ayl

A harmonic function U on the Riemann surface defines an analytic differential (1.3.2)

dw = ax dz

2

(ax - i a)

dz.

Starting on the other hand from an analytic differential dw = (u + iv)dz, its real and imaginary parts give the harmonic differentials (in the sense of Section 1.2) (1.3.3)

at = udx - vdy, fl = vdx + udy.

In fact, the Cauchy-Riemann equations connecting u and v express the solenoidal and irrotational character of the vector fields. The complex unit i may be regarded as a symbol which distinguishes

the positions of the components of the combination u + iv. Multi-

plication of u + iv by i replaces u by -- v and v by u. In other words, i sends a harmonic function into its conjugate. We could avoid the use of complex numbers entirely by introducing the following * operator acting on differentials. Given at = aldxi + asdx2, we define (1.3.4)

*a = - adxl + - aldx2.

We observe that, in (1.3.3), fi = *cc; in other words, the real and imaginary parts of an analytic differential are a pair of conjugate harmonic differentials. We have, for general a and fi,

ELEMENTARY TOPOLOGY

§ 1 41

17

**a = (1.3.5)

a *a =

(a, + a2)dxldxa,

(compare [1]). A differential a with dot = 0 is said to be closed. By (1.2.7) and (1.2.9) we see that a is harmonic if at and *a are closed.

In the case of a scalar g,, we set *p = gpdxdy

(1.3.6)

while, for a differential of degree 2, say cdxdy, *cdxdy = c.

(1.3.7)

In particular, (1.3.8)

**q

whenever q, is a scalar or a differential of degree 2. 1.4. ELEMENTARY TOPOLOGY OF SURFACES

As the title of this book indicates, we shall be concerned primarily

with finite surfaces, that is to say with surfaces which are finite complexes in the sense of topology. A finite surface is one which can be triangulated into finitely many simplexes such that two simplexes are either disjoint, or one is a side of another, or they have a common side which is their intersection. Finite surfaces are completely classified. In fact, a classical theorem states that any finite surface is obtained from the sphere by cutting out holes and attaching handles and cross-caps. In other words, any such surface may be mapped topologically onto. the sphere with a finite number of holes,

handles and cross-caps. To attach a handle, we may cut out two holes with boundaries Ci and C. and then identify Cx and C2 in such a way that, if Cl is traversed in the positive sense, the botmdary C2 is traversed in the negative sense. To attach a cross-cap we have only to cut out a "circular" hole and then identify diametral points on its boundary. Let m be the number of boundary curves (holes), h the number of handles, and ,c the number of cross-caps. A surface is orientable if the number c of cross-caps is zero; otherwise

it is non-orientable. If it is orientable, the genus is equal to the

18


[CHAP. I

number h of handles. If it is non-orientable, each handle may be converted into two cross-caps and the genus is defined to be equal to the total number of cross-caps when all handles have been converted, therefore equal to 2h + c. The surfaces therefore have three topological invariants: orientability or non-orientability, genus, and the number m of boundary curves. A 2-simplex is oriented by ordering its vertices, say (POP,.P2), a 1-simplex is oriented by ordering its end-points, say (POP1), and

a 0-simplex is oriented by attaching a + or - sign. If a2

=

+ (PoP1P2)

is an oriented 2-simplex or "triangle", then - a2 = - (PoP1P2) = + (P1POP2) Similarly, if al = + (POP1), then

- a1 = --- (PoP1) = + (PiPo) We form chains from the oriented simplexes of the surface. A kchain (k = 0, 1, 2) of a finite surface consists of the finitely many k-simplexes belonging to the surface, each with a definite orientation and with a definite multiplicity. A k-chain Ck is written (1.4.1)

Ck=uiai+u2o+...+.ucr

where the uk are integers and a = ock is the number of k-simplexes belonging to the surface. If, for example, ul = 0, then the simplex does not occur. If ui = 2, of occurs with multiplicity two while,

if ul = - 2, the oppositely oriented simplex occurs twice. The boundary of all consists of its (k - 1)-dimensional sides, each having

the induced orientation. We denote the boundary of all by ask. For example, aas = a(POP1P2) = (P1P2) - (POP..) + (POP1),

ao' = a(P1P$) = (P!) -

The boundary aCk of Ck is defined to be (1.4.2)

ack = Xuxaox.

(Pl)

§ 1.4]

ELEMENTARY TOPOLOGY

19

If the boundary of a chain vanishes, the chain is called a cycle.

It is easily verified that a2a = a(ad) = 0, and therefore that a(W) = 0. Thus, any boundary is a cycle. The addition of k-chains is commutative, associative and distributive. The chains

Ck = u1Q1 + ... + uaaa`, Dk = y1Qi + ... + V e.'

have the sum

+ + (ua + va)c. Ck + Dk = (u, + If a k-cycle Ck is the boundary of a (k + 1)-chain, it is called vl)ak}

a bounding k-cycle or it is said to be homologous to zero; in symbols C k P-+ 0.

The residue classes of the group of k-cycles with respect to the subgroup of bounding k-cycles are the homology classes, and they form the elements of the k-dimensional homology group. In the case of a Riemann surface it is only the 1-dimensional homology group which is significant. We remark that on a connected complex (of arbitrary dimension) the 1-dimensional homology group is the Abelianized 1-dimensional homotopy group. By formula (1.2.5) applied to a 2-chain C2: (1.4.3)

fdoc = Cs

J

a.

act

Here a = aldxl + a2dx2 is a 1-form. Let (1.4.4)

P(a, Cl) = fx

be the period ofd around the cycle C'. From (1.4.3) we see that if a is closed and if C' , 0 (in which case C' = aC2), then P((X, C') = 0. Thus the periods of closed differential forms depend only on the homology class of the cycle C'. In particular, the periods of harmonic differentials depend only on the homology class. For further details concerning the elementary topology of surfaces

we refer the reader to the literature (in particular [8]). We remark that in many applications it is preferable to consider chains whose coefficients are real numbers, which are not necessarily integer,


20

[CHAP

I

for example in applications involving integration over chains on a manifold. Here we require only the simplest results of the classical topology of surfaces. 1.5. INTEGRATION FORMULAS

In this section we bring together the formulas of integration which

we shall have occasion to use in the sequel. The scalar product of two differentials a = aldx + over a domain (1.5.1)

b1dy, fg = a2dx + b2dy

offcc. a Riemann surface is defined to be

(a, f3) =

*fi = a

J

(aas + blb2)dxdy = (p, (X).

a

We observe that (1.5.2)

ffg.' =(a,>g)

(*a, a

a

a

by (1.3.5). The scalar product of two scalars p and W is defined by the formula (9', V) = fc' - *y = frptpaxdy. a

a

Let ip be a function which is twice continuously differentiable and set (1.5.4)

Alp = ax9 +

=4

8

where z = x + iy is a uniformizer. We have (1.5.5)

dzdz = (dx - idy) (dx + idy) = 2idxdy.

We shall often denote dxdy by dA (element of area). Then dydx = -dA. It is apparent that (1.5.6)

AVdxdy = - 2i

aza2V

az

dzdz

INTEGRATION FORMULAS

§ 1.5]

21

is a differential of degree 2 and that (qi, dp) = fqiiipaxay. s

We observe that

A = *d*d. Let -p be a function (scalar), a a differential, both assumed to oe continuously differentiable. Then we have the formula

fq, . a.

(dip, *a) - (q , *da)

(1.5.7)

86

This follows at once from (1.2.5) by choosing a2 = 3 and replacing

a by 4p . a. Taking a = *4, V a scalar, we obtain the standard unsymmetrical Green's formula, namely (1.5.8)

(dqp, 4) ± (p, AV) = fc, *dp

where, in the more usual notation, (1.5.9)

f9'. *di=-fp (dx_dY) -- f p M

ds.

as

asp

Here ds is an element of arc length on the boundary and 8/8n denotes

differentiation with respect to the normal which points to the left of the vector (dx, dy). The symmetrical Green's formula is

i *dg)).

AV)-(v, Ago) = f (4, so

Finally, if we take at = dip, then (1.5.7) becomes, since d (4) = 0,. (1.5.11)

(dye, *dV)

fT dp. so

All the above integration formulas are extremely trivial, and may be proved directly. However, the notation we have used has certain advantages.

We shall be concerned mainly with analytic differentials. An

22


[CHAP. I

analytic differential dl has the form

df = du + idv where dv = *du.

(1.5.13)

Thus

dl = adx + bdy

(1.5.14)

where a, b are complex and

a=

(1.5.15)

04 dx

-- i

av dx

-I- iav b- au T dy

We have

df = du - idv = adx + bdy

(1.5.15)

where a, l are the complex conjugates of a, b. By (1.3.4),

*df = dv + idu = idf,

(1.5.17)

and it would therefore be natural to define (dl, dg) = fd/. *dg =

if

a

a

dl . dg

= i f f'g'dzdx = 2 f f' a

g'dxdy

a

as the scalar product of two analytic differentials. Here dl r f'dz. However, for formal reasons we take instead (dl, dg)

=

2

f d t *dg a

ft' g dxdy. a

We observe that (1,5.19)

(df,. dg)

((dg, dl))Where (a)- denotes the complex conjugate of a. Also (1.5.20) (td f, *dg) = (dl, dg).

INTEGRATION FORMULAS

§ 1.5]

23

Writing

N(dl) = (df, dl),

(1.5.21)

we have

N(df) =

(1.5.22)

2

f dfdf

f I f' ladxdy.

a

9

In fofmulas (1.5.18) and (1.5.22), we understand that dxdy is the area element. Thus (1.5.23) N(df) Z 0. We note the useful identities (1.5.24)

5d/dg = f fdg,

Jd14 = -f dgdf = -f gdf,

and (1.5.25)

fd f dg = 2f udg = 2i b

85

f vdg-,

dl = du + idv.

Be

REFERENCES

1. L. AHLFORS, "Open Riemann surfaces and extremal problems on compact subregions," Commentarii Math. Helvetics, 24 (1950), 100-134. 2. E. BELTRAMI, Ricerche di analiss applicata alla geometria, Opere I, 107-198, Hoepli, Milan, 1902. 3. R. COURANT, (a) tYber konforme Abbildung von Bereichen, welche nicht durch alle Riickkehrschnitte zerstiickelt werden, auf schlichte Normalbereiche,"

Math. Zeit., 3 (1919), 114-122. (b) "The conformal mapping of Riemann surfaces not of genus zero," Rev. Unsv, Nac. de Tucuman, Ser. A, II (1941), 141-149. (c) Dirichlet's Prsnesple, conformal mapping, and msnsmal surfaces, Interscience, New York, 1950. 4. P. HARTMAN and A. WINTN$R, "On the existence of Riemannian manifolds which

cannot carry non-constant analytic or harmonic functions in the small," Amer.

Jour. of Math., 75 (1953), 260-276. 5. A. HtYRWITZ and R. COURANT, Funhtionentheorie, Springer, Berlin, 1929. 6. O. D. KELLOGG, Foundations of potential theory, Springer, Berlin, 1929. (Reprint,

Murray Publishing Co.). 7. F. KLEIN, On Riemann's theory of algebraic functions and their integrals, Macmillan

and Bowes, Cambridge, 1893 (translation by F. Hardcastle of Klein's book Uber Riemanns Theorie der algebraischen Functionen and ihrer Integrale, Teubner, Leipzig, 1882).

24


[CHAP. I

8. S. LEFSCHETZ, Introduct2on to topology, Princeton Univ. Press, 1949. 9. L. LICHTENSTEIN, (a) ,Beweis des Satzes, dass jedes hinreichend kleine, im wesent-

lichen stetig gekriimmte, singularitatenfreie Flachenstiick auf einer Teil einer Ebene zusammenhangend and in den kleinsten Teilen ahnlxch abgebildet werden kann," Abh. der Preuss. Akad. der W2ss., Phys.-Math. Cl 1911, Anhang (1912). (b) ,Zur Theorie der konfornien Abbildung. Konforme Abbildung nichtanalytischer, singularitatenfreier Flichenstiicke auf ebene Gebie;.e," Bull. intern. de l'Acad. des Sci. de Cracovvv, Ser. A, 1916, 192--217, (1917).

10. H. WEYL, Die Idee der Riemannschen Fldche, Teubner, Berlin, 1923. (Reprint, Chelsea, New York, 1947).

2. Existence Theorems for Finite Riemann Surfaces 2.1. DEFINITION OF A RIEMANN SURFACE

In this chapter we establish the existence of analytic functions and differentials on Riern.ann surfaces. The proof of the existence is based on the concept of a Hilbert space of differentials, in the metric of which orthogonal projection is possible. The method of

)rthogonal projection, which is closely related to the classical Dirichlet's Principle, leads to the existence of harmonic and analytic 3ifferentials. It is sufficient to apply these methods to the particularly simple case of a closed orientable Riemann surface. Any compact Riemann surface 93which 1 has a boundary or is non-orientable may be :overed by a symmetric closed orientable Riemann surface called its "double". We shall construct the functionals of AR (such as the

;green's function) in terms of the functionals of its double. In Section I.3 we arrived at the concept of a Riemann surface. Now we give a more precise definition. A Riemann surface 9t is a connected complex, to each point :p of which there is a neighborhood which is mapped one-one and bicontinuously onto an open domain

Af the complex z-plane. The variable z belonging to the point $ is called a local uniforrnizing parameter or, more shortly, a uniformizer. If the point q lies in this neighborhood of P and if z' is a uniformizer

at q, then z and z' are related by a direct or indirect conformal mapping. That is, z' is an analytic function of z or z (the complex -onjugate of z) which has a non-vanishing derivative at P. If the Riemann surface 931 is a finite complex, we say that it is a. finite Riemann surface. In this case, corresponding to each boundary point p of 932, we suppose that there is a boundary uniformizer which

maps a subdomain of 9 near P onto a domain of the upper halfplane Im z > 0 in such a way that the boundary of 9t goes into a Segment of the real z-axis. Every finite Riemann surface is compact [25)

26

EXISTENCE THEOREMS

[CHAP. II

and is bounded by m closed curves, m z 0. It is proved in topology that a finite Riemann surface I is topologically equivalent to the sphere with m holes cut out and with h handles and c cross-caps

attached. If c > 0 the surface is non-orientable, otherwise it is orientable. If V is non-orientable the genus is 2h + c while if U't is orientable the genus is simply equal to h, the number of handles. When 0l is orientable we shall retain only one or the other of the

two classes of uniformizers which are related to one another by direct conformal mappings and the other class will be rejected. We remark that a finite Riemann surface cannot have a single isolated

point (puncture) as part of its boundary, The unit disc is a finite Riemann surface, but the unit disc punctured at the center is not. From the point of view of the systematic theory of Riemann surfaces, it would be desirable to define a Riemann surface in a somewhat more general way and to show that this definition is equivalent to the one given above (see [15]). However, since we shall be concerned primarily with functionals of finite Riemann surfaces,

we do not enter into these finer topological considerations here. If z = x + iy is a single-valued analytic function on the Riemann surface and if the value of z at the point p of the surface is a, we say that P lies over the point a of the sphere. In this way we are led to a realization of the Riemann surface as a multi-sheeted surface,

spread over the z-sphere, which is conformally equivalent to the original surface. In general, we shall say that two Riemann surfaces are conformally equivalent if there exists a topological mapping of one surface onto the other such that any analytic function or differen-

tial on the one is carried into an analytic function or differential on the other. We must distinguish the essential properties common to all realizations from the non-essential properties associated with particular ones. For example, the genus is an essential property while the kind and position of the branch points of a multi-sheeted surface are non-essential properties. A common realization of a finite Riemann surface of genus zero

is a multiply-connected domain of the complex z-plane having m analytic boundary curves. Near a boundary pount za of such a domain we may use a boundary uniformizer s which coincides on the boundary

curve with the arc length parameter. A half-neighborhood of the

§ 2.1]

DEFINITION OF A RIEMANN SURFACE

27

domain at zo is mapped by s onto a half-neighborhood bounded by a segment of the real s-axis. Any finite Riemann surface may be represented as a disc of the complex plane having 2n arcs on the boundary of the disc identified in pairs (see end of Chapter 8). It is a classical theorem in the elementary topology of surfaces that any finite surface is equivalent to the disc (topological polygon) with pairs of arcs on the boundary of the disc identified. The identification in topology is accomplished by a topological map of one arc upon the other, a pair of corresponding

points in the mapping being regarded as equivalent. In the case of Riemann surfaces we must show that the indentification can be established by one-one conformal maps. For example, consider the unit disc I z I < 1 of the complex zplane and let the upper half of the circumference I z I = 1 be identified with the lower half by the direct conformal mapping which sends the point z into the point 11z. The resulting surface is topologically equivalent to the sphere, and it is made into a Riemann surface by introducing appropriate uniformizers. At an interior point of the disc the variable z itself is a uniformizer. At a point P. of the surface which corresponds to a pair of identified points z,, 1/zo on I z I = 1, zo 0 1, - 1, we define a uniformizer in the following way. Let 9l be the half-neighborhood at zo which is the intersection of the circle I z - zo I < e with the disc I z I < 1, and let % be a similar half-neighborhood at 1/zo.. We set

z - zo,

t-

1

z

-zo,

z eJt1 ZE%s.

Then the half-neighborhoods 911. and I2 fit together in the plane of

t to form a full neighborhood of t = 0, so t is a uniformizer at the point po of the surface. At the point of the surface corresponding to z = 1, we define a uniformizer by first mapping the circumference I z = 1 into a straight line by the transformation w

2

2+1-1

_ 1-z 1+ z

in which z = 1 goes into w = 0 and in which the half neighborhood

28

EXISTENCE THEOREMS

(CHAP. II

% = [ I x I < 1] n [ I z - i I < e] goes into a half-neighborhood lying to the right of w = 0. Set t =

(2.1.2)

wa _

l 11

-z12 l

1+z

In the plane of t the neighborhood 92 appears as a neighborhood of t = 0 cut along the negative real axis, opposite points on the two edges of the cut corresponding to identified points z, 1/z near z = 1. The identification of points is therefore achieved by erasing the cut to give a full neighborhood of t = 0. Thus t is a uniformizer at the point of the surface corresponding to z = 1. A similar uniformizer may be defined at z = - 1. With these definitions of uniformizers, the surface is made into a Riemann surface. From the topological point of view, the projective plane is represented as the unit disc with diametrically opposite points of its boundary

identified. Now let this identification be accomplished by the indirect conformal mapping which sends the point z of I z I = 1 into the point - 1/2. Let 91 be the half-neighborhood 1z I < 11 n [ I z - ze I < Q] at zo, 912 a similar half-neighborhood at - 1/zo, and define Jz (2.1.3)

r =

xo,

x e 921

- -z zo,

z E 922.

1

Then z maps the union of 921 and 5912 onto a complete neighborhood

of the origin and is therefore a uniformizer at the point P. of the surface corresponding to the pair of identified points zo, - 1/90. With these uniformizers the surface becomes a closed non-orientable Riemann surface which is topologically equivalent to the projective plane. The simplest example of a non-orientable.finite Riemann surface with a boundary is provided by the Mobius strip. The Mbbius strip may be obtained by cutting a hole out of the projective plane. Consider also the circular ring 1 < I z I < R of the z-plane. The Mobius

strip is obtained from this by identifying the points z and - R/z. This clearly represents a non-orientable surface, because in the identification a small circle oriented clockwise at z goes into a small circle oriented counterclockwise at - R/z. By cutting the ring along

DOUBLE OF A FINITE RIEMANN SURFACE

§ 2.2]

29

the real axis in the z-plane, and joining the two halves together along

corresponding boundaries, it may be verified that the familiar Mobius strip is obtained. Thus the ring 1 < I z I < R with points z and - R/z identified is a canonical form for the Mobius strip. 2.2. THE DOUBLE OF A FINITE RIEMANN SURFACE

A finite Riemann surface has a "double", which we presently define. The double of a multiply-connected domain of the plane was

first introduced by Schottky [11], and it was later used by Picard (see [9]) and by Klein [5], the latter of whom extended the concept to general Riemann surfaces. The double z of a finite Riemann surface )l may be defined as follows. Two points oft are associated with, or lie over, each interior point of 2 and one point of is associated with each boundary

point of 9. Two disjoint neighborhoods of lt lie over each neighborhood of an interior point of )t. If z is a uniformizer at an interior point of 9A, then z is a uniformizer in one of the two associated apneighborhoods and z (the complex conjugate of z) is a uniformizer in the other. If z is a boundary uniformizer at a boundary point of 't, then a uniformizer at the corresponding point of z5 is given by the variable which is equal to z in one sheet of 15 and equal to z in the other. Only direct conformal transformations of the uniformizer are permitted on 15. If 9J is orientable and has a boundary or if 92 is non-orientable, 15 is a closed orientable Riemann surface; if is a closed orientable manifold, 15 consists of two closed orientable YI

surfaces.

To obtain a more complete description of 15, let us assume first that D1 is orientable and that it has a boundary. In this case we suppose (by discarding #'one class of uniformizers) that two uniformizers which belong to the same neighborhood of Ul are related by a direct conformal mapping. Now let }" be obtained from TZ by introducing the class of uniformizers which was discarded for V. The double 1 is obtained by identifying corresponding boundary points of V and t. If p and are identified boundary points of )1 and ft let z be a boundary uniformizer of 0 at p. Then z is a boundary uniformizer of 9 at , and we take as uniformizer on t5 at the

s0

point p =

EXISTENCE THEOREMS

[CHAP 11

the variable r defined by Iz in 932

z in D. Second, assume that 9R is a non-orientable finite Riemann surface. To each point of 9A there are two classes of uniformizers, the unifor-

mizers within each class being related to one another by direct conformal transformations. Let 91 be the relatively unbranched two-

sheeted orientable covering of M. Two points of 91 lie over each point of 931 and one class of uniformizers is associated with one of these points, the other class with the other. Then 92 is a connected neighborhood space. For, the curve on % lying over a 1-cycle on 931 along which the orientation is reversed leads from one sheet of 91 to the other. In fact, 92 is an orientable finite Riemann surface. If 9l1 is closed, the double 16- of 932 is defined to be 9. If 9)1 has m boundary curves, m > 0, then 91 has 2m boundary curves and two boundary curves of %t lie over each boundary of 932. If a boundary curve

of D1 is oriented and if the two overlying boundary curves of 92 are given corresponding orientations (defined by the continuous mapping of 92 onto 931), then 91 lies to the left of one of the curves

and to the right of the other. We now identify the two boundary curves of 9 lying over each boundary curve of 932 (a pair of equivalent

points lying over the same boundary point of 912), and we obtain in this way a closed orientable Riemann surface which is defined

to be the double of V. The two points p and

of zg- which lie over the same point of . If p corresponds to a boundary point of fit, then p = . Thus at a point p of which corresponds to 912 are called conjugate points of

a boundary point of 932 we may use as uniformizer either the variable z or its complex conjugate z. The correspondence between conjugate points of g- defines a one-one indirect conformal mapping of onto

itself. If this mapping is S, then S2 = I, the identity mapping. Klein [5] calls a closed orientable Riemann surface symmetrical if, as in the case of , there is a one-one involutory indirect conformal mapping of the surface into itself. If we identify conjugate points of a symmetrical surface, we obtain a finite Riemann surface 9 whose double is the given symmetrical surface. To compute the genus G of the double g-, we assume first that 9A

§ 2 21

DOUBLE OF A FINITE RIEMANN SURFACE

31

is not closed and orientable and we make a simplicial decomposition of Tt. This division of 9N may be carried up (in the usual way) to g-, and we then apply the Euler characteristic formula to . Each simplex which is interior to 5911 gives rise to two simplexes of while each simplex lying in the boundary of 9A gives rise to one simplex of . Since the number of 0-simplexes on any boundary component

of 9N is equal to the number of 1-simplexes on that boundary, the boundaries of 9N contribute nothing to the Euler characteristic N of 9)2 and we see that the characteristic of is equal to 2N. The genus G of is therefore equal to N + 1. But if 931 is equivalent to the sphere with h handles, c cross-caps and m holes we have

N = 2h + c + m- 2, so

G = 2h+c+m-1. If 931 is closed and orientable, we define G = 2h. To obtain a formula for G which will be valid in all cases, we let

R° be the number of components of the double (0-dimensional Betti number). Then R° is equal to 1 unless 931 is closed and orientable

in which case R° = 2. The formula (2.2.1)

G=2h+c+m+R°-2

is valid for every finite Riemann surface 911. The double of a simply-connected domain with boundary is the sphere, while the double of a multiply-connected domain of. genus zero with m boundaries is the sphere with m -1 handles. The double of the Mobius strip discussed in Section 2.1 is the torus. We now define what is meant by an analytic differential dZ" of dimension v on a closed orientable Riemann surface . It is a rule which associates to each point of a meromorphic function g(-c), z a local uniformizer, such that (2.2.2)

d,," = (dr')", dZ" = g(r)dr'', is invariant to direct conformal transformations of the uniformizer.

In other words, if t is another uniformizer at the same point of then dZ" _ dZv dt\v (2.2.2)

(_)

g(r) =gilt)

-r} '

,

32

EXISTENCE THEOREMS

[CHAP II

where gl(t) is the meromorphic function assigned to the uniformizer t.

If v is zero, g is invariant under direct conformal changes of the uniformizer and is called a function of . if 3 is symmetrical and is the double of a finite Riemann surface JI, then to each differential dZ" of we can associate a conjugate differential dZ". If p, p are a pair of conjugate points of, we define

dZ"(p) = (dZ''(m-.

(2.2.3)

In other words, if z and (2.2.3)'

d0"(15)

dz"

are uniformizers at

-

d

dzv

J

(i)"

then

dz I

If a the factor (d(;)-/(dz))'' is equal to unity. If dZ''(p)-dZ" (p), we say that dZ" is a differential of 9A (of dimension v). Thus the differentials of 1't are characterized by the property that they take conjugate values at conjugate places of and are real on the boundary of V. The operation of forming the conjugate of a differential will be

called the `- operation'. The operation is of order 2 - that is, the square of the operator is the identity. If )l is non-orientable, the parameter transformations are both direct and indirect. In fact, at each interior point there are two classes of uniformizers which correspond to conjugate points of g-.

If dZ'' is a differential of 9)1, then at one point it takes the value dZ" and also the value (dZ")-. We shall therefore agree that in the interior. of 9)t the differential dZ1' depends on the uniformizer in such a way that it remains invariant under direct conformal transformations of the uniformizer and goes into the complex conjugate under an indirect conformal transformation of the uniformizer. On a non-orientable surface invariance means "invariance up to the complex conjugate". As we have already remarked, a differential of dimension 0 is called a function. Differentials of dimension 1, 2, and - 1 will be called linear, quadratic and reciprocal differentials respectively. In

general, differentials are defined only for integer values of the dimension v although it is possible to extend the definition to fractional values of Y. An ambiguity then arises in (2.2.2)' owing to the presence of the factor (dt/dr)".

HILBERT SPACE

1 2.3]

33

There are also differentials of integer dimension which are multivalued on the manifold. For example, a linear Frym differential is multiplied by a factor of modulus unity when we pass around a closed loop of the manifold which is not homologous to zero (see [11]). However, we shall be concerned for the most part with differentials which are single-valued. A harmonic function u on a closed orientable surface a gives rise to a linear analytic differential of , namely dZ _ 1 8u u 1. (2.2.4)

dx = (du)

2 (TX

-i 2!

If u is single-valued on a and has only polar singularities (including logarithmic poles), we say that u is a harmonic function of a. If is symmetrical and is the double of a surface l)l, we say that u is a

harmonic function of D1 if it is single-valued and if either dZ or idZ is a linear differential of V. Suppose that 0 has a boundary and that z in formula (2.2.4) is a boundary uniformizer. If idZ is a differential of t we see that 8u/8x = 0 on the boundary, while if dZ is a differential of W1 then 8u/ey = 0 on the boundary. According

as idZ or dZ is a differential of 1, we say that u is a harmonic function of $t of the first or of the second kind. If t is not a closed orientable surface, the harmonic function of V of the first kind which has a source at a point q of 8' and a sink at the conjugate pointq will be called the Green's function G(P, q) = G (P, q, "q) of 9R. Let C be a uniformizer at q, C (q) = 0. Then g is a uniformizer at q and we suppose that G is normalized such that

I + regular terms (near q);

G = log (2.2.5)

G = log

I + regular terms (near The harmonic function of the second kind which has a dipole singularity at a point of SR is of importance in conformal mapping. I

2.3. IIILBERT SPACE

A set H of elements /, g.... is called a Hilbert space if it satisfies the following postulates:

(1) H is a linear space. That is:

EXISTENCE THEOREMS

34

[CHAP. TL

(a) There exists a commutative and associative operation, called

addition and denoted by +, such that f + g belongs to H if / and g are both elements of H. (b) There is a field of numbers A, called multipliers, such that for each element / e H and each A the element A/ is defined and belongs to H. This scalar multiplication satisfies the usual associative and distributive laws. It is required that 1 / = f. If this field is the field of real numbers, H is called a real Hilbert space; if it is the field of complex numbers, H is called a complex Hilbert space. No fields other than the field of real numbers and the field of complex numbers are considered here. (c) There exists a null element which, in general, is denoted by zero despite the fact that it is not a number but an element of H. It satisfies the requirements

f +0=f, On the left side of the third formula the symbol 0 is a number; elsewhere in the formulas it is to be interpreted as zero element. (2) H possesses a metric which measures the "length" of each element. That is, for each pair of elements f, g of H there exists a complex number (/, g) with the following properties: (a)

(b)

(Af, g) = A(f, g)

(f1 + f2, g) _

(f1,

g) + (fa, g)

(d)

(g, f) _ ((f, g))(f, f) z 0

(e)

(f, f) = 0 if and only if f = 0.

(c)

In the case of a real Hilbert space we assume that (f, g) is real so that (c) becomes (g, g). It is usual to call

Ilfli= the norm or "length" of f. However, we shall find it slightly more convenient to use as norm the quantity (2.3.1)

N(f) = (f, f) = 11 f (3) H has infinite dimension. To each positive integer n there 112.

§ 2 -al

HILBERT SPACE

exist n elements f 1, f2,

,

f n which are linearly independent; that is

f Anfn = 0

A1ft + 1212 +

only if AI=A2=.

35

=An==0.

(4) H is separable. That is, there exists a countable set of elements of H, say /1, f2 ' - in, - , which is everywhere dense in H. Given any element g e H and any positive number e, there exists an element

fv of the set such that N (g - f,,) < e. of (5) H is complete. That is, if a sequence f1, fz, , f,t, elements of H satisfies the Cauchy criterion that N(f, - f,,) tends to zero as ,u, v become infinite, there exists an element / e H such

that N(f - f,) tends to zero. The great analogy of the structure of H with that of a linear vector space of finite dimension is obvious. Assume that the space is complex. From (2) we have (2.3.2)

N(A/ ± Fug) = I A 12N(f) + 2 Re {A/-z (t, g)} + I i 12N (g) z 0

for every pair of complex numbers A, Iz. We conclude that (2.3.3)

(f, g)

N (g)

ti

This is the familiar Schwarz inequality. In particular, taking I = 1,

,u = -- t in (2.3.2), we obtain

N(/ - g) = N(f)-2 Re{(f,g)}+N(g)

N(f) + 2 1 (f,g) j+N(g) N(f) + 2 '/N(f)-VN(g) + N(g) _ {VN(f) + )}2 by (2.3.3). Thus we have the triangle inequality

v'N(f - g) < v'N(f) + '/N(g) The inequality (2.3.3) shows that we may define the angle 0 bet weet two elements /, g by the familiar formula (2.3.4)

(2.3.5)

cos 0 =

W,

g)

VN(g)

In particular, we call two elements f, g of H orthogonal if cos 0 = 0, that is if (f, g) = 0. By (2e), the only element which is orthogonal

to itself is the null element.

EXISTENCE THEOREMS

36

[CHAP. II

An example of a real Hilbert space is provided by the real differentials a = adx + bdy

which are assumed to be square integrable on a Riemann surface 9)1. We recall (see Section 1.5) that the scalar product of two differentials

a, Ii is defined to be (2.3.6)

((X,

fl) = 1 a - * fl. ant

The completeness of this space follows from the Riesz-Fischer theorem.

We shall find it convenient to use differentials and derivatives simultaneously. A differential d f may be written in the form dl = f'dz and f' is the derivative connected with the differential dl. While we defined scalar multiplication only for differentials by (df, dg),

it will sometimes be convenient to denote the same expression also

by (f', g').

An example of a complex Hilbert space is provided by the square integrable complex analytic differentials f' (p) on the Riemann surface

9R, that is by the analytic differentials f' (p) defined on 9)1 which possess a finite norm (2.3.7)

N(f') = Jit' I2dxdy = 1 Ì

f' Izdzdz.

2z .1 9R

T1

Let f;,(p) be a sequence of differentials in this Hilbert space which converge to an element f'(p) of the space. Let Po be a fixed interior point of 2, and let z = z(p) be a local uniformizer in the neighborhood of pa such that z(0) = 0. We then have the Taylor developments (2.3.8)

-o

z'',

each of which converges in a circle I z I < r, * being a number which is independent of u. We have

HILBERT SPACE

§ 2 31

3?

00

(2.3.9) N(ff, -

f') Z n 2?

1

ov+1

I bu, 12 r2('+i) Z

x

1

bur I2 rscr+2)

v+1

for v = 0, 1, 2, . The convergence of f to f' in the metric of the Hilbert space implies, therefore, the convergence of the coefficients in the local Taylor development of to the corresponding coefficients of f. Thus Hilbert space convergence implies uniform convergence of the differentials in any compact subdomain of 01 lying in its interior. For brevity, we call the latter type of convergence "Vitali convergence". On the other hand, Vitali convergence does not imply convergence in the metric of the complex Hilbert space even if we assume that N(ff,) S M2 (2.3.10) where M is independent of ,u. However, assuming (2.3.10), we can

show that for any fixed differential g' of the space we have lim g') = 0. (2.3.11) u-*w

In fact, let l' be a compact subregion lying in the interior of 0. Then fµ converges uniformly to a limit f' on X and we have M2.

N2M,(f') = Jim u-*w

Since It' is an arbitrary compact subregion, we conclude that N,M(f') < M2.

(2.3.12)

Now, givens > 0, let V' be chosen such that NV_W (g') =

(2.3.13)

f

I g' I2dxdy < s2.

a-TV

Holding m' fixed, we choose a number no = no(e) so large that < s2 (2.3.14) for ,u

(2.3.15)

no. By the Schwarz inequality and (2.3.13), (2.3.14), we have g')9oa-w I

I

{VN(g') + 2M} s.

Since a is arbitrary, the statement (2.3.11) follows.

EXISTENCE THEOREMS

38

[CHAP II

We are thus led to the concept of weak convergence in the general theory of Hilbert spaces wherein a sequence of elements /F, is said to converge weakly to an element / if, for each fixed element g of

the space, we have

lim(f,,-f, g)=0. It -- W

We recognize that Vitali convergence is a special instance of weak convergence.

In the Hilbert space of analytic differentials /'(P) on an orientable Riemann surface fit, an important but extremely simple inequality deserves to be singled out. Namely. if j'(p) is a given differential which at Po has the development E av z'', Y=o

I z I < r,

where z is a uniformizer in a neighborhood of p0. z (po) = 0, then

N(f') Z x E

v.QV + 1

I a, I2r2(v+1) z n l a0 2r2.

Since ao = /'(P.) (differentiation being expressed in terms of the chosen uniformizer z), the inequality may be written in the form (2.3.16)

I

f'(Po) 12 5

2ZN(/')

This inequality provides a bound for the local value of a differential

in terms of its norm. The question arises as to the best possible number k(1io) in the inequality (2.3.16)'

I f'(Po) 12 S k(p0)N(f')

Since f'(fio) is a differential, we see that k(po) I dz 12 is invariant; in other words, k(po) transforms like I /'(fio) I2. The number k(po) will be determined in Chapter 4, and it is connected with an important functional of the Riemann surface. Let us return to the general Hilbert space H. In it there always exists a complete orthonormal set of elements f1, f 2, , that is a

set such that (2.3.17)

(fIU fl) = 61

HILBERT SPACE

§ 2.3]

39

and such that every element / E H can be represented in the form (2.3.18)

a,

1 = E avfv, V=1

= (f,

1v)

In fact, by the separability axiom (4) there exists a countable set of elements ',, which is everywhere dense in H. This set can be orthonormalized by the Gram-Schmidt procedure which is a stepby-step process. After p1, 9'2, - ., q,n have been replaced by n orthonormal elements f1, f2, the element f+1 is defined by the formula fv)fv

V-1 (2.3.19)

fn+1 =

n

E

fv)1ti)

v=1

1

12

It is clear that fn+1 is orthogonal to all the preceding elements fv, v = 1, 2, .. , n, and that N(fn+1) = 1. The final set (f,) obtained

in this way is linearly equivalent to the initial dense set Given any element f E H, let us t Y y to approximate it by linear combinations of the first n 'elements f i, f 2, . , /,'In fact, in the combination E arfr, r-1

let us determine the a, in such a way that the approximation error n

n

y -I

v=1

(2.3.20) N(f-Eavfv)=N(1)-2Re { ,(f,fr)}+. ja,12 v=1

is a minimum. Since n

(2.3.20)' N(f-Eavf,)N(f) rml

/ 1 ((f,1.) I2+EIav12, r=1 n

n

r-1

we have n

(2.3:21)

N(f - Eavfv) V-1

n zN(f)-EI(f,fr) 12

+-1

with equality if and only if. a, = .(f, f.) We observe that the minimizing coefficients a,,are independent (2.A.22)

EXISTENCE THEOREMS

40

[CHAP II

of n. Moreover, by (2.3.21), 12< N(f)

(2.3.23) v=1

(Bessel's inequality). Thus the series W 1T11.)f, ZTll

r (T,

,-1

represents an element of H. Since its partial sums approximate f better than any other partial sums, and since the f,, are linearly equivalent to the q', which are everywhere dense in H, we conclude that lim N(f n-s oo

n

-E (f, f,)f,) = 0. V=1

That is (2.3.24)

f = V-1 E (f, f,)f,

and formula (2.3.18) is established. 2.4. ORTHOGONAL PROJECTION

Let H be a Hilbert space, F a complete linear subspace of H. Then F is either a unitary space of finite dimension or it is a Hilbert space. We have the fundamental decomposition formula (2.4.1)

H = F + G

where G denotes the orthogonal complement of F with respect to H.

Let h be an arbitrary element of H. Formula (2.4.1) states that (2.4.1)' .

h = f -F- g

where f e F, g e G and (/, g) = 0. To prove (2.4.1)', we seek an element / E F which lies nearest to It in the sense of the metric, that is, which minimizes the norm N(h - /). If d is the greatest lower bound of N (h - f ), f E F, we have the following inequality of B. Levi:

(2.4.2) N(JS 1/N(h- f1) -d +

f,) -d.

Here fl, /2 are arbitrary elements of F. To prove this inequality, let d and p be a pair of numbers (real or complex depending on the

ORTHOGONAL PROJECTION

§ 2.41

41

character of the Hilbert space) such that A + µ = 1. If f1, fE belong to F. so does Afl +,uf2 and we therefore have

N(h - (Afi + /42)) z d.

That is,

N(A(h-fl) +,u(h-f2)) = I AI2N(h-fl) + 2Re{2u(h-fl, h-f2)} + I'a N(h - f2) z d, 12

or (2.4.3)

IAI2[N(h-fl)-d]+2Re{A [(h-f1, h-f2)-d]} + Ii' it [N(h - f2) - d] 0.

The inequality (2.4.3) remains. true without the condition that A + ,u = 1, and is therefore valid for arbitrary numbers A, µ. As in (2.3.2), we conclude that (2.4.4)

I (h-fl, h-f2)-doss

[N(h-fl)-d]

' [N(h-f2)-d].

Hence

OSN(fj7-f2)=N((h-f1)-(h-f,)) = N(h-f1) -d- 2[(h- f 1, h-/2)-d] +N(h-t,)-d

[N(h-f,)-d]

SN(h-f1)-d

+N(h-t,) -d

={VN(h - f1) - d + and this is (2.4.2).

-d);,

Let f,, be a sequence of elements of F such that lim N(h - f,,) = d. By (2.4.2) we have

lim N(f,, - f,) = 0. Thus, since F is complete, P"-'othere exists an element f e F such that f = lim f,,. It is clear that (2.4.5)

N(h - f) = d.

Finally, let f o be an arbitrary element of F. Then f + / o is also an element of F and we have

(2.4.6) N(h-f-efo)=N(h-f)-2Re{e(h-f, fo)}+Is12N(fo) zd.

EXISTENCE THEOREM$

42

[CHAP. II

We conclude from (2.4.5) and the arbitrariness of s that (2.4.7)

(h - f, fo) = 0,

fo a F.

Thus h - f = g is an element of G, the orthogonal complement of F, and formula (2.4.1)' is proved. In the case of a linear vector space of finite dimension, the above process corresponds to orthogonal projection of a given vector h onto a linear subspace F. Hence the name "orthogonal projection" has also been applied to the process in a Hilbert space. 2.5. THE FUNDAMENTAL LEMMA

On a Riemann surface differentiation is expressed in terms of a

local uniformizer z = x + iy. If the derivatives of order k of a function T on the surface are continuous when expressed in terms of one uniformizer, they are continuous when expressed in terms of 'any other. A function 9P on the surface is said to be of class Ck if it is continuous together with its derivatives up to the order k. In particular, q' is of class CO if it is continuous. A differential a = adx + bdy is of class Cl if its coefficients are of class Ck. The class of a differential is clearly independent of the particular uniformizer used in determining it.

Let j be a subdomain of the Riemann surface V. The carrier of a continuous function on I is the smallest closed set of points outside of which the function vanishes. If rl is a function on which vanishes outside a compact subdomain of 3, we say (after L. Schwartz and de Rham) that 21 has a compact carrier. Let us understand by equality of functions on that the functions differ at most in a set of measure zero. We remark that a set of measure zero in the plane of one uniformizer will appear as a set of measure zero in the plane of any other uniformizer. Measure is understood to be Lebesgue measure. In the existence proof for harmonic differentials on a Riemann of

surface based on the method of orthogonal projection, the following lemma is of fundamental importance: LEMMA 2.5.1. A scalar 4p of class L2 satisfying the equation (2.5.1)

(-P, Arl)a = 5cvi17axay = 0 a

§ 2.51

THE FUNDAMENTAL LEMMA

43

for every scalar rt of class C° with a compact carrier is equal to a harmonic function on 5. If (2.5.1) is satisfied for a wider class of functions 27, say n of class

Cr, r z 2, the conclusion follows a fortiori. The proof which we give of this lemma follows closely along the lines of the proofs given by Weyl [14b], Kodaira [6] and de Rham [10].

We observe first that the lemma is trivial if T e C2. For since n has a compact carrier, we have by Green's formula (1.5.10) (AT, ,) = (w, An). Thus (AT, rl) = 0 for every n of class C°° with a compact carrier. Since AT is continuous, we conclude that AT = 0, that is T is harmonic. Now assume only that q3 e D. Without loss of generality we may suppose that J lies in the domain of a local uniformizer z = xx + ix2. For simplicity let x = (x1, x2), y = (y1, y2), and write dx = dxidx2. Let e be a small positive number. We denote the distance between the

points x and y by r(x, y) and define 0 if r(x, y) > e e

(x, Y) =

For e/2 S r(x, y) S e, we suppose that 0 < e(x, y) S 1 and that Q is chosen in such a way that it is of class C° and symmetric. Let (2.5.2)

w(x, Y)

=

I e(x, y) log r(x y)'

and write (2.5.3)

Aw(x, Y), x

y(x, Y)

0

Y

, x=y

where A 9 is the Laplace operator with respect to x. Then y (x, y)

is of class C, s and it is identically zero for r (x, y) < e/2. Let , be the subset of j whose points are at distance more than e from the boundary of , and let u be a function of class C°° with carrier in e. Then (2.5.4)

,p(x) _ f w(x, y)µ(y)dy a

EXISTENCE THEOREMS

44

[CHAP. II

has a carrier in I since co(x, y) = 0 if r(x, y) > e. Moreover, y, is of class C°° and we have dxq,(x) = -,u(x) + v(x) (2.5.5) where

v(x) =

(2.5.6)

f

Y(x, Y)u(Y)dy-

C

The formula (2.5.5) is a consequence of Green's formula applied to

a small sphere about the point x. We observe that v e C. Choosing n = gyp, 77 e CaO, we have by hypothesis

(4P, An) = -

f 9 ,udx + f c vdx = 0. S

0

That is, f/Acvcix= 59(x){fY(x. Y)AA(Y)dyI dx (2.5.7)

a

a

C

= f u(Y) {Jv@r , 9

Y)99(x)dx I dy.

4

Since formula (2.5.7) is valid for every choice of the function ,u, # e C°°, which has a carrier in a we readily conclude that (2.5.8)

90(y) =

f y(x, y)92(x)dx a

in 1. Formula (2.5.8) shows that ip is equal to a function of class C°° and hence 97 is equal to a harmonic function in e. Since e is arbitrary,

we arrive at the lemma. Z.G. THE EXISTENCE OF HARMONIC DIFFERENTIALS WITH PRESCRIBED PERIODS

We suppose that the Riemann surface 9X is closed and orientable,

and we denote by D the space of all real differentials a = aldx + a2dy

(2.6.1)

on

1 such that

(2.6.2)

N(a) = (a, a) = f (ai + 4)dxdy < oo. IR

§ 2.6]

DIFFERENTIALS WITH PRESCRIBED PERIODS

45

As remarked in Section 2.3, D is a real Hilbert space. We shall say that a differential a is closed if a is of class C1 and if da = 0. We shall say that a is exact if there is a single-valued function y, on

l of class C2 such that a = dy,. An exact differential is closed but not conversely. Let F be the linear subspace of D composed of differentials a which satisfy (2.6.3)

0,

exact.

The condition (2.6.3) is fulfilled by all closed differentials a. For suppose that a is closed. Then by Stokes' formula the period

P(a, K) = f a K

depends only on the homology class of the 1-cycle K. Moreover, there exists locally a function q, of class C2 such that a = dq,. The function 4' can be continued over the whole of Wt, but it will not generally be single-valued on M. Let h be the genus of )Jl, and let K1, K2, - -, K2h be a homology basis of 1-cycles for the surface composed of h dual pairs K21 1, K2l,, a = 1, 2, , h, such that

K = K1 + K2 - K1 - K2 + .. +K2h-1 +K2h -K2h-1 -K2h bounds a subdomain , of O2. The subdomain , is obtained by cutting in the usual way along the cycles K1, K2, - -, K2h. Writing (2.6.4)

P',=foc KI,

=JK

dg,,

we see that the value of p at any point of the surface 97t is determined up to a linear combination of periods of the form m1P1 + m2P2 + ... + m2hP2h where the mF, are integers. By (1.5.11), Jr

(2.8.6)

(dT, *dy,)=

cdy,=- E [P2j dy, p1 K K2K-1

- P., f 4]. K21,

Here the right side is zero since all periods of y, vanish. Thus the condition (2.6.3) is satisfied by closed differentials oc. Conversely, if

EXISTENCE THEOREMS

46

[CHAP. II

a e F, a of class C1, then we see at once from (2.6.3) that da = 0. The subclass of F composed of differentials a which satisfy (2.6.6)

(a, *#) = 0

for any closed differential fi will be denoted by E. Since fi = dip, where V is an integral with periods which may be arbitrarily prescribed, we conclude from (2.6.5) that if a is closed, then a = dp where 4p is single-valued. Thus (2.6.6) is fulfilled by all exact different-

ials. Conversely, if a e E, a of class C', then a is closed, hence exact.

We shall establish, by the method of orthogonal projection, the fundamental decomposition formula (compare [14b]) (2.6.7)

F=E+H

where H denotes the space of harmonic differentials on J2. It is clear that E and H are orthogonal spaces since a harmonic differential x is characterized by the conditions that x and *x are closed.

Hence, if a e E, we have by (2.6.6)

(a, x) = - (a, *x,.) = 0 where X, = *X. If, in particular, we select from F any closed differential, formula

(2.6.7) tells us that this closed differential is equal to a differential of E plus a harmonic differential. The differential of E must be of class C' since it is equal to the difference of two differentials, one of which is C', the other harmonic. Since this differential of E is closed, it is exact, and we conclude that the harmonic differential has the same periods as the given closed one. In particular, there exist harmonic differentials having prescribed periods.

Let y be any differential of F. By the method of orthogonal projection (Section 2.4), we see that (2.6.8)

Y = a -}- x

where a e E and x is orthogonal to all elements of E: (2.6.9) (x, al) = 0, ale E. Now let n be a single-valued function of class C3 on 9)1 whose carrier lies in the domain of a local uniformizer z = x + iy. Let z' = x' + iy' be any other uniformizer valid in a neighborhood

§ 2.6]

DIFFERENTIALS WITH PRESCRIBED PERIODS

47

contained in this domain, and define a??

(2.6. 10) 4' = ax

_

ai, dx'

an dy'

+ ay' ax'

an ax'

V

an ay'

ax' ax ay ax' ay + ay' ay Since ' and V do not depend on the choice of the uniformizer, they are functions on ))2 of class C2 which give rise to the exact differentials dq, and dye. Taking al = dq,, we have by (2.6.9) (2.6.11)

J al ax + a2 aa y R

_ (x,

dxdy

52

where (2.6.12)

Taking (26.13)

x = aldx + a,dy. = dip, we have by (2.6.3) 9 = (x, *dp) _ J

(- ai

5m

y

+ a2 a 1

dxdy.

a We suppose that the integrands in these integrals are expressed in terms of the particular uniformizer used above to define q' and y,. Then a9 ay

_

a2n

axay

-

_ clip

ax'

Hence, subtracting (2.6.13) from (2.6.11), we obtain (2.6.14)

J al (ax +'alyl) dxdy = SR

a1Lh dxdy = 0.

a

Applying Lemma 2.5.1, we conclude that al is harmonic. Similarly a2 is harmonic. In particular, x e C1.

For every exact differential 4, we have by (2.6.9) and (2.6.8) (2.6.15) (x, dp) = 0, (x, *dg,) = 0. Applying (1.5.2) and (1.5.7), we see that these conditions become (2.6.16)

0 = (*x, 4) _ (w, *dx),

(2.6.17)

0 = (x, dp) = - (9,, *d*x).

Since q, is an arbitrary function of class C2, we conclude that dx = 0, d*x = 0. (2.6.18)

EXISTENCE THEOREMS

48

[CHAP. II

In other words, x is a harmonic differential and the decomposition formula (2.6.7) is therefore proved. We remark that a harmonic differential on a closed surface cannot be exact unless it is zero. For harmonic differentials are orthogonal

to the exact ones. We are thus led to the conclusion that a harmonic differential without periods is zero. 2.7. EXISTENCE OF SINGLE-VALUED HARMONIC FUNCTIONS WITH SINGULARITIES

In order to obtain harmonic functions and differentials with singularities, it is necessary to modify the preceding argument by choosing, instead of y, a differential which does not satisfy (2.6.3). Let po be an arbitrarily chosen point of the closed orientable surface Ul, and let z = x + iy = re;* be a particular uniformizer at po which is valid for I z I S b, b > 0. Let 0 < a < b, and define (after Weyl [14a]) cos

r 9)

(2.7.1) 0,

r cos q a2

r S a,

0

elsewhere on V.

Now let (2.7.2)

9= h(r,T), 05r
0. We shall show that I I (C1) -I (g) I can be made arbitrarily small by choosing I Im small; this is equivalent to the asserted

continuity of h(q) on

.

We divide the semi-circle j_ into three parts: ft, (_-S3) n 9 - . We shall show that the contribution of each region to the integral I (C) - I (sl) tends to zero as C approaches the origin. Let us choose, in fact, an arbitrarily small but fixed number e > 0. We choose the radius Q of so small that the integral

f

dxdy

1

,z

IZ

is smaller than E2. We observe that the integral converges since

- R - A forms a horn-angle at the origin. For I C I < Q and Re c = 0, we find by a simple geometric argument that

(alog Z-I)2(°g Ix-12s 8 ax

J

ay

J

1z12

ralog I z-C1 l2

(alog Iz-C11)2

ax

ay

J+t

8

J S1.

BOUNDARY-VALUE PROBLEMS

§ 2.8]

55

Thus, we can estimate the integral

z-I

f ax LEI ax n JD

aV- a logIz_C I )1dxdy

\loglz-C11 + ayay

z-C1

J

by the Schwarz inequality, and we find that it is smaller than As, where A is of the order of magnitude of Consider next the integral extended over - -i. Since is now kept xed, we assert that the expression )s

l ylogf x-CrI

tends uniformly to zero in _ - 9 -, as C converges to the origin. Hence, the contribution of this domain to I (C) -I (P,1) can be made smaller than a by choosing I Im C I small enough.

To estimate the remaining integral, which is extended over 9, we integrate by parts. But first we cut out a neighborhood of the origin by means of a small circular are c around the origin of radius 8, since we are not sure that y, and its first partial derivatives are continuous up to the boundary C of 9. Let the truncated domain which is obtained by removing the interior of c from 9 be denoted by ft'. Integrating by parts, we have

Dst.( ,log1Z_11)=-f [loglz(2.8.11)

-f it,

-

1-k]ands

à,

l

l

k]

wh ere k is an arbitrary constant of integration. Since C and C1 are inverse points with respect to S@, the ratio I Z - C L/ I Z - C1 I is a constant, depending on C, for z on the circumference of R, and we choose k to be just the log of this value. Thus in the integral over aft' only the integration over c remains. Following a well-known argument, we can show that the integral over c tends to zero if the radius 8 of c approaches zero through a suitable subsequence. In fact, suppose that there exists a positive

constant A such that (2.8.12)

A = dt,t. Finally, the differential dQ,0,l(p) defined by (3.1.25) is of the third kind and its integral has a single-valued real part. We denote by dw,o,l(p) the differential obtained from dQ,e,l(p) by adding a

RELATIONS BETWEEV I'1 r 'Js"Rk NV TIAL S

74

[CHAP. III

linear combination of basis differentials of the first kind in such a way that the periods

Pp _ i d V1( `T'2p-s

vanish. We have two ways of normalizing differentials of the second and third kind; by requiring that their integrals have single-valued

real parts or vanishing periods with respect to the cycles K2,,_1. The first type of normalization is called ,real normalization", the second "complex". 3.4. PERIOD RELATIONS

be the surface obtained fl om by cutting it along the cycles (3.1 19). The integral of any differential dl of the first or second kind is single-valued in , and we may therefore consider the integral Lfx

f f1d/2 M

where df1 and df2 are an arbitrary pair of such differentials. By Cauchy's residue theorem,

f fldf2 = E I(K,,, K,) f dfl f dfs 2h

(3.4.1)

K,,

et7

_

R,

If df1 ' f aft- f df2' fdjlj = 2ni (sum of residues of f1df2ina). K2k-1

R2k-l K2k

K2k

Taking first dl, = dZ, d/2 = dZ,, we see that h

z (ru, 2k-lrs, 9k - I , 2k-lr,4.2k) = 0 k-l The vanishing of the imaginary part of this expression gives the symmetry law (3.4.2)

Re

f dZ = Re f dZ,, µ,

K,

1 ,2 , . . . ,2 A .

xµ

We have used here the fact that Im r,,,= I(K,,, K,). This symmetry law is also an immediate consequence of (3.2.8) and (3.1.16).

PERIOD RELATIONS

§ 3 4]

75

Taking d f 1= dwu, d f 2= drv,,, we obtain

Jaw,. =

(3.4.2)'

J

, h.

dwti, ,u, v = 1, 2,

xu

X,

Next, let df 1 = dTQ') (p) = dT ? (p), d f2 = dZZ. In terms of the particular uniformizer 4, we have for p near q + regular terms) dC.

dT4{

Formula (3.4.1) gives 11 j rp 2k-1-

j dT4')

I.

Taking real parts, we have fdT Q') x,,

= 2ri r (

1 1

!

d"Zk(q)

)

dqr

x2x-z

xzx

(3.4.3)

}

dg)

1

_ -2ni (y

1) !

In'

1, 2, .

. ,

2h.

J

If we choose df1 = dg"), d/2 = dW, we obtain

fd) _ -2ni (r--1)I

(3.4.3)'

1

d'w,u(q)

dq'

Third, take dl, = dS2Q1QS d/2 = dZ, Since f1 is single-valued in I only if we cut along a path connecting q1, q2, we have in place of (3.4.1) the formula

J /1d/2 +

(3.4.1)'

ae

f gs o

f ld f 2 = 2ni (sum of residues).

q,

Here oft denotes the jump of f1 from left to right across the cut joining q1 and q2. We have

fq.afldf2 = 2ni J'aZ,. = 2ni[Z,(g2) - Z,(q1)]. 91

Qz

Thus (3.4.1)' becomes r9s

dD,3Q (p) = 2ni Im

(3.4.4) s,,

J

ql

dZ;,.

RELATIONS BETWEEN DIFFERENTIALS

76

d/2 = dw,, we find that

In a similar way, taking d f 1

f dw.,.,

(3.4.4.)'

[CHAP. III

2aci

K2 f,

fdwn. 91

Now take df1 = dTQ'), d/2 = dT( ). The imaginary part of equation (3.4.1) then gives

Similarly d't°lr)

dr_L') (Q)

d pr

dqr

If we choose d/1 = dTQ') (p), d/2 = dS2,1°., formula (3.4.1) gives (taking imaginary parts) (3.4.6)

Re{TQ')(41)

-

T')(4a)} = Re {

(4) /

(r - 1) !

dqr

while the choices d/1 = dtQ'> (p), dt2 = dcv°1°, give (3.4.6)'

ta' (41) - t( r) (42) = (y

Finally, let df1 = d

D,1,.,

1

1)'

d/2 = dS2°1°a; we obtain (P2)} = Re{Qm19,(41)

(3.4.7)

(4)

r

Q.,,,(/Z9)},

while, if d f1 = dw91D,, d/2 = do)°1 °,, then (3.4.7)'

wal°,(P ) = W9192(41) -ws1v,(g2)

Formulas (3.4.7), (3.4.7)' constitute the law of interchange of argument

and parameter. 3.5. THE ORDER OF A DIPPERENTIAL

The order of a differential is the difference between the sum of the orders at its zeros and the sum of the orders at its poles. Given any linear differential dZ of , its order is given by the formula (3.5.1)

ord dZ = 2(h -1).

In order to prove (3.5.1), let z be any non-constant function of

.

We observe first that, for each complex number a, dz/(z - a) is a differential of the third kind whose residues are the orders of the

§ 3 51

ORDER OF A DIFFERENTIAL

77

function z - a at its zeros and poles. Since the sum of the residues of this differential is zero, we see that z - a takes the values a and oo equally often on . Thus z takes each value on the same number of times, say n times. Let us say that a point 16 of t where z takes the value a lies over the point a of the z-sphere. Then t is realized as an n-sheeted relatively branched covering of the z-sphere. Fewer than n points 15 of lie over finitely many values a; these values a correspond to points of 5' where the function z is of higher multiplicity than 1. Let PO

be a point on N where z takes the value a exactly m times. If t is a local uniformizer at P0, we have z = a + Cmtm + Cm+itm+1 + ..., C,,n zA 0.

Hence (z - a)'Im is also a uniformizer at p0 and the point po over a is a branch point of order m - 1. The differential dz has a zero at p0 of order m - 1. If z takes the value oo at PO exactly s times, then

z= I (co+cit+...),co30, so (1/z)ii' is a uniformizer at

P0.

In this case dz has a pole of order

s + 1 at CO. The sum

V=E(m-1)+E(s-1) 0

00

is called the branch number. We shall now prove the formula (3.5.2)

V=2(h+n-1).

Let the z-sphere be triangulated in such a way that z = oo and the points lying under branch points are vertices of triangles. Moreover, suppose that at most one vertex of a triangle has branch points lying over it. This triangulation can be carried over to the covering Over each triangle of the sphere there are n triangles of W, over each side of a triangle of the sphere there are n sides of 16. Let a2, a1, 0 ° be the numbers of triangles, sides and vertices of W, and let a2, a1, a° be the corresponding numbers for the sphere. Then a2 = nag, al = nal. (3.5.3) If v points with branch orders r1-1, , ry - 1 lie over z = a, then


78

[CHAP. III

v=n-[(rl-1)-F-...+ 0"-1)7

and hence

a° = na° - V.

(3.5.4)

But (3.5.5)

a° - a' + at = 2, 0 ° -'al + a2 = 2 - 2h;

therefore

2-2h = a2-a1 + a° = n (a2-a; + a°) -V = 2n-V. The order of dz is equal to

E(m-1)-E(s+ 1) =V-2Es. 0

co

00

Since z takes each value (including oo) n times, we have

Is = n. Thus the order of dz is equal to V - 2n = 2(h - 1). Since the ratio of two differentials is a function and the order of a function is zero (as we have seen), every- linear differential of g- has the same order,

namely 2(h- 1). Since the v-th power of a linear differential is a particular kind of differential of dimension v, and since the ratio of two differentials

of the same dimension is a function, we conclude at once from (3.5. 1) that (3.5.1)'

ord dZ' = 2v (A - 1).

3.6. THE RIEMANN-ROCH THEOREM FOR FINITE RIEMANN SURFACES

A divisor on the double

of a finite Riem,ann surface 912 is a 0-cycle

D = Emp, with integral coefficients m;, consisting of a finite number of points P,. We define the order of a point p, to be 1, ord ps = 1, and the order of D to be Em, ord p; = Em;. The set of all divisors form an Abelian group, the divisor group. To each differential dZ° of gg- which is not identically zero we associate a divisor s 7n;1; where m, is the order of dZ' at P_. The divisor of a function / of will

be denoted by (/) and two divisors D and D' are said to be linearly equivalent if there is a function f of , not identically zero, such that

§ 3.61

RIEMANN-ROCH THEOREM

79

D - D' = (f). A class of equivalent divisors (divisor class) which contains the divisor D will be denoted by (D). All divisors in (D) plainly have the same order which is called ord (D). Clearly, the divisors of linear differentials dZ all lie in the same divisor class which 0 will be denoted by (W). Given a divisor D = Zmp, we write D if all m, Z 0. The set of all functions f of such that (f) + D z 0 will be denoted by F(D), and the real dimension of the linear space

F(D) will be called the dimension of (D): dim (D) = dim F(D).

If (f) + D z 0, we say that f is a multiple of - D. If D = Emip; is a divisor of the double , we call D = the conjugate divisor. Here p denotes the point of which is conjugate to p. A divisor of u will be called a divisor of SR if and only if it is equal to its conjugate. A divisor D of 9J2 therefore has the form

D = Em;Pf where Pi = Pi if p,; is a boundary point of N and Pi = p; + pi if p, is an interior point of 9)1. Two divisors D and D' of 9Tt are called linearly equivalent on 9R if D - D' _ (f) where / is a function of fit, and the real dimension of the linear space of

functions / of 9N such that (f) + D z 0 is the dimension of (D). We have the following theorem ([4]) in which divisors, differentials

and dimension refer to 91: THEOREM 3.6.1. For any finite Riemann surface (3.6.1)

9Tt,

dim (D) - dim (W - D) = ord (D)-G+ R°

where

(3.6.2)

G = 2h+c+m+R°-2,

(3.6.3)

ord (W) = 2(G - R°).

We remark that dim (W - D) = dim { dZ I (dZ) z D}. In fact, let dZ° be an arbitrary linear differential. If f is a function with (/) Z D - (dZ°), then dZ = / dZ0 is a differential with (dZ) Z D. Conversely, if (dZ) z D, then f = dZ/dZ° is a function

with (f) z D - (dZ°). If 9N is closed and orientable, (3.6.1) is the classical RiemannRoch theorem. If we assume that (3.6.1) is valid for closed, orientable surfaces, the validity of (3.6.1) in general is readily proved.

80


[CHAP III

For, let a divisor belonging to a function of 02 (or ) be called a principal divisor of 9971 (or 3). The group of the principal divisors of 9A is equal to the intersection of the group of principal divisors of 2 with the divisor group of 9)1. For, let D be a divisor of 9t which is a principal divisor of . Then D defines f, where l is a function of . The conjugate function f has the divisor b = D, so j// is

equal to a constant, 7 = at, say. Forming the conjugate, that is, applying the -operation to the equation f = al, we obtain / = 41; hence as = I a 12 = 1. But then g = Val is equal to its conjugate and is therefore a function of 9)1. Since g has the divisor D, we see

that D is a principal divisor of R. , f,, be linearly indepen-

Now let D be a divisor of 9't and let /1, /2,

dent functions of 9)t in the real sense such that

f = al/1 + ... +

ak real,

is the most general function / of 91 satisfying (f) + D dim (D) = ,u. Let g = c1/1 + + c fo, ck complex.

0. Then

We have to show that g is the most general function of t satisfying

(g) + D Z 0. It is clear that (g) + D z 0. Let g be any function

of 1J satisfying (g) + D

gl =

0, and let

be its conjugate. Then

g+ g

g-g

2

, 92 =

2i

are two functions of t which are equal to their conjugates and are, therefore, functions of 9'1. Since (g) + D ? 0 and (g) + D z 0, the same is true of gl and g2. Hence g,=ai/1+...+a,l,,,

g2=b1/1+.+b/,,,

where ak and bk are real. Thus

g = g1+ig2=

(a1+1b1)/1+...+ (a, +

and the validity of formula (3.6.1) for closed orientable surfaces therefore implies its validity for an arbitrary finite Riemann surface. For completeness, we indicate briefly how the formula (3.6.1) for a closed orientable surface 0 may be deduced from the period rela-

RIEMANN-ROGH THEOREM

§ 3.8]

81

tions of Section 3.4 (see [6]), and for simplicity we assume that

D= p1+. P2+ .. + P0-Q1-Q2-....._Q' where Pi = P + 1' Q, = qj + q, and the points p{, q{ are distinct points of Tt. Let (3.8.4)

, dt,,, are the normalized differentials of the second kind defined in Section 3.3 and c1, c2, , cf, are ,u complex constants.

where dt91, dtq,,

Then dt has double poles at p1, p2,

, p. and its periods Pµ all

vanish. The remaining periods will vanish provided that the following

equations are satisfied: dw1

dw2

dw1

dw1

C1dpl+.C2dp2+...+C1,apK=0,

+ dw2 + C2dw2 aPa + ....'f C0 a pµ = 0,

................

(3.6.5)

C1 dp1

C

dw dw _+cg--!

1 dP1

dwh

+-...+C d h=0. /1 dp,,

P2

This follows from equation (3.4.3)'. If equations (3.6.5) are satisfied, the integral t of dt is a function of ; it will vanish at q1, q2, , q,

if we can choose a constant of integration co such that co + c1t1(g1) + c2t0=(q1) + ... + C,tl,,(g1) = 0, (3.6 6)

co + c1tn1(g2) + CA,, (q2) + ... + c),, (q2) = 0, .

.

.

.

.

.

.

.

.

.

C0 + C1tv1(q,) + c2t9, (q,)

.

.

.

.

.

.

.

.

.

.

+ ... + ct, (q,) = 0.

The u + i constants co, c1, c2,, cA therefore must satisfy h + v linear homogeneous equations. If the rank of this system of h + v equations in IA + 1 unknowns is r, then there are u + 1- r linearly independent solutions in the complex sense; that is (3.6.7)

dim (D)=2(,u4-1-r)=ord (D)-2h±2+2(h+v-r).

Assume first that v = 0. Then there are no equations (3.6.6) and the transposed system corresponding to (3.6.5) is


82

dw1 Yi

dw2

+ Y2TP1 + dw1

dw2

dw.,

dw2

+ Yh

dwh

_ 0,

dw,

... +YhdP2 Yi-+Y2--+ dp2 d02

(3.6.5)'

[CHAP. III

0,

dwh

+ Yh dpµ = 0.

We see that (3.6.8)

dim(W-D) = 2(h-r) = 2(h+v-r)

since v = 0 by assumption. In this case formula (3.6.1) follows from (3.6.7) and (3.6.8). Next, assume that v = 1. Then there is one equation (3.6.6) which merely determines co in terms of c1, c2, , C,,. We therefore ignore this equation and consider the h equations (3.6.5) in the ,u unknowns c1, c2,

(3.6.7)'

, c,,. If the rank is r, we have

dim (D) = 2(µ - r) = ord (D) - 2h + 2 + 2(h - r).

But the number of linearly independent differentials which are multiples of D is equal to the number of linearly independent differentials which are multiples of P1 + - - + P,, = D + Q1. For the sum of the residues of a differential must equal zero. Hence we again have (3.6.8) and therefore (3.6.1).

Finally, suppose that v > 1. We then subtract the v - 1 last equations (3.6.6) from the first and use formula (3.4.6)'. We obtain

the v - 1 equations Ci

(3.6.6)'

C1

L"'2 + C2

dP, T

dP 1 + c2 dP2 2 + c[W4hQr

ci

'I'S + . . . + C

dpi + ca

A14s = Q

dp

dP2

dp, = ... + cµ -4192

dwwl,, C"

0,

=p dp.

CONFORMAL MAPPINGS

§ 3.71

83

plus the one equation f co + c1t,1(g1) + c2tv,(g1) + ... + ct,, (q1) = 0.

(3.6.6)"

The equation (3.6.6)" determines c° and will again be ignored. We therefore have the system of h + v - 1 equations (3.6.5) and (3.6.6) in the it unknowns c1, c2, , c,,. If the rank is r, we have (3.6.7)"

dim (D)=2(4a-r)=ord (D)-2h+2+2(h+v-1-r).

The transposed system is dw 1 Y1

d

+ ... +

dw1 Y1

02 + dw1

Y

dwaa

dw h Yh+1

dwh

... + YA dp2

+ Yh+1

d

+ ... + Yh+y-1

dW ala,.

1

dcoQ a,

d-2

dwh

&0q a

dp,.

dpv

-f-

dpi

= 0,

+ Yh+,-1 dP2

+. Yh+ti-1 dal'" =

0.

Since every differential dZ which has poles at the q, may be written

in the form YIdw1 + ... + Yh dwh + Yh+Idwala, + ... + Yh

a c,

we have (3.6.8)'

dim (W -D) = 2(h + v- 1-r),

and formula (3.6.1) follows from (3.6.7)" and (3.6.8)'. 3.7. CONFORMAL MAPPINGS OF A FINITE RIEMANN SURFACE ONTO ITSELF

Klein [2] classified finite Riemann surfaces by means of their algebraic genus G, where G is given by (3.6.2). There are precisely seven surfaces with G :!!g R°, namely:

h= 0, c=0, m = 0 (sphere) h = 0, c = 0, m= 1 (disc) h = 0,"c = 1, m = 0 (projective plane)

G = 0,


84

[CHAP. III

h=1, c=0, m=0 (torus) h = 0, c = 0, m = 2 (ring or doubly connected domain)

G_

h = 0, c = 1, in = 1 (Mobius strip) h = 0, c = 2, m = 0 (Klein bottle or non-orientable torus) Let a be the dimension of the space of all classes of conformally equivalent finite Riemann surfaces of the same topological type, and let a be the number of parameters of the continuous group of conformal mappings of the surface onto itself, The formula (3.7.1)

a - e = 6h + 3c + 3m - 6

which was given by Klein [2], has already been discussed in Section 2.11 where, however, we made use of a fact which depends on formula

(3.6.3), namely if w = u + iv is the function of Section 2.11 with a simple pole at one point of l (u being single-valued on), then the total multiplicity of the zeros of dw in the interior of 9R is equal to 2h + c. As remarked in Section 2.11, the differential dw has two simple zeros on each boundary component of 7l. Since dw has order - 2 at one point of lfl, we see that the number of zeros of dw in, the interior of t is 2h if l)l is closed and orientable and 2h + c otherwise, and hence the statement follows. We have the following theorem: THEOREM 3.7.1. The number e of the parameters of the continuous group of con f ormal mappings of a finite Riemann surface onto itself is different from zero only in the following seven exceptional cases where

(3.7.2)

6 for h = 0, c = 0, m = 0 (sphere) 3 for h = 0, c = 0, m = 1 (disc) 3 for h = 0, c = 1, m 0 (proj ective plane) 2 for h = 1, c = 0, m = 0 (torus) 1 for h = 0, c = 0, m = 2 (doubly connected domain) 1 for h = 0, c = 1, m = 1 (Mobius strip)

1 for h = 0, c = 2, m = 0 (Klein bottle). This theorem is an immediate consequence of the following general theorem (due essentially to H. A. Schwarz) which we state without

proof (for a proof see [6]); THEOREM 3.7.2. The group of the con f ormal mappings of an orientable

§ 3.8]

RECIPROCAL AND QUADRATIC DIFFERENTIALS

85

Riemann surface onto itself is discontinuous apart from the following seven exceptional cases: sphere, simply or doubly punctured sphere, disc, punctured circle, doubly connected domain, torus. In fact, if IR is orientable, Theorem 3.7.2 is immediately applicable.

If V is non-orientable, we apply Theorem 3.7.2 to its double a. If the conformal mapping oft onto itself carries the point p into the point q, we extend the mapping to by sending the point into the point q. We therefore conclude that e = 0 unless 1R is one of the surfaces (3.7.2), and we then examine the cases. The only cases (3.7.2) which need to be mentioned are the nonorientable ones. The projective plane arises from the sphere by identification of diametrically opposite points. The group of conformal mappings onto itself is therefore the group of spherical rotations. If the Mobius strip is represented in the normal form discussed in Section 2.1 we see that the only conformal mappings onto itself are given by w' = eiB w and w' = R es°f w, 0 real. Finally,

the Klein bottle arises from the w-plane punctured at infinity by identifying points which are equivalent with respect to the group of transformations

(3.7.3) w-*w+m+nicw, wow-{- 2 +m+nia, w> 0. The relatively unbranched two-sheeted orientable covering of the Klein bottle is obtained by identifying points which are equivalent with respect to the subgroup w --* w + m + nuo. 3.8. RECIPROCAL AND QUADRATIC DIFFERENTIALS

The everywhere finite reciprocal differentials dZ-1 of 9a2, z a local uniformizer, dZ-1 = r(z)dz-1, are connected with the infinitesimal conformal mappings of t onto itself. This is intuitively clear. For let z be a local uniformizer valid in the neighborhood of a point of V. Under the infinitesimal mapping

the point with the parameter value z goes into the point with the parameter value z + er(z) where a is an infinitesimal real quantity. If z' is another uniformizer valid in the same neighborhood, then z' is analytic in z or in ,f. Assume for simplicity that z' is analytic in z.

86


[CHAP. III

Then (neglecting infinitesimals of higher order) dz'

z' +srl(z') =z'(z+er(z)) = z' +sdz r(z), so

rl(z')

r(z)

dz'

dz

Furthermore, if z is a boundary uniformizer, then z is real at a boundary point p. Since a boundary point goes into a boundary point, z + sr(z) must also be real. Thus r(z)/dz is real on the boundary and is therefore a reciprocal differential of fl. We show now that (3.8.1)

P = dim (- W).

By (3.6.3),

ord (- W) = 2(R° - G). Since the order of an everywhere finite reciprocal differential must be non-negative, there are no everywhere finite reciprocal differentials when G > R°. Formula (3.8.1) is therefore- proved in this case. Taking (D) = (W) in (3.6.1) we see that the number of linearly independent everywhere finite linear differentials of Yl is equal to dim (0) + 2(G - R°) - G + R°

G

since dim (0) = R°. If G = R° we therefore have R° everywhere finite linear differentials (up to a constant factor) whose orders

are equal to zero. Their reciprocals are the only everywhere finite reciprocal differentials and we have dim (- W) = R°. From (3.7.2) we see that (3.8.1) is true in this case. Finally assume that G = 0, in which case the double 2 is the z-sphere. All reciprocal differentials of t must then be equal to a rational function of z

times dz-1. Bearing in mind that 1/z and not z is a uniformizer at oo, we see that the everywhere finite reciprocal differentials of the sphere

have the form a + bz + cz$ (3.8.2)

dz

where a, b and c are arbitrary complex constants. Hence if DI is

the sphere dim (-W) = 6. If 0 is the unit circle, the reciprocal differentials still have the form (3.8.2) but they must satisfy the

§ 3.8]

RECIPROCAL AND QUADRATIC DIFFERENTIALS

87

additional restriction that they are real on I z I = 1. It is then seen that the everywhere finite reciprocal differentials of the circle are linear combinations (with real coefficients) of the three differentials

i dz, i

(3.8.3)

dz `z

+ z )' dz \z - z I

Hence dim (-W) = 3 in this case. Finally, the projective plane is the unit circle with diametrically opposite points of its circumference identified. The three differentials z

z(z

dz dz

11

T zj

z(z Zdz

(

z

take the same or conjugate values at diametrically opposite points and are therefore reciprocal differentials of the projective plane. Hence dim (-W.) = 3 in this case also. By comparison with (3.7.2) we see that formula (3.8.1) is valid when G = 0. Finally, taking (D) = (2W) in (3.6.1), we obtain

dim (2W) - dim (- W) = 3(G - R°) since ord (2W) = 4(G - R°) by (3.6.3). Thus, by (3.6.2), (3.8.4)

dim (2W)-Q= 6h+ 3c + 3m-6. Comparing with (3.7.1) we see that 13.8.5) dim (2W) = o. This equation shows that the quadratic differentials are connected with the moduli of the finite Riemann surface R. REFERENCES 1. K. HENSEL and G. LANDSEERG, Theorze der algebrazschen Funktionen einer Variabeln,

Teubner, Leipzig, 1902.

2. F. KLEIN, Rzemann'sche Flkchen I and II, Vorlesungen Gdttingen, Wintersemester 1891-1892 and Sommersemester 1892. 3. C. NEUMANN, Vorlesungen iuber Rzemann's Theorie der Abel'schen Integrate, Teubner,

Leipzig, 1884.

,Extremale quasikonforme Abbildungen and quadratische Differentiale", Abh. der Preuss. Akad. der Wzss., Math.-Natures. Kl. 1939,

4. O. TEICHMbLLER,

22 (1940). 5. K. I. VIRT-4NEN, tVber Abelsche Integrate auf nullberandeten Riemannschen Fl9chen von unendliehem Geschlecht," Ann. Acad. Scz. Fenn., A. 1, 56 (1949). 6. H. WEYL, Die Idee der Riemannschen Flache, Teubner, Berlin, 1923. (Reprint. Chelsea. New York, 1947).

4. Bilinear Differentials 4.1. BILINEAR DIFFERENTIALS AND REPRODUCING KERNELS

In this section we make some preliminary remarks concerning the present chapter, which centers around the bilinear differentials and, in particular, reproducing kernels. In the next section we define the Green's and Neumann's functions, and we then express as many of our functionals as possible in terms of these basic functionals. In particular, we express the differentials of the first kind in this way, as well as the mapping functions corresponding to the canonical slit domains of Chapter 2. We then define certain linear spaces of differentials on a Riemann surface, and construct the reproducing kernels for each of them.

The underlying purpose of this chapter is thus to express the various functionals in terms of one functional, namely the Green's function. In subsequent chapters we shall be concerned with the dependence of these functionals on the Riemann surface, and the results of the present chapter will enable us to confine our study of the dependence essentially to the Green's function. We call a complex analytic expression A (P, q) which depends on two points P, q of a Riemann surface a bilinear differential if A((, q)dzdg

is a conformal invariant, z and C being local coordinates of the points

and q respectively. In other words, 2(P, q) transforms like a linear differential of each argument and is therefore a double covariant vector.

If 9)l is a closed orientable Riemann surface, the expression z

(4.1.1)

vao

)

apaq

is a bilinear differential. Here dD (88]

is the differential of the third

REPRODUCING KERNELS

§ 4.1]

89

kind defined in Chapter 3. This bilinear differential is symmetric in p and q, has a double pole for p = q, and, it should be remarked, is independent of the parameter point q0. In order to prove these statements, we use the law of interchange of argument and parameter in the form (compare (3.4.7)) D..(P) - D...(PO) + ('SAQQO(

)) - (D.. (Po))

_ Q,0(q) -Q,0(go) +

(Q,,0(q))-- (Qvn,(go))-.

Differentiating this identity /with respect to p and q, we find that a2Qv90(q)

(4.1.2)

apaq

apaq

That is, (4.1.1) is a symmetric bilinear differential independent of qo and po. For p in a neighborhood of q, we have, in terms of the local uniformizer z(p), (4.1.3)

D..(p) = -log [z(p) - z(q)] + regular terms

and hence

d D..(p)

=

regular terms. [z(p) - z(q)]2 + Let us consider now the case of an orientable finite Riemann surface 9't with boundary C. If we complete t to its double a-, we `may consider. the class of differentials which are regular analytic on . This class has a finite basis, namely the G basis differentials of the first kind of . In order to develop a significant functiontheory for the Riemann surface 9)t, we introduce a wider class of differentials, namely those which are regular analytic in the interior of 9t but which are not necessarily defined on the boundary of 9)t (4.1.4)

apaq

or elsewhere. Because of their restricted domain of definition, we call these differentials "interior differentials" on I't. The interior different-

ials with finite norm over 9)1 form a Hilbert space M which will be one of the main objects of study in the sequel. Let be the double of fit, and QQ,Q(p) the above integral o{ the third kind of . In the theory of the Hilbert space M, the following bilinear differential plays a fundamental role: (4.1.5)

LM(P, q)

1 a2 D. (p) n apaq .

BILINTEAR DIFFERENTIALS

90

[CHAP. IV

This bilinear differential is symmetric in p and q, and, for p near q, has the development (see (4.1.4)) (4.1.5)'

LM(P, q) =

z(q)]2

+ regular terms.

n [z(p) As a differential of P defined on SR, Lm (p, q) has to be distinguished from LM(P, q), which is everywhere regular in E2 because its double pole lies in 1. While LM (p, q) obviously satisfies the symmetry law (4.1.6)

LM(P, q) = LM(q, p),

we now want to prove the law of Hermitian symmetry for the differential LM(p, q), namely (4.1.7)

LM(P, q) = (LM(q, p))-.

By (4,1.6), (4.1.7) is equivalent to (4.1.8)

LM(P, q) =

q))

In order to prove (4.1.8) we make use of the fact that the differential dD. (p) has imaginary periods on Re {QQa(p)

-

.

Hence the expression

represents a single-valued harmonic function on . We observe that the singularities of the two terms cancel at q and also at, q. Applying%

the maximum principle to this regular harmonic function on the closed Riemann surface , we see that (4.1.9)

.Sl;;'(p)

where C is independent of

= (.Q"4))- + C Differentiating this identity with respect

to P and "q, we find that (4.1.10)

a2Q (p) apaq

T

26(_) l

(-4ai )

In this formula conjugate uniformizers must be used at conjugate places. Finally, (4.1.8) follows from (4.1.5) and (4.1.10), and this proves the Hermitian symmetry of the kernel LM(P, q). Let 5 be a point on the boundary C of V. Since _ we have the identity LM(P, q)dz =

q)dz.

§ 4.1]

REPRODUCING KERNELS

91

We use the law (4.1.8) and obtain (4.1.11)

LM(P, q)dz = (LM(p, q)dz)-, p e C.

Formula (4. 1.11) may be interpreted as follows: Given any fixed point q of 2, we can find a pair of differentials taking conjugate values on C. One differential has a double pole at the point q, while the other is regular throughout Ill. It is easily seen that the differentials LM (P, q) and LM (fi,q) are, for fixed q, conjugate differentials of in this dependence upon p (Section 2.2). The differential LM(p, q) in its dependence upon P belongs to the

Hilbert space M. We shall show in Section 4.10 that it has the reproducing property (4.1.12)

(dl, LM (p, j))

1'(q)

for any dl a M, and on the other hand, (4.1.13)

(df, LM(P, q)) = 0. In formulas such as (4.1.12) and (4.1.13), we should write (dl, LM (P, q)dz) but, for simplicity, we drop dz. It will be clear which

variable refers to the integration. In the case of plane domains the bilinear differential -LM (1, q ) has been extensively studied, and is called the reproducing kernel -of the class M (see [2a, b, c, d], [3a], [4], [6a], [10], [13], [14b, c]). Even in this special case, it was pointed out (see [3b], [14b]) that the introduction of the kernelLM(p, q) completes the kernel theory in a symmetric manner. However, it has been customary to regard the two kernels as essentially different, although the theory of the double of a domain shows that they are the same. In the special case of a plane domain we may replace the abstract ports p, q by their coordinates z and C. Consider the interior of the circle + z I < r; then the double may be chosen as the z-sphere and we choose y2

(4.1.14)

x = -. z

The integral of the third kind on the sphere is given by Da (z) = log

x

+ constant,

BILINEAR DIFFERENTIALS

92

[CHAP. IV

and hence we obtain

(4.1.15)

LM(z,

1

1 C)2.

(z

Let us compute LM(z, ) by means of (4.1.14) and (4.1.15). Using the transformation law for differentials, we have i

LM(x, )d

(z

Thus, using

as uniformizer at , we find that

Z)2d

1

y2 2

-7 2

(4 1 16)

LM (x ) - 01(y2 - zC)2.

If TZ is an arbitrary multiply-connected domain of the z-plane, we can write (4.1.17)

L (x,

C)2

+ l(z, C)

where l(z, C) is regular analytic in M. The expression l(z, C) plays

an important role in the theory of conformal mapping of plane domains (see [3b], [6a, b], [14b, d]). The representation (4.1.17) is possible because every plane domain is, by definition, imbedded in the complex plane (Riemann sphere), and we can in this case, split off the singularity of the L-differential belonging to 9N by sub-

tracting the L-differential of the sphere. The theory of the logarithmic potential and of analytic functions in plane domains is essentially based on the imbedding in the complex plane. This geometrical fact has the analytical aspect that the elemen-

tary singularities are logarithmic and rational in the large as well as in the neighborhood of the singularities. Thus the principal parts may be subtracted off in the large, leaving a function everywhere regular

in the interior of the domain. In place of subdomains of the sphere, we shall consider subdomains

of an abstractly given Riemann surface N. This more general imbedding leads to new results even in the case of plane domains (see [15a, b]). We decompose a differential on alt into a differential on 91 with singularities, plus an interior differential on Wt. For example, let LM(p, q) and 2M(¢, q) denote the bilinear differentials

§ 4.2]

GREEN'S AND NEUMANN'S FUNCTIONS

93

for V and g{ respectively. We can then write (4.1.18)

LM(z, q) = £°M(', q) -4-1M(P, q), where ZM(P, q) is an interior bilinear differential on

7t.

In this chapter attention is confined to the domain functionals and, in particular, to the elementary singularities belonging to one given Riemann surface. The comparison of the functionals of two Riemann surfaces, one of which is imbedded in the other, will be taken up in the following chapter. 4.2. DEFINITION OF THE GREEN'S AND NEUMANN'S FUNCTIONS

When V is a domain of the complex z-plane, the Green's function G (z, g) of 31 is defined to be the (single-valued) harmonic function on ll which vanishes on the boundary of Dl and which has a logarith-

mic pole at the point C such that the difference G (z, 4) - log

1

remains regular at C. Neumann's function N(z, ') (Green's function of the second kind) has the same logarithmic pole at C as the Green's function and has on the boundary of 9Jl a constant normal derivative. Green's function is uniquely determined by Tl while Neumann's function is determined only up to an additive real constant. Green's function is a harmonic function of the first kind. Neumann's function, on the other hand, is not of the second kind and is not even a conformal invariant. However, the difference N(z, C) -N(z, CO) is an

invariant which is a harmonic function of the second kind. We therefore redefine the difference N(z, C) -N(z, Ca) to be the Neumann's function of D7Z. This difference will be denoted by N(z, C, CO).

We suppose first that J1 is an orientable finite Riemann surface of genus h (h handles) with m boundary components, m > 0. The denote double IN of Jl then has genus G = 2h + m - 1. Let the integral of the third kind of which is normalized by the condition that the real parts of its periods vanish. We define the Green's function G (P, q) of t by the formula ([15a, b] ) (4.2.1)

G(', q) =

1

2

{tea"v(p) -"


94

[CHAP. IV

Let us make the general remark that D.. (P) + -Q.,,. (P) = C,

where C is a number independent of p, for arbitrary fixed points q and q0. In fact, this combination of integrals of the third kind is regular on the whole surface since, by the definition of Q, its poles cancel. Moreover, since the real part of each term is single-valued, the real part represents a bounded, regular, single-valued harmonic function on a closed surface, and is therefore a constant. Thus, using (4.1.9), we have (4.2.2)

Slv"a(

)=-Q

c1 _ - (? (fl)- + c2

where c, and c2 are numbers independent of p. Hence G(p, q) = Re {De.(p)} + cs, and Tm G, as function of p, is therefore constant. On the boundary of 9)l, p = so G (p, q) vanishes when p is on the boundary C of V.

Therefore, Im G = 0 in p and q. By the law of interchange of argument and parameter (4.2.3)

G(p, q) = G(q, p),

and it follows that G(p, q) = 0 whenever p or q is on the boundary C of Dl. When p is near q, q in Tt, z a uniformizer at q, we have (4.2.4)

(4.2.5)

G(p, q) = log I x(p)

1

x(q)

`+ regular terms.

Thus G has all the properties of Green's function for plane domains.

We observe that

G(p, q) _ -G(p, q), G(p, 4) _ -G(p, q). The Neumann's function of the surface shall be defined as

(4.2.6)

1Y

1

(4.2.7) N(p, q, q0) = 2 {nar,(p) + D (p) + Qav,(p) + As a function of p, Im N is constant by (4.1.9) and, since the function has been defined in (3.1.25) only up to an additive


§ 4 21

95

constant c(q, qo), we may determine the latter such that Im N - 0. We ha`, e (4.2.8)

N(p, q, qo) = N(p, q, qo), 1\''(Y, q, qo) = N(p, q, qo).

Therefore, if z = x + iy is a boundary uniformizer, we see that aN(P q, qo)

(4.2.9)

a

=0

Y

on the boundary C of T2. In other words, N has a vanishing normal derivative on the boundary C. Let z be a uniformizer at q, zo a uniformizer at qo. Then (4.2.10)

1

N(p, q, qo) = log I z(p)

z(q) I + regular terms,

N (p, q, qo) = log I zo (p)-zo (qo) I -I- regular terms,

asp approaches q and qo respectively. Finally (law of interchange of

argument and parameter) (4.2.11) N(p, q, qo) -N(po, q, qo) = N(q, p, po) -N(qo, p, po) It should be remarked that N (p, q, qo) is defined only up to an additive real constant c (q, qo). However, the combination (4.2. 3 1) is a uniquely

defined real-valued function on g-. We observe that the expressions az{N(p, q, qo)-N(po, q, qo)} apaq

= aaN(p, q, qo) , apaq

and

a2G(p, q)

apaq

are analytic on a (apart from the obvious singularities). From (4.2.1)

and (4.2.7) we have a2G (p, q) apaq

(4 . 2 . 12)

_

- 2 apaq

a2N(p , q, qo )

(4.2.13)

1 a2QQQ (p)

apaq

=

1 a2.0

'

(p)

2 'apaq

'

Now the difference (4.2.14)

A (p'

q)

a2G(p, q) = apaq

a2N(p, q, q0) apaq

=

1

2

azQQa(p)

{ apaq - apaq

is an everywhere finite bilinear differential on the double

.

By


96

[CHAP. IV

(4.1.2) A(p, q) is independent of qa and in its dependence on p, is a

differential of the first kind: C

(4.2.15)

=

A16 q)

where dZl, dZ2

u=1

yrA (a)

dZk(p dp

)

, dZG are a complex basis for the differentials

of the first kind on

. Since X(p, q) = A(q, p) (more properly, A(p, q)dzdC = A(q, p)dCdz), we see that

(4.2.16)

Z yu(q)

da

=

)

El y;u(p)

d

Let pi, p2, , Pr, be points of T1 which are chosen such that the determinant dZZ,(pk

dpk

does not vanish. This choice is obviously possible because of the linear independence of the dZ, (see [17]). Solving the system of equations Ir yp(q)

as1

dZ,,(pk) dpk

G

= µ-1 E y,.(pk)

dZ"(q)

k

= 1, 2, ..., G,

dq

for the y,, (q) we obtain (introducing a factor r/2 for later convenience) (4.2.17)

y,(q) =

dZ,,(q)

Ec

2 _1 " dq

Substituting from (4.2.17) into (4.2.15) we find that n E cµr dZ,(p) dZ,(q) (4.2.18)

A(p, q) =

2/1',_1

dp

dq

Since A(p, q) = A(q, p), we have (4.2.19)

c,,, = C' M.

In view of formula (4.2.13) the quantity a2N(p, q, qo)

apaq does not depend on the point qo. Therefore we may replace qo by qo

in this term, and then by (4.2.6) and (4.2.8), we have


§ 4 2]

a2G(p, q)

a2G(p, q)

apaq '

apaq

a2N(p, q, qo) apaq

-

Thus a2G(p, q)

apaq

_

(4.2.20)

, 211,,-,

c

-

97

a2N(p, g, qo)

apaq

a2N (p, q, q0)

--

apaq

ydZm (p) dZ,(q) dp

dq

Since G and N are real-valued functions, we have

2(p, q) = (2(q, p))-

and, hence, (4.2.21)

c,,,,= CIA = c,,,,

by (4.2.19). Therefore the c,,,, are real. By (4.2.14), (4.2.18) and (4.2.20):

(4.2.22)

l

a2G(p, q) apaq

= a2N(p, q, q0) +?r E

a2G(p, q)

_

apaq

dZ,(q) dp

2

a

dq

dZ,,(p) dZ,(q) dq apaq apaq 2 F dp The formulas (4.2.22) show how Green's and Neumann's functions are related, and that if one of these fundamental functions is known, a2N(p, q, qo)

c

,,,-j

the other can easily be computed with the aid of the integrals of the first kind. Thus the boundary value problems of the first and second kinds for the Laplace equation appear as equivalent problems. This fact is a consequence of the Cauchy-Riemann differential equations, in virtue of which the first boundary value problem for a harmonic function is equivalent to the second problem for its harmonic conjugate. The Green's function G (p, q) of an orientable surface with boundary is symmetric with respect to its two arguments and is skew-symmetric

in its dependence on conjugate points of the double. It thus has two symmetries. In the case of a closed orientable surface one of these symmetries is lacking, but the symmetry which arises from the law

of interchange of argument and parameter is still present. Thus, although a Green's function in the strict sense does not exist, a


98

[CHAP IV

function can be defined which incorporates the symmetry in argument

and parameter and this function is the analogue of the Green's function. In fact, if 1t is closed and orientable, let po and qo be fixed points of 9)2 and define (4.2.23)

V (P, po; q, qo) = Re {SZQQO (p) - Qaco (po)}.

By the law of interchange of argument and parameter (4.2.24)

V (P, P ; q, qo) = V (q, qo; p, P0)

Moreover, we clearly have V (P, p0, q, qo) = - V (PO, P; q, qo), V (P, po; q, qo) = - V (P, po; qo, q) The symmetry properties of the function V (P, PO; q, qo) are therefore

(4.2.24)'

the same as those of the Riemann curvature tensor. We shall find it convenient for purposes of analogy to extend the definition of V if one or more of its arguments lie on the conjugate surface 01. We set (4.2.24)"

V (p, po; q, qo) = - V (p, po; q, qo);

similar relations for the other arguments are implied by (4.2.24) and (4.2.24)'. In view of the formal similarity of V(p, po; q, qo) to a Green's function, we shall call it the Green's function of the closed orientable surface 9)'t. Moreover, whenever the dependence on the parameter points po, qo is not involved, we shall write (4.2.25)

G(p, q) = V (p, po; q, qo)

with the understanding that when p and q are interchanged, so are PO and qo. Thus (4.2.26)

G(p, q) = G(q, p)-

By (4.2.24)" we have also (4.2.26)' G(p, q) = - G(p, q). We must bear in mind the singular behavior of the Green's function for p near qo, q near po. In this connection, the following electrical analogy is useful. The double ty of 932 is disconnected and consists

of two components 9X and !0. Let small holes be opened at the

§ 4 21


99

points q0 of I)1 and 40 of S, and let these two holes be joined by a thin tube. Imagine that an electric charge of magnitude + 1 is placed at the point q, an electric charge of magnitude -1 at q. Since the lines of force issuing from q must all enter the tube, the mouth of the tube at q0 behaves like a negative charge. If G* (p, q) is the Green's function of the surface S))l* obtained from 2J by opening a hole at q0, we shall

show in Section 7.4 that (4.2 27) G* (p, q) -G* (p, qz) - . V (p, po; q, q1) -V (qo, po; q, q1)

as the holes shrink to the point q0. In formula (4.2.27), qi is an arbitrary parameter point. Thus the choice of the function V (P, po; q, qo) as a Green's function is justified both on physical and

on mathematical grounds In the case of a closed orientable surface (m + c = 0), we define

the Neumann's function to be the Green's function when both points p, q he in 1) or FA, and to be the negative of the Green's function when p lies in V, q in !1l or p in !FA, q in 9R. Thus we define (4.2.28)

N(p, q, q0) = V (P, po; q, q0),

and we impose the symmetry condition (4.2.29)

N(p,q,go) =N(p,q,g0)

It remains to define a Green's function for non-orientable surfaces. Since a non-orientable surface with or without boundary has a connected double which is closed and orientable, the Green's function

of a non-orientable finite surface may be defined by the formula (4.2.1). However, on a non-orientable surface Tt there are closed paths lying in its interior such that each of the two overlying paths on the double t joins conjugate points, and this means that the Green's function defined by (4.2.1) reverses its sign on such closed paths (on which the orientation is reversed). On the other hand, the Neumann's function defined by (4.2.7) is single-valued. It is in some instances desirable therefore to define a single-valued Green's function G8 (p, q) for non-orientable surfaces. In the case of

a closed non-orientable surface, there is no single-valued Green's function G,(p, q) with pole only at q. For such a function, G,(p, q) = = G,(p, q) should hold, but then G,(p, q) would have two poles, each with residue + 1 on the closed orientable double. This is im-

100


[CHAP. IV

possible. If the single-valued Green's function has poles at q and q0, it must coincide with the Neumann's function (4.2.7). We therefore

take this function to be the single-valued Green's function of a closed non-orientable surface.

If the non-orientable surface 931 has a boundary, let 9t be its two-sheeted 'orientable covering (see Section 2.2). The genus of T is 2h + c - 1, and the double of 93'1 is formed by identifying the two boundary points of 91 lying over each boundary point of R.

Now the number of boundary curves of 9't is 2m, where m is the number of boundary components of R. If, instead of identifying corresponding boundaries of 9`t, we form the double of 91 in the usual way, we obtain a quadruple covering £ of the surface 931, the genus of the quadruple being 4h + 2c + 2m - 3. Let the two points of 9t which lie over a point of 931 be denoted by p, ,, and let their conjugate points on the double of S91 be p, p respectively. We observe that (4.2.30) _ .

The four points of the quadruple Q which lie over a point of 9)'t therefore are In terms of the quadruple, we define a single-valued Green's function G,(p, q) by the formula G,(p, q) = 4 a3(P) + +Qaff(() + 11 Here the .Q's denote integrals of the third kind on the quadruple whose periods are pure imaginary. We observe that (4.2.31)

(4.2.32)

q) = G,(p, q), G,(j5, q) = -G,(p, q), G,(j, q) = G,(q, P) Let Gi be the Green's function for 91. Since conjugate points of

the double of 9t are q, q, we have (4.2.33)

G, (P, q) = i {D.1 (P)

Because of the symmetry, (4.2.34) G,

q)=G, (p, j), G1(',

q), G1( ,q) _ -G1( ,q)

y 4.3]

DIFFERENTIALS OF THE FIRST KIND

101

It is clear that {G1(p, q)

G3(5, q)

(4.2.35) -=

q)

4)}

G1(P, q) + G1(p, q)

We observe that the Green's function (4.2.1) for Dl has the form (4.2.36)

G(p, q) = GI (p, q) - G1(P, 4 ). For the difference of the left and right sides is a regular harmonic function of P (for fixed q) which vanishes on the boundary of V. We define the Neumann's function of a non-orientable surface ,)2 to be the function (4.2.7). Because of its symmetry property, it is single-valued on V. 4.3. DIFFERENTIALS OF THE FIRST KIND DEFINED IN TERMS OF THE GREEN'S FUNCTION

In Chapter 3 we defined basis differentials dZ for a closed orientable

surface by interpolating vortices along the cycles of a homology basis. That is to say, we formed differentials with poles of residues + 1, - 1 at the vertices of 1-simplexes. By adding the differentials belonging to the 1-simplexes of a 1-cycle, we obtained a differential

of the first kind. If, in the case of an orientable surface with boundary, this interpolation along a cycle a is carried out using differentials from the Green's function, the symmetry of the Green's function automatically makes a corresponding interpolation along the conjugate cycle a of

the double. Thus interpolation with the Green's function yields differentials which take conjugate values at conjugate places and therefore belong to the surface. This interpolation will be carried out along each cycle of a homology basis, which consists of G cycles

- , Kr, G = 2h +,m - 1. For a closed orientable surface, the differentials obtained in this way agree with the differentials dZ, defined in Chapter 3. Non-orientable surfaces differ from orientable ones in that they are not imbedded in their doubles, but the differentials of such surfaces still take conjugate values at conjugate places of the doubles. In terms of the surfaces themselves, we require (see Section 2.2)

K1, -


102

[CHAP. IV

that the differentials are invariant under direct conformal transformations of the local parameter, and go over into conjugate complex values under an indirect conformal transformation. This requirement forces us to interpolate using differentials formed by means of the Green's function (4.2.33). The differentials of the first kind belonging to a non-orientable surface are defined only on the Betti group which

has G generators, G = 2h + c + m - 1. If m = 0 there is one torsion coefficient of value 2, while if m > 0 there is no torsion coefficient (the cycle which, taken with multiplicity 2, bounds on the closed surface is now homologous to the sum of the m boundaries).

We thus interpolate along the G cycles K1, , KG of a Betti basis using the Green's function (4.2.33), the K,, forming closed curves on the double. In all cases we therefore definer

Z,,(q) = - 2 aJ

(4.3.1)

K

a$G(p,

q)

apaq

d

or

(4.3.1)' Im Z,, (q)

1

q) ds9

= 2= Kf µ

_ 2a '

an

f

aGap q) dp

2n K K

where afan denotes differentiation with respect to the normal which

points to the left with respect to the oriented cycle K.. Here we use the fact that (4.3.2)

aGap q) dp

= 2 dG(p, q) +2i

an9

q) ds,.

Let T (p, q) denote the analytic function of p whose real part is the

Green's function G(p, q). For future reference we note that, by (4.3.1)', (4.3.3)

P(dT, KO) _

JdT = - 22ci Im Z,.(q). Kµ

Also, in the case of a non-orientable surface, by (4.2.34), (4.2.35), and (4.3.1), (4.3.4)

where

Z,'(q)=Z;,,''(q) +Z,:1Y(q), ImZ,,(q)=ImZal)(q)-ImZ;,1)(4),


§ 4.3]

(4.3.5)

Zj1)'(q)

2

q)

fa

apaq

x,,

103

dp,

q) is the analytic function whose real part is G,(p, q), we have in a similar way If

f

(4.3.6) P(dT K,,)= dT$ = - 2ni {Im Z,1)(q) + Im ZE,1)(q)} x,,

= - 2ni {Im Zµ1} (q) - Im 2;,1) (q)}.

Finally,

(Zµ(q))- _

-

2

f

xJ

2 d f (dG - aG dp) dq ap J x

(azG(P.q`q) dp)

xf

2 f a2G(p, q) dp

x a aq

ac J

a2agq) dp =

Z1', (q).

Thus (4.3.7)

dZ,,(q) = dZ,,(q)

and the differentials dZ, belong to 9 (Section 2.2). We observe that (4.3.8)

Z,.(q) = ZA(q) = - x f

za, q dp

x1.

where K,, is the cycle on the double conjugate to K,,. For orientable surfaces with boundary we choose on J`t a canonical basis K1, K2, , Kr, satisfying the following two conditions: K2h-1 belong to a subdivision S of the double, (i) K1, K3, K2, K41 , K2h belong to the dual subdivision *S, and the remaining cycles K2h+1, , K2,,+,,,-, are each homologous to a boundary component; K2,) =-0 (ii) for 15 ,u, v S 2h we have l (K2,.-,, K2,) = 1, I (,u 0 v), I (K2 1, K2,-1) = 0, I(K,,,, K20 ) = 0. The cycles K2,+1, ,

K2h+,1 will be called boundary cycles. For closed orientable surfaces we saw in Chapter 3 that (4.3.9)

(dw, dZ,) = - P(dw, K,,) = -

f dw x,,

104


[CHAP. IV

for any differential dw of the first kind. This formula carries over at once to orientable surfaces with boundary, and is valid for interior differentials with finite norms (space M). It is sufficient to prove (4.3.9) for interior differentials which are regular in the closure of

2R. If Kµ is not a boundary cycle, Im Z. vanishes on the whole boundary and the argument is similar to that for a closed orientable surface while, if K,, is a boundary cycle, it may be assumed to coin-

cide with a boundary component of the surface. Then Im Z,, is single-valued in the interior of the surface and vanishes on each boundary component except K. where it has the value unity and we have (dw, dZ,,)

J

Im ZZ, - dw = -

Kµ

dw. J

As in Chapter 3, we write (4.3.10)

(dZ,,, dZ,)

J

dZ,, =

K,

Then r = (r,,,)- and (4.3.11)

Im r, = I(K,,, K,).

If 9t is a non-orientable surface, let 91 be its two-sheeted orientable covering which has 2m boundary curves, m 0. A canonical basis K1, K2, - , KG for 9)t will consist of one cycle from each dual pair of cycles belonging to 91, m of these cycles being boundary cycles of 91.

In the case of a non-orientable surface IR the scalar product (dw, dZ,,) is not defined, but (4.3.12)

[dw, dZ,,] = Re {(dw, dZ,,)U}

has a uniquely determined value. We suppose that each differential dw remains invariant under a direct conformal transformation of

parameter and that it goes into the complex conjugate under an indirect conformal transformation. The differentials dw defined in the interior of 9't are differentials in the ordinary sense in the interior of the two-sheeted orientable covering % of 931, but they have the special property that they take conjugate values at conjugate places of 9t. In particular, the space M of interior differentials of 0 is just the

§ 4 4]


105

space of differentials in 9't having finite norms and taking conjugate values at conjugate places of 91. We have (4.3.13)

[dw, dZ,,] = 2 Re{(dw, dZ,,)R} = 2 (dw, dZ,,)91,

*here the subscript SJt denotes that the scalar product is to be extended over 9t. If K,, is not a boundary cycle,

f dw=-Re f dw, K,,

while, if K. is a boundary cycle and dw is regular on the boundary,

then K = K,, and [dw, dZ,,]

2

f (dw + dw)

Re

K

f

dw.

K,,

Thus (4.3.14)

[dw, dZ,,]

Re f dw.

Re{P(dw, K,)}

K

We write [dZ,,, dZ,] = 1',,,.

(4.3.15)

Then I',,, = I',,,, and (4.3.16)

Im 1',,, = I (K,,, K,) = 0.

For simplicity of notation, we shall henceforth write (dw, dZ,) for [dw, dZ ], but we shall understand by the scalar product over a non-orientable surface one-half the scalar product extended over 92 4.4. DIFFERENTIALS OF THE FIRST KIND DEFINED IN TERMS OF THE NEUMANN'S FUNCTION

For reasons of symmetry, we also introduce differentials based on the Neumann's function defined by (4.2.7). We proceed in a manner entirely analogous to that followed in the preceding section

where the definitions were based on the Green's function. We define: 2

Z,'(q) = E- Kf K

'

&11N (P,

apaq

q0)

dp


106

[Cxwp IV

or

Im Z* (q) =

1

2n

(4.4.2)

r aN(p, q, qo) an KJ

= 2i -

d.5,

f'

2

faN(p,q,q) p

dp.

From (4.4.2) we see that Z*'(q) = 0 if K,, is a boundary cycle. Therefore, there are 2h differentials (4.4.1) if the surface is orientable, 2h + c - 1 otherwise. We have WW (P, q, qo) d) apaq J

2J

=-

K,,

_- 2 d

('

dJ

(_a)

K,,

2

2 (' a2N(p, q, qo)

a K,, J

Ja'N(p, q, go)dp

c K,.

a aq

d_-Z'(q)

apaq

The relation

dZ* (q) _ - dZ*(q)

(4.4.3)

shows that idZ* is a differential of T't. We have also (4.4.4)

qO) d p Z ' (q) = Z"(g) = - f aQN 4'

8,,

This formula, together with (4.4.3), shows once more that Z*' (q)

0

for a boundary cycle since, for a boundary cycle, we may assume that K,, = K,,. Let (4.4.6)

2

X

f

as[G(p, q) +IV (p,g, g0)3d

K,,

By (4.3.7), (4.3.8), (4.4.3) and (4.4.4),

(4.4.6) WW(q) =

2 [Z;,(q)-

N(p, q, q0)} dp "(q))= _ f a'[G(p, q)-Oaq fps

and, from symmetry,

4

PERIOD MATRICES

§ 4 51

(4.4.6)' Wj'j (q) =

[Z,(q)-Z (q)] 2

- 11

107

q)a pN(p, q,go)]dp. a2[G(p,

Kµ

Thus

dZµ = dWµ + dWµ

(4.4.7)

and dZ* = dWµ - dW;r .

(4.4.8)

The differentials W' (q) are normalised differentials of the first kind

for the closed orientable double. In fact, the. Green's function V (p, p0; q, q0) of the closed surface, as defined in Section 4.2, satisfies 2 a$V apaq

= a2[G(p, q)+N(p, q, q0)] apaq

Thus the differentials W' (q) given by (4.4.5) are exactly the differen-

tials Z(q) defined by (4.3.1) for the double, on the cycles K. If the surface is closed and orientable, we see from (4.4.6)' and our definitions in Section 4.2. that

dW; - 0.

(4.4.9)

Hence by (4.4.7) and (4.4.8): (4.4.10)

dZ, = dWµ,

(4.4.11)

dZ* = dWµ.

The vanishing of dWp, is an expression of the fact that the double is disconnected. 4.5. PERIOD MATRICES

If TZ is an orientable surface with boundary, let xi,xj, ..,xGbe arbitrary complex numbers. Then by (4.9.10) G

(4.5.1)

unless

G

N( E xxdZp) = Z 1 xµz, > 0 p. -1

p-1 G

(4.5.2)

ZxpdZZ-0. p-1


108

[CHAP. IV

The identity (4.5.2) implies that G

(4.5.3)

F(5)

Z x,, Im

µ-1

depends analytically upon . Since is constant where on each boundary component of TOZ, it is identically constant on Mt. The relation (4.5.3) is then valid by continuation over the double . By computing the increments of (4.5.3) around the cycles Kr, we conclude that x,, = 0, v = 1, 2, , 2h. For v = 2h + i, -, G, we have Im Z. = 2 Im W,, (modulo constants) by (4.4.7) and the same reasoning, applied now to the double a in place of TI, shows that Thus x,=0, G

E r,,,x,.x, > 0

(4.5.4)

f4, -1

= xG = 0. In particular, the matrix I I rKy II unless x1 = xz = is non-singular and the same is true of any principal submatrix , n, 1 s n S G. Hence the dZ" 11, u, v = 1, 2, (u = 1, 2, . , G) are a real basis for the everywhere finite differentials of )1, and a complex basis for the corresponding differentials of the double If 0 is non-orientable, let x1, x2, xG be real. We have II

r.,,,,

G xAdZ,,)

(4.5.5)

N(Z

µ-i

G

=E

>0

p, v+1

unless G

Z xµdZ,, aE 0.

(4.5.6)'

K-1

This last formula implies that G

(4.5.7)

E x,, Im Z.. - constant A-1

on the double, and we again find that x1. = xa =

= xG = 0. In all three cases (closed orientable, orientable with boundary, non-orientable) the matrix II Re F. I I is non-singular, while if m + c > 0 the matrix IIr,,, II is non-singular. If the surfgce is nonorientable, Im r,,, = 0.


1 4.6]

109

4.6. RELATIONS BETWEEN THE GREEN'S AND NEUMANN'S FUNCTIONS

If m + c > 0, the surface has a Green's function (4.2.1) and a Neumann's function (4.2.7) which are connected by the relation (4.6.1)

a1G(r, q) = a2N(P q, q0) apaq

apaq

+ 2 ,.

G

µ,Z14W)Z, q,,

where the c,,, are real coefficients, c,,, = c,,,. This formula was established in the case m > 0, c = 0; but the proof when c > 0 is the same. We now evaluate the coefficients c,,. By (4.4.6) and (4.4.6)' G

1i'o(q) _ _ '

2 .I c,.,P

Ko) Z,(q),

1

(4.6.2)

9 E cP(dZ,,, K,,) Z,(q)

W(q)

N, V-1

Adding the equations (4.6.2) and using (4.4.7), we obtain G Z.,(q)==_ G (4.6.3)

I c,,,Re{P(dZ,,,K0)}Zq(q)= E c,,,Rel'Z;(q).

v, ,=1

Is,

V=1

Since the Z,,(q) are linearly independent, we conclude that G

I c,,, Re 1',,,

That is,

µ-1

(4.6.4)

c,,, II=II Re -P,., Subtracting the equations (4.6.2) and using (4.4.8), we have G

(4.6.6)

Z; '(q) = i Z c Im {P(dZ,,, K,)}Z,(q). P, V-1

The differentials WQ(q) are the differentials ZQ(q) for the closed orientable double, and we therefore have (4.6.6)

Im P(dWQ, K,) = I(KQ, KQ).

By (4.4.7)

Im P(dZ,,, K,) = Im P(dW,, + dW- , K,) (4.6.7)

if c = 0, = I(K K,, -f- %v) _ 1 I K,, K,,), if c > 0.


110

[CHAP IV

Substituting from (4.6.7) into (4.6.5), we see that 2h

2 Z cµ.I (Ka, K3)Zv(q), c = 0, m > 0 (4.6.8)

Z;'(q) =

2h+c-i

i Z c,,v I (K K,,) Zy (q), c > 0. µ.v=1

We observe that Zu*'(q) = 0 for a > 2h(c = 0), a > 2h + c - 1 (c > 0). 4.7. CANONICAL MAPPING FUNCTIONS

If the finite Riemann surface 9)1 is of genus zero, the uniformization principle states that 9)t can be mapped conformally onto a subdomain

of the closed w-plane whose boundary consists of m rectilinear segments parallel to the real w-axis (Hilbert canonical domain). If 9) is of higher genus, a mapping of this sort onto a subdomain of the closed plane (sphere) is impossible for topological reasons. However, in Chapter 2, we pointed out that the Hilbert parallel-slit domain has an analogue for higher genus in which there appear, in addition to the boundary slits, a finite number of other slits where

edges are suitably identified such that the resulting domain has the required genus. Let us recall how such a function is constructed. Let be the double of R and let u,. = uQ, be the single valued harmonic function on t with a dipole singularity at the point q of 9R which, expressed in terms of a particular uniformizer at q, C(q) = 0, is such that the difference (4.7.1)

uQ - Re

1

is regular harmonic in a neighborhood of q and vanishing at the point q itself. Then uQ is unique. Let x Q = uaz be the corresponding

harmonic function with a dipole singularity at the conjugate point q of a which is defined in terms of the uniformizer T at j. Then (4.7.2)

ua(b) = u).

Writing (4.7.3)

'ua (p) = ua(1) + u (A

§ 4.71

CANONICAL MAPPING FUNCTIONS

ill

we see that (4.7.4)

ua fit') = ua ( )

Thus u* is a harmonic function of D of the second kind with a vanishing normal derivative on the boundary of Dt, and the conjugate harmonic function v has a constant value along each boundary, by the Cauchy-Riemann equations. We denote the analytic function whose

real part is at by f,t, and it is clear that (4.7.5)

The periods of df, are pure imaginary. Since dfQ is real on the boundary of STZ, it follows that for the boundary components C,,, (4.7.6) P(df,, C,) = 0, Y= 1, 2, -, m.

Any determination of f, therefore maps C, onto a finite rectilinear segment parallel to the real axis in the plane of f,. Moreover, each choice of uniformizer C leads to a different function with these properties. All possible functions f, can be constructed from the two particular functions f,t and The function (4.7.7)

maps the boundary components C, of 9)t onto slits which are parallel

to the imaginary axis, and it has a single-valued imaginary part. More generally, the function 0 real, defines slits parallel to the direction eie. Let to = e`B(f,t cos 0 - ig,t sin 0), (4.7.8)

esB f, ,:et(p),

t go

= e(g,r cos 0 - if,t sin 0) .

The image of a boundary component C, by f° is a slit parallel to the direction e`° while the corresponding image by g° is parallel to the direction iei8. We observe that fat + gat = to + g°

(4.7.10)

Since

(f,t cos 0 - ig,t sin 0)} is everywhere regular and single-valued on the double Re

, we see


112

[CHAP. IV

from the maximum principle for harmonic functions, that it is a constant. Hence (apart possibly from a constant) (see [7]) (4.7.11)

eisf.,eist - ese(fft cos 6 + fa,tt sin B) = fs.

Now any determination of the function 1

(4.7.12)

2

I tf.t + gut}

maps a boundary component C, of Dt onto a curve which is cut at most twice by a line parallel to the real or the imaginary axis in the image plane and, by (4.7.10), the same property is true of a line having arbitrary inclination. It follows that the image of each C, by the function (4.7.12) is a convex curve without double points. In the case when t is of genus zero it can be shown that the mapping defined by (4.7.12) is schlicht (see [14a], [6a]). If the genus is greater

than zero, the mapping cannot be schlicht on topological grounds.

Let _ + i-, be a uniformizer at q, C(q) = 0. Then W'91

aN(p, q, q1)

(P)

is a single-valued harmonic function of p on the double a. When p is near q, let C be chosen as the local coordinate of p. Then, near q,

= Re (+) + regular terms,

u-.d (p)

and similarly near

If z = x + iy is a boundary uniformizer of

)t,

it is clear that awl.d

_

0

ay

on the boundary. Let Fat - uA(t') + iv-9a (p)

be the analytic function with real part uA (p) . Since the function Re {Fgt(p) --- / (p)} is single-valued and regular on the double B-, it is equal to a constant. Therefore (apart possibly from a constant)

F.t(P) = /.t(6J-In other wands,

§ 4.7)

CANONICAL MAPPING FUNCTIONS

(4.7.13)

Re fat (p)

113

_ - aN(p,a q, qj)

Hence Re fa,;r(p) = aN(p, q, qj)

(4.7.14)

Thus {aN(P. q, qz)

Re far(p) + i Ref4i tr(P) _

a$

(4.7.15)

4i)1 - i aN(p,,q, an !

_-2aN(p,q,qj)_-2aN(p,q,qj) ac aq

(4.7.15)'

Re far(p) - i Re

,a,,c(p)=-2

aN a, q, qx) q

or

2aN(p,q,qj)

Ref() +iImgar(p)

(4.7.16)

(4.7.16)'

Re far(p)

-

i Im gar(f')

aq

_ - 2 aN(p,aqq, qj)

Let z = = x + iy be a uniformizer at p. Differentiating (4.7.16) and (4.7.16)' with respect to fi and observing that

td/

ax (4.7.17)

(p), dz

a Im

gar(p)=Imdgac(p)

ax

dz

far(p)=-ImdfQa(p),

ay Re

y

Im gar(p)

= Re

we obtain (4.7.18)

dfar(p) dp

+ agac(p)

dlar(p)

dgar(p)

dp

dp

ap

4

a2N(p, q, q:) apaq a2N(p,

q, q1)

apaq

Formulas (4.7.13), (4.7.14), (4.7.18), (4.7.18)' show the close relation between the canonical mapping functions and the Neumann's function. This relation may be explained by the fact that the canonical functions have constant imaginary part on the boundary coEn-


114

[CHAP. IV

ponents of SJ7t, and that their real parts have therefore vanishing normal derivative. 4.8. CLASSES OF DIFFERENTIALS

Let K,,, 1A = 1, 2, - - -, G, be a canonical basis for the 1-cycles as described in Section 4.3, and let M be the Hilbert space of differentials

d f which are regular at each interior point of DI and have finite norms over 0. If TZ is non-orientable, the differentials are further required to assume conjugate values under an indirect conformal change of uniformizer. Let F(i1, iE, be the subclass of M composed of differentials d f for which

,

(4.8.1)

u=

(df,dziI)=0,

nSG.

The smallest of these subclasses is F(1, 2, - , G), and we denote it

by the letter S. We begin with the following lemma: LEMMA 4.8.1. If any differential of the first kind of the double belongs to S, then it is identically zero. PROOF: The lemma is immediate if 9Xt is closed and orientable. If it is not, let dw1, dwa, -, dw1 be any basis for the everywhere

finite differentials of 3 (differentials of the first kind). Since dw1, dw., - - , dwG are linearly independent, we can orthonormalize

them over Dt by the Gram-Schmidt process and we may therefore assume that (4.8.2) (dw,,, dw,) = d,,,, u, v = 1, 2, -, G. Every differential dw of the first kind belonging to i3 has the form G

dw = E c,,dw,,. I

1

If this differential has vanishing periods, then (4.8.3) µ= 1, 2, - - , G. (dw,dZ,,) = 0, Since each dwQ is a linear combination G

dwQ=EyQ,dZ,,

it follows that

-1

CLASSES OF DIFFERENTIALS

g 4.8]

cQ=(dw,dwQ)=0,

115

e=

Now let G

dw = E c,,dZ,,

(4.8.4)

.U-1

belong to F(i1, is,

,

Then Cr

Cr

(4.8.5) (dw, dZiy) = - E cf,(dZ,,, dZ{y) = X cµrµsY=0 µ-1

,

v=

1, 2,

., ra.

Thus (4.s.6)

Eci1' /s s

E'crQ t,y Q

where the sum on the right runs over the set of integers which are complementary to 11, i2, , i,,. If 11 r I) (y, v = 1, 2, -, n) is non-singular, the equations (4.8.6) always have a non-trivial solution (c{l, , c;') no matter how the values cQ are prescribed, and in this case the dimension of the space of differentials of t which belong is precisely G - n. Thus if m + c > 0, in which to F (il, 12, is non-singular, the basis for the differentials of t of case I I r{p;r I I the first kind in F(i1, has the form

,

,

dw,., dwy,

(4.8.7)

The differentials dZ, , µ = 1,

,

dwr

,,.

, n form the orthogonal comple-

ment of the differentials (4.8.7). In general, any differential d t of g- of the first kind can be represent-

ed in the form df = dw -}- dW

(4.8.8)

- , and dW belongs to the orthogonal complement, that is (dw, dW) = 0. This follows by orthogonal projection, which in this case is trivial since the spaces have finite dimenwhere dw e F (il, i$,

sion.

If (4.8.9)

l(K,Kjy) = 0,

we say that the class F (ii, i2,

1u, v = 1, 2,

,

., n,

is symmetric. We see by

(4.3.10) and (4.3.11) that for a ,symmetric class, the matrix I f r{,,,y

is real, symmetric and non-singular.


116

[CHAP. IV

The period conditions imposed on the class F(11, i2, (4.8.10)

P(dg, K='`) = 0,

if c= 0

Re {P(dg, Ki..)} = 0,

if c > 0

(IL

, i,4) are

= 1, 2, ,n ).

Suppose that m + c > 0 and let dg be a meromorphic differential satisfying the period conditions (4.8.10). Write G---n

d t = codg + E cdwo. 'Uai

The condition that the periods of dl vanish on the G - n cycles K,4,

p 0 ii, i2,

, i,,, give G - n conditions in G - n + 1 unknowns

c,4, and therefore these equations have a non-trivial solution with co 0 0. It follows that G---n

(4.8.11)

dg = df + E b,4dw,4 µe1

where

if c=0

JP(dfK) = 0,

I Re {P(df, K,4)} = 0, if c > 0 A meromorphic differential satisfying the conditions (4.8.12) will be

said to be single-valued on 9t. On an orientable surface 932 we may also define the class of differentials dg satisfying (4.8.13)

Re {P(dg, K{,)} = 0,

,u = 1, 2,

, n.

, i,4). By the argument given above, we see that there are precisely G - n real linearly independent basis differentials (4.8.7) in G. The class F(11, i8, , i!4) on a non-orientable surface 9X may now This class will be denoted by G(11, is,

be interpreted as follows. Let % be the two-sheeted orientable covering of fit, and let G(11, i2, , i.) be the class G on 92. Given any differential dg of G, let dg(p) = (dg($))be its conjugate. The elass F is the subclass of G composed of differentials dg which satisfy dg = dg.

§ 4.9]

BILINEAR DIFFERENTIALS FOR THE CLASS F

117

4.9. THE BILINEAR DIFFERENTIALS FOR THE CLASS F

In the orientable case an expression LF(p, q) is called a reproducing

kernel of a class F of differentials dl on D1 if: (1) For each fixed q in the interior of X11, LF (p,q) as a differential

of p belongs to F. (2) LF(p, q)dzdZ is invariant; that is, LF(p, q) is a bilinear differential.

(3) For each df a F, LF(p,q) has the reproducing property (4.9.1)

(df,

dl (q).

A reproducing kernel is unique. For let L*F be another. Then N(LF - LF*) = (LF - LF, LF) - (LF - LF, LF) = 0. Hence the difference LF -LF is identically zero. We remark that property (2) follows from (1) and (3). In fact, by (4.9.1) we have (4.9.2)

(LF(p, f'), LF(p, q)) _ - LF(q, F) and this relation implies (4.9.3)

LF(q, P = (LF(r, q))

We shall see that the reproducing kernel LF(p, q") exists for all classes F, but it is clear from (4.9.1) that LF(p, q) is different from zero if and only if the class F contains elements dl which are not identically zero. Heretofore the kernel has been defined mostly for plane domains, and it has been customary to call the Hermitian bilinear differential - L (p, q )dzd the "kernel function." From the reproducing property of the kernel LF (p, q) belonging to an orientable domain I, we may derive an important formula for representing L F (p,q ) in terms of any complete orthonormal system for the class F. Since LF(p, "q)dzdT, as a differential

in p, belongs to F, we may develop it in the form (4.9.4)

LF(p, 'q)dxdT = .E a.(q)dCd .(p). 'n-1

Here the Fourier coefficient a (q )d is given by (4.9.5)

an(q)d

(LFC-b, q)d--d, d9 (p)) = (dgPn(p), LF(p,

(dT.(q))-

118


LCHAP. IV

by virtue of the reproducing property (4.9.1) of the kernel L. Introducing (4.9.5) into (4.9.4) we obtain (4.9.6)

0 E

LF(p, 4)dzdC

n-1

Thus, LF (p, 4) is indeed the kernel of the orthonormal system of differentials The explicit formula (4.9.6) clarifies the significance of the reproducing property of LF and gives, on the other

hand, a convenient tool for constructing the kernel LF in terms of

any complete orthonormal system in F. In the case of a closed Riemann surface, the space F has finite dimension while, if the surface has a boundary, there are infinitely many independent interior differentials, and F has infinite dimension. When 5771 has a boundary, it turns out that, for fixed q in 5B1, the bilinear differential LF(p, q) is regular everywhere on except at

p = q where it has a double pole: regular terms. LF(p, q) = n [x(p) x(q)]2 + Thus LF(p, q) is a singular bilinear differential on 5371, with a double pole at p = q. Because of the singularity, LF(p, q) is not an element of F although it satisfies the period relations for F. Further, for each dl e F, we have (4.9.7) (df, LF(p, q)) = 0, that is, LF(p, q) is orthogonal to the elements of F. These properties determine LF(p, q) uniquely, for if LF is another, then L,, -- LP*. is an

1

element of F and we again have N(LF - Ls) _ (LF - LF, LF) - (LF - - LF, LF) 0. We shall take the singular bilinear differential LF(p, q) as fundamental, rather than LF(p, q). When the double g of 1Y is connected

they are, of course, the same and the choice of the factor 1/= in normalizing the singularity is dictated by the form of the reproducing property for LF(p, q) In the case of the singular bilinear differential, the scalar product in the preceding statements has not yet been defined so we begin by defining a scalar product valid for differentials df, dg whose singu-

larities are included among a finite set of points qf, i = 1,

, n.

BILINEAR DIFFERENTIALS FOR THE CLASS F

§ 4.9]

119

This definition must satisfy the requirement that it reduces to the ordinary one when both differentials are regular. Assume that 9A is orientable. Let V,, be a disc of radius a, in the plane of a local uniformizer at q,, with center qt, and let J't' denote the union of the 2Q,, i = 1, , n. We define

(df, dg) = lim f f'(p) (g'(fi)) dA (4.9.8)

TI-+rt'

= lim (dl, dg)m_u, n

as E a, approaches zero, provided that this limit exists. It is easily :=1

seen that the extended definition of the scalar product retains the properties (2) (a) - (e) of Section 2.3. We shall also interpret the integral

f f'(p) (g'(p))-dA

a

as a limit according to the above definition whenever the integrand is singular. The singular differentials dg to be considered below are regular on

t except for a double pole at a point q of V. In particular, dg is C(q) = 0.

without residue at q. Let C be a uniformizer at q,

Then, except for a constant factor,

g' =

1

+ regular terms

< b, where b is sufficiently small and near q. Let A be the disc I fixed, while 9)1' is given by I C I < a < b. We denote the boundaries of these domains by Co and C' respectively. Then the integral of dg is defined up to a constant and single-valued in fit,-9l', and we may verify that the scalar product (4.9.8) is defined when df is regular in

fit. In fact, lim

f /'(p) (g'(p)) dA = f /'(p) (g'(p)) dA + lim f f'(p) (g'(p)) dA = f f'(p) (g'(p)) dA + U-0,

2i

f gdf -lim Ce

2i

f gdf C'


120

[CHAP IV

using (1.5.24), and it is clear that

f gdf = f (g(0))`/' (P)dC = o (1) C'

C'

as a tends to zero, so the limit exists. We note that the additive constant in g does not affect this result since dl has vanishing period around Co and C' if b is sufficiently small. Thus the scalar product in (4.9.7) is defined. Analogously, we may verify that (dl, dg) is defined when dl has a singularity of the above type at a point r distinct from q. In particular (4.9.9)

(LF (P, q), LF (P, r) ),

r

q,

is defined by (4.9.8). The value of this product will be given in (4.12.3).

On the other hand, N (dg) _ (dg, dg) is not defined by (4.9.8) and a further extension is necessary. This extension will be based on the limit of the product (4.9.9) as r tends to q (see Section 4.12). In extending the definition of scalar product in the case of nonorientable surfaces 1)1, we note that the differentials df, dg take conjugate values at conjugate places q, q of Jl, so we may choose conjugate regions 9't' and §Z' containing the singular points q,, q{. Then we define

(df, dg) = lim 2J I f'(p) (g'(P))-dA (4.9.8)'

9t-'-fit' =lim 2 (dl, dg)%_

,

,,

The analogue of L,, (P, q) in the case of a non-orientable surface X71

is the unique kernel L. ,(p, q, q) satisfying (4.9.1)'

(df, L,) = - Re {f' (q)}

for every df e F. The singular bilinear differential LF(p, q, q) with singularities at P = q and p = q is again orthogonal to all d t e F: (4.9.7)'

(dl, LF(P, q, q)) = 0.

4.10]

BILINEAR DIFFERENTIAL FOR THE CLASS M

121

4.10. CONSTRUCTION OF THE BILINEAR DIFFERENTIAL FOR THE CLASS M IN TERMS OF THE GREEN'S FUNCTION

We shall construct the bilinear differential Lg of the class F in terms of the Green's function and the differentials of the first kind, and we begin with the case in which F = M. We suppose first that JJ is orientable and has a boundary. We set 2 a2G (p, q)

LM (p, q) _ - n

(4.10.1)

1 aQS2ac (p)

v

apaq

apaq

Then by (4.2.6) we have (4 10.2) LM

f) _ 2 a2G(p, q) ---(p,q)=--X2 a2G(p, a apa apaq

1 a'Q (p) apaq

We have to show that LM(p, q) is orthogonal to the differentials df of M and that LM(p, q) reproduces them.

When p and q lie in the same neighborhood, let both of these points be expressed by the same uniformizing variable. If C(q) = 0, we have by (4.10.1) (4.10.3)

+ regular terms.

LM (0, q) _ R

< a,

Hence, if fit' is the disc (4.10.4)

(df, Lm (P, q))

= If f f'(p)(LM(p, q)) dfdz + o (1) M-W

as the radius a tends to zero. Now assume that d f is regular analytic on the boundary C of P. Since aG(p, q)laq" is single-valued on SR and equal to zero for p on the boundary of !R, integration by parts gives

I (4.10.5)

J a-SRI

f'(p)(LM(p,

q))"dtdz = - -1. fr() m-+R' fl(P)

Since

aq

q) dC

a2G(p,

q) dzdx


122

aG(p, q) aq

(4.10.6)

2C

[CHAP. IV

+ regular terms

near q, we have (4.10.7)

f f'(p) aG

, q)

dc =

..

f f'(p)

c

a

+ o(1) = o(1).

Hence the orthogonality property (4.10.8)

df a M,

(df, LM (p, q)) = 0,

is proved subject to the restriction that / is regular on the boundary C of an.

If dl is regular on the boundary, the reproducing property of LM(p, q) is proved in an analogous fashion. In fact, since LM(p, q"")

is everywhere regular in fi't, we have

= 2i f f(p) (Lm(p, q))-dzdz + 0(1).

(4.10.4)' (dl, LM(p, q))

lot-W

Integrating by parts, we obtain

f

2i f f,(p)(LM(p, q))-dzdz = (4.10.5)'

,

aG C,

/I(P)

q)

a2 Gagq)

dzdz

dC

q

Near q we have instead of (4.10.6) (4.10.6)'

aGd

,

q)

= 1 + regular terms,

q so

(4.10.7)'

f

&i'

aG(p, q)

dC = f(q)

Letting a tend to zero, we obtain the reproducing formula (4.10.8)'

(df, Lm (p, q)) = -f'(q) subject to the restriction that f' is regular on the boundary C of R.

§ 410)


123

We now remove this restriction. Given the point q interior to V, we construct a sequence of finite Riemann surfaces 3tµ satisfying the following conditions: ' (1) 9)1k together with its boundary lies in the interior of $.)t.

(2) Each 9't,, contains the point q in its interior. (3) 9X,, C ,+1 and, given any point p interior to 9R, there is a positive integer go = ,ua(p) such that p e 9)2,, for ,u z go. (4) At each point of 9't,, the uniformizers are admissible uniformizers of 9)1.

The sequence Et is easily constructed using the level lines of the Green's function G(p, q) where q is the given interior point of V.

In fact, we define 9t,, to be the subdomain of points p of TZ for which

G(p, q) z

e

where s is a positive number. If a is small enough, then for each positive integer ,u the level curves G (P, q) = --/,u consist of precisely

m analytic Jordan curves each of which is homotopic to precisely one boundary curve of 9)t, and as u tends to infinity these curves approach the boundary curves of Ft. Let p be a boundary point of X12,,, z a uniformizer of J)1 at p. In the plane of z a subarc of the boundary of P. through p appears as an analytic arc bounding a half-neighborhood of points belonging to 1.. By the Riemann mapping theorem the half-neighborhood can be mapped onto a subdomain of the upper half-plane of a variable t such that the analytic boundary arc goes into a segment of the real t-axis. It is then clear

that t is a boundary uniformizer of 9t,,' at p, and that t is also a uniformizer of 9)t at p. We shall prove the validity of (4.10.8) for a general dl e M. The corresponding proof for (4.10.8)' is similar. Let 9R' be a fixed uniformizer disc I C I < a at q with boundary C. Let 9)Zo be a compact subdomain lying in the interior of 9)1 and containing 9)1' in its interior.

There is a 1u1 such that 12o is a compact subdomain of P.,, for ,u Z µl. In Tto - ' the Green's function G. (P, q) of V2,,, together with its derivatives, converges uniformly to the Green's function G(p, q) of 212 and its corresponding derivatives. It follows that the regular sequence IL,( j6, q) - L (p, q)} converges uniformly to zero in

BILINE4R DIFFERENTIALS

124

(CHAP. IV

Wt. and that {LM) (p, q) aG0 (p,q)laq} converges uniformly on C'. Further (4.10.9)

(d/, LM) (p, q))vtµ = 0

by (4.10.8), since df is regular on the boundary of D1.,. Then (4.10.10)

(df, LM(p, q)) = (df, LM(p, q))vt-v,, (df, LMT (p, q))v:µ vt, -i- (df, LM( , 4) -LMT (p,

By the Schwarz inequality (

4 10.11 )

Jf'()(L(, 4)) dA

2

Wt-a'

SO-WO

DO-IRO

2dA.

(f

, q)

M

Here f I LM) (p, q)I2dA

S fL)(P,q)J2dA = M"-M'

W"-"O

f L ) (P, q) aG"a

q4

C'

where K is independent of u and R0, by the uniform convergence on C'. Hence 2

(4.10.12)

(L(") (P, q)) -dA

if

SK

I f'(p) 12dA.

J

Analogously, (4.10.13)

I

f f'(p) (L. (p, q))-dA a-tee

Dt'-'9Ro

where

k=51LMP q) ZdA. Letting u tend to infinity, we obtain from (4.10.10) (4.10.14)

(df,LM(P,q))J'

K' 'f l /I (p) I2dA,

where K' is independent of. Letting (4.10.8) for any df of class M on U.

approach fit, we obtain

We have at once from (4.3.1), (4.10.1) (4.10.15)

P(LM(p, q), K,,) = Z,,(q)-

§ 4101


125

This relation remains valid if we replace q by q. We remark that LM(p, q) is symmetric in p and q and that LM(p, q) satisfies the law of Hermitian symmetry. If T is closed and orientable, we again define LM (p, q) in terms of the Green's function for 9J, using the first equality of (4.10.1). The appropriate definition for L. (P, q") is then (4.10.2) in view of (4.2.26)'. Since the essential properties of the Green's function are the same in

this case, all the properties of these bilinear differentials follow as before, but without the necessity of considering the behavior of dl on the boundary. I f the surface M is non-orientable, let a2apa,q)

Lax(p,q) =- 2

(4.10.16)

apaq

be the bilinear differential for the orientable covering 91. Then it is

clear that LM{p, q, q) = LM) (p, q) + LM() (p, q) _ (4.10.17)

2

a2Gi(p, q)

-7r I

apaq

a2G1(p, q)

+

apaq

By (4.2.34) LM)(fi, q) = (LM) (p, q))-

and hence (4.10.18)

LM(fi, q, q) = (LM(p, q, q))-.

Thus LM(p, q, q) is a differential on 91 in its dependence on p which

takes conjugate values at conjugate places. In its dependence on p the differential (4.10.19)

Lm (p,

Lm(l)

+ Lm") (P. 4)

is everywhere regular on 9t.

It is obvious that (4.10.20)

(dl, LM(p, q, q))5t =

(4.10.21) (df, LM(p, q,

2

2

for every differential of class M.

(d/, LM(p, q, q))% = 0,

(dl, LM(p, q, q))91= - Re{/'(q)}

BILINEAR DI FFERER TIALS

126

[CHAP. IV

By (4.3.5) (4.10.22)

P(Litii (b, q), K,,) = Zµl)'(q)-

Therefore by (4.3.4) (4.10.23)

P(LM(p, q, Q), K,4) = Z,( ,1)'(q) + ZZ1)'(q)

4.11. CONSTRUCTION OF THE BILINEAR DIFFERENTIAL FOR THE CLASS F

Suppose first that the surface is orientable. Let n

(4.11.1)

LF(p, q) = LM(p, q) + Z Yµ,Zi,,(p)Zsy(q) /1'7-1

where the coefficients are to be determined such that e = 1, 2, ..., (4.11.2) P(LF (, q), K{,) = 0, By (4.10.15) and (4.3.10) the period conditions (4.11.2) give Z' (q)

-

n

Z Y4wfiui QZt,(q)

= 0,

q = 1, 2, ... n,

and, because of the linear independence of the Z,, (q), we therefore have (4.11.3)

1, 2,

..., n.

µ-1

Thus LF(p, q) exists if and only if the matrix I I r....Q I I is non-singular,

and then (4.11.4)

11 n,

II=II.r,,,II_1-


Y,, = (Y,,.)+,

which implies Hermitian symmetry for LF(P, q).

If it exists, the expression LF(p, q) has the desired properties. In fact, if dl e F, then (4.11.6)

(df, LF(p, q)) = (df, LF(q, p)) = (df, LM(p, q)) = 0 by (4.10.8) since, by hypothesis,

(df, dZtµ)=0,

µ=1,2,,n.

§ 4.11]

BILINEAR DIFFERENTIAL FOR THE CLASS F

127

Similarly

(df,LF(p,q)) =-/'(q)

(4.11.7)

The bilinear differential LF always exists if there is a boundary, for in that case the matrix I I I'{,,, 11 is non-singular. But by (4.11.5)

we shall have (4.11.8)

LF(p, q) = LF(q, P)

if and only if the class F is symmetric (see Section 4.8). If the Riemann surface is closed, the situation is quite different for I I -r,, , I I may be singular. However, the bilinear differentials for the symmetric classes always exist since I I Re r,,, I I is non-singular. Of the symmetric classes on the closed surface, two deserve to be singled out, namely the classes F (1, 3, . , G-1) and F (2, 4, , G), G = 2h (where his the genus). It is sufficient to consider F (1, 3, G - 1). Since, for fixed q, LF(p, q) is a differential of the first kind whose periods P,, around the cycles K2,,_1 all vanisli, we conclude

,

from (3.2.3) that LF (P, q) = 0. Thus by (4.11.1) (with q replaced by q) h

(4.11.9)

E Yµr Z1µ 1(p) 4-1(q)

LM(P, q)

µ, V"1

where (4.11.10)

I I-i On the other hand, LF(p, q) for fixed q is a differential of the second I

I Yµ

21-1

kind all of whose periods P, vanish, and for p near q we have, provided p and q are expressed in terms of the same local coordinate

x, z = x(p), x(q) = 0: (4.11.11)

LF(p, q) _

+ regular terms.

Thus (4.11.12)

,(p) LF(p, q)

n

and by (4.11.1) (4.11.13) LM(p, q)

dtap) µ.

p

1(p)Z',i.-1(q)

Computing periods in (4.11.9) and (4.11.13) around a cycle K


i28

[CHAP IV

we find using (3.4.3)' and (4.3.10) that h

(4.11.14)

(4.11.15)

_

Z (q)

= 2zzv (q)

Yµvrsµ-i,,4Z,-x(q)

h

ZZ(q) µ, v-i

If the double were connected, formula (4.11.13) could be obtained from (4.11.9) by analytic continuation, and similarly (4.11.16) could be obtained from (4.11.14). The fact that the double is disconnected makes the formulas with q in place of q quite different. Now suppose that the surface l't is non-orientable and set (4.11.16) LF(p, q, q) = LM (p, q, q) + Z y14,Z{µ (p) Re{Z; (q)} µ, V-1

where the real coefficients yo, are to be determined by the conditions (4.11.17)

Re {P(LF(p, q, q), K,Q)} = 0,

e

= 1, 2,

, n.

By (4.3.14) and (4.10.23) n

Re {Z{)' (q) + 411), (q)} - E

Re {Zs, (q)} = 0,

µ, s-i

1,

n.

That is, by (4.3.4), (4.11.18) Re {Z,' (q)} -- E

Re {Z,, (q)} = 0,

µ,V-L

Because of the linear independence, we have + YM.x'{µ{Q = a,,Q, µ=L

v,

= 1, 2, . .., yy,

and therefore, since (I r{,,{Q I I is always non-singular, (4.11.19)

Il Yµ.11=11 r="{,

II-1.

In view of the fact that the bilinear differential for a non-orientable R is a simple linear combination of the differentials for its orientable covering, we shall henceforth assume that the surface $t is orientable

when the contrary is not explicitly stated.

§ 412]

PROPERTIES OF THE BILINEAR DIFFERENTIALS

129

4.12. PROPERTIES OF THE BILINEAR DIFFERENTIALS

If the Riemann surface has a boundary, let p be a boundary point.

Then p = $ and (4.12.1)

LF(P, q)dzdC = LF(p`, q)dzdC = LF(p, q)dzdC.

If q is on the boundary, we have in the same way (4.12.1)'

LF(p, q) dzdC = LF(p, q)dzdg = LF(p, q)dzdC-.

Formulas (4.12.1) and (4.12.1)' are consequences of the fact that LF(p, q)dzdC is invariant to changes of the uniformizer and, at a point of the boundary, either the variable or its complex conjugate can be used. Using conjugate uniformizers at conjugate places, we define 4F(p, q) = (LF((, q))'Since

(LF(p,q)) =LF(q,fi) we have, replacing p by fi, (LF(fi, q)) = LF(q, p);

that is LF(p, q) = L1, (q, P).

If, in particular, the class F is symmetric, then LF(p, q) is the same as LF (p, q), and in this case LF (p, q) is a bilinear differential of Tl

which is real when both p and q are on the boundary. Since the regular differential is an element of F, we have by (4.11.6) and (4.11.7),

(LF(p, F). LF(p, q)) = 0 and

(LF(p, f), LF(p, q)) = - LF(q,

F).

We observe that these integrals are not obtained from each other by replacing q by q, even if Dt has a boundary. The reason is that the integi als are discontinuous because of the singular behavior of the integrands at the boundary. In particular, LF(q, q) z 0. (4.12.2) N(LF(p, q)) We now consider the scalar product of two singular differentials.

_-


130

[CHAP. IV

Let q and r be distinct points in the interior of 9A. We shall prove that (4.12.3)

LF(9, q) = (LF(P, q), LF(p, r)).

The scalar product on the right is to be interpreted in the sense of (4.9.8).

We prove (4.12.3) first for the case in which F = M. Let C be a uniformizer at q, q a uniformizer at r, and let, SJat, be the uniformizer circles I C I S a, I n 1 5 a at q and r respectively. Denoting the boundaries of M, !)l, by CQ, C, respectively, we have 2

_ -!

(LM(p, q),LM(p, r))

=

12

f

aG

, q) aq

a2G (P, q)

apaq

(LM(p, r))-& _! j'

1 1aG(p, q)

(4.12.4)

(LM(p,

ci J

aq

(LM(P,r))-dA.+o(1)

aG(P, q)

(Lm (P,

-dC-

r})-dry -{- 0(1).

C,

Since aG(t, q)/aq vanishes when p is on the boundary C of R, we have only to consider the remaining two integrals. For p on CQ,

8G('

(4.12.6)

aq

q)

= ! + regular terms,

(LM(p,r))-= 60+5id+

(4.12.6)

Therefore aq, q)

1. f

(4.12.7)

(LM(P, r))-dc- = o(1)

Cc

as a tends to zero. Finally (4.12.8)

c

, q (Lm (P,

J aG

C,

Jaca4 q)

a(agg)

+-

C,l

r))-d

+ a2G ragg) + ...} . a (r, q) +o(1)

t

$

_-LM(j,0+0(1)

I

q


§ 4.12]

131

as a approaches zero. Substituting from (4.12.7) and (4.12.8) into (4.12.4), we obtain (4.12.3) in the case F = M. Consider now the general case. Writing (4.12.9)

LF(p, q) = LM(p, q) + AF(p, q)

where AF(p, q) is the bilinear sum of differentials occurring in (4.11.1), we have (LF(p, q), LF(p, r))=(LM(p, q), Lm (p, r)) + (A,, (p, q), AF (p, r)) since the cross terms vanish by (4.11.6). By the case already proved, the first term is simply LM(f, q). For the second, we have (Ay (p, q), AF (p, r)) (4.12.10)

_Z

=

Z Yµ,YiQ2,,(q)Zi,,(Y) . (Z,,,(p),Z (p)) (q) Zta

Z Yo.Zf, (q) Zj', (1)

=AF(r,q). Here we have used the fact that the matrices 1+ y,. II and are inverse. We now consider the definition of a norm for differentials dg which are regular on Tt except for a double pole without residue at a fixed

point q of R. The scalar product in (4.12.3) is not defined by (4.9.8) if r = q, and we now define (4.12.11)

N(LM(p, q))

(LM(p, q), LM(p, r)) = LM(q, q). = hm f -iq

We note that LM(q, q) = LM(q, q") S 0 by (4.12.2). That is, the norm

defined by (4.12.11) is real. However, the extended norm may be negative in the case of singular differentials. For any d/ E M, we have (4.12.12) (df+LM(p, q), df +LM(p r)) = (d/, d/)+(LM(p, q), Lm(p, r)) using (4.10.8), so we define

N(df -f LM(p, q)) = N(df) + N(LM(p, q)). If dg is regular on Ut except at q, where it has the same singularity as Lm (p, q), we may take df = dg -LM(p, q). Then (4.12.13) gives (4.12.13)

(4.12.14)

N(dg) = N(dg --LM(p, q)) + N (LM(p, q))


132

[CHAP. IV

In particular, we have defined (4.12.15) N(LF(P, q)) = N(LF(p, q) - LM(p, q)) + N(LM(p, q)). On the other hand, the identity (4.12.3) implies that we should have (4.12.15)' N(LF(p, q)) = lim (LF(p, q), LF(p, r)) = LF(q, q).

To verify that the result (4.12.15)' agrees with (4.12.15), we note that (4.12.10), in which we may take r = q, may be rewritten as (4.12.16) N (LF - LM) = LF (q, q) -- LM (4, q) = LF (q, q) - N (LM) using the definition of dF in (4.12.9). Similarly, for dl e F we have

(4.12.12)' (dt+LF(p, q), df+LF(p, r)) _ (df, df)+(LF(p, q), LF(p, r)),

so we should have (4.12.14)'

N(dg) = N(dg - LF(p, q)) + N(LF(p, q))

for singular differentials satisfying the period conditions for the class F. Again the definitions are consistent. In fact, for the regular differential dg -Lm (P, q) we have

N(dg-LM) =N(dg-LF+LF--LM) =N(dg-LF) +N(LF-LM) = N(dg - LF) + N(LF) - N(LM) using the orthogonality properties of LF and LM, and (4.12.16). Further, since N(dg - LF) z 0, we derive from (4.12.14)' an important characterization of the singular bilinear differential for a class F: the singular bilinear differential LF(p, q) minimizes the norm

among all differentials dg having the same singularity at q and satisfying the period conditions for the class F. Further, the norms of all normalized singular differentials are bounded from below by LMIq", q), by (4.12.14).

If dl e S is regular on the boundary C of X12, the scalar product on

the left in (4.12.12)', with F = S, may be integrated by parts. If C. and C, are uniformizer circles of radius a about q and r, respectively,

and if Ws(p, q) is the integral of L8(p, q): awe(p, q) ap

=L(pa) s ,


§ 4.123

183

then

(dl + Ls (p, q), df + Ls (p, 7)) 1

2i

, (f(p) + 's(p, q)) (f'(p) +L.(p, r))-dz C

+

2i

f (f(p) + W8(p, q)) (f'(p) +Ls(p, r))'d C,

(4.12.17)

+

2i f(f(P) + Y. (p, q)) (f'(P) +Ls(p, r))-dn+o(1) C, 1

2t

f (f(p) + 71.(p, q)) (f'(p) +L8(p, C

letting a tend to zero. Letting r tend to q and choosing dl = dg - Ls (p, q), we obtain the simple formula N(dg)

(4.12.18)

2i f g dg, c

valid for normalized singular differentials dg which have vanishing

periods and are regular on the boundary C of Ill. The reproducing kernel L, (P, q') also gives the solution of certain

minimum problems. For arbitrary df a F, we have by (4.11.7), (2.3.3), and (4.12.2) (4.12.19)

(d/)2 1 (df,

L. (p, q)) 12 s N(df)N(LF(p, q))

- LF(q, q)N(df) and there is equality in (4.12.19) if and only if f' (p) is proportional to LF (p, q"). Thus the best value for k (PO) in (2.3.16)' is - LM (-ho, fio)

Analogously, the best value when dl is restricted to be an element of a class F is - LF (po, fio)

When L,(p, q) is not identically zero, it can be defined in terms of a minimum problem suggested by (4.12.19). Let dfo be the 0, differential which minimizes N(df) over all df e F with dl (q) normalized in terms of a particular uniformizer at q so that 1'(q) = 1.


134

(CHAP. IV

The minimizing property implies that d10 is orthogonal to dh (p )

df (p) - f'(q) dfo(p), for arbitrary dl a F, since dh(q) = 0. Thus -d/0 (p) /N (d/0) has the reproducing property and must coincide with

LF(p, q)dz. In particular, the minimizing differential dfo may be expressed in the form dfo = LF (p, 4) dp,

(4.12.20)

LF(q, q)

using (4.12.2).

Let F1 and F2 be two classes of differentials on D1 such that (4.12.21)

F1 c F2

and the bilinear differential exists for each class. In particular, LF,(p, q) E F2.

By (4.12.19) and (4.12.2) we have (4.12.22)

0

LF, (q, q) z LF, (q, q) .

Since S is a subclass of every F while M contains every class F, we have 0 Z Ls(q, 4) z LF(q, q) z Lm (q, q) (4.12.22)' q) minimizes the norm among all On the other hand, differentials dg having the same singularity at q and satisfying the period relations for F2. Since LF2(p, q) is among the dg we have (4.12.23)

LF,(q, q) 5 LF,(4, q)

and (4.12.23)'

LM(q,q) S4 (4,q) 5Ls(4,q)

The quantities LF(q, q") and LF(q", q) need not be equal unless F is a symmetric class.

These inequalities may be generalized at once to the Hermitian quadratic forms c

(4.12.24)

.S LF(q, , q,)x,. z,

and

a

E' LF(q,, gµ)xrx;.

For example, we have from (4.12.23)' Q

(4.12.25)

L P. V-1

_

s

Q

e

LF(g,, 1g. -1

s E L8(g q, )x,9,. M. V-1

§ 4.12]


135

Now, by (4.12.2), (4.12.26)

Gi L lgl'l --N(Exa LF( qv)J = µ,s-2 V-1

o.

Thus the first Hermitian form (4.12.24) is non-positive and, if F is a

symmetric class, the came is true of the second. When t is orie4itable and has a boundary, the bilinear differentials for the class S may be given a geometric interpretation, In fact, the integral / of each differential dl e S is single-valued on WI and, unless d f

vanishes identically, gives a mapping of WI onto a Riemann surface

spread over the complex plane. Conversely, each such mapping f corresponds to a differential d t in S. In particular, since such mappings exist, the class S contains elements d f which do not vanish identically and L. (p, q") is non-trivial. For each mapping the internal area of the image domain is given by N (d f ). Let q be a fixed point interior to WI. 0, normalized Then if we consider the set of all mappings with dl (q) in terms of a particular uniformizer at q so that 1'(q) = 1, the mapping

which minimizes the internal area is given by

dl _ Ls(p, q) dp Ls (q, q)

and the value of the minimum area is 1

(4.12.27)

A

Ls (q, q)

The singular differentials dg whose periods vanish also correspond to mapping functions g. If dg = du + idv is regular on the boundary C of 9)1,

N(dg) = __ (4.12.28)

=

2i

1

2

f gq c

4i

fd

c

12

I

g

+ 2 f (udv -- vdu) c

(udv -- vdu) c

since g is single-valued.

Let E, be the image of C, by g, and let E, be oriented such that it is described positively by w = g(p) as p describes C, in the negative

1313


[CHAP. Iv

sense. We define Qy (w0) to be the order of L',, with respect to the point w0. By (4.12.28) (4.12.29)

N(dg)

'

J

Q (wo)dA

where each integration in the sum is over the whole wo-plane. We

therefore say that --- N(dg) is the area external to the image of Jt by dg. In the class of all mappings dg, normalized to have the same singularity at q as Ls(fi, q) 134, the norm is minimized by Ls(j, q). That is, the mapping which gives the maximum external area is given by

dg = Ls (P, q) 134,

and the maximum external area is (4.12.30) Ls(q, q), E using (4.12.15)'. Multiplying (4.12.27) and (4.12.30) we obtain

AE = L8(q, q)

(4.12.31)

La (q, q)

which is a generalization of a known identity for multiply-connected domains of the plane (see [6a], [14a]). In fact, the class S is symmetric whenever the surface D1 is of genus zero, in which case Ls (q, q) = L8(q, q) and AE = 1. We remark that, in the case of genus zero, the function g corresponding to the maximum external area is schlicht

and maps )l onto a subdomain of the sphere. As another application of the bilinear differentials we give a generalization of the Poisson formula which permits the representation of an analytic function inside a circle by means of the boundary values of its real part. Let T1 be an orientable Riemann surface with boundary C and q an interior point of P. If f (p) is single-valued on

9 and regular on C, then we may integrate by parts in the regular scalar product in (4.11.7), with F = S, to obtain (4.12.32)

f'(q) = -" (df, Ls(p, q)) = 2i f f'(p) (Ls(p, q))`dzdz 1

f /(P) L.(q,fi) di. C

APPROXIMATION OF DIFFERENTIALS

§ 4.131

137

' is the disc I C I < a, where C is a uniformizer at q, _ C(p), g(q) = 0, the formula (4.11.6) yields

Similarly, if

0 = (Ls(q, P), df) = 2i f Ls(q, p) (1'(p)) dzdz + o(1) (4.12.33)

=

j'Ls(q , )(f( ))-dx- 2i fL. (q,

(p)) -dC + o (1)

c,

=

2i fLs(q, p)(f(p)) dz,

letting a tend to zero. Since Ls(q, fi)dz = Ls (q, p)dz on C, we obtain by adding (4.12.32) and (4.12.33) (4.12.34)

f'(q) =

f L. (q, p) Re {f(-P)} dx. C

This is a generalization of Poisson's formula, for the case of an arbitrary surface with boundary. 4.13. APPROXIMATION OF DIFFERENTIALS

In the following chapter certain formulas will be established for

differentials regular on the boundary, and their validity for any differential with finite norm will require an approximation argument which we now supply. In Section 4.10 we approximated the bilinear differential by the corresponding differentials of domains fit,, lying in the interior of the given surface. Now, given any differential df of class F, we define a sequence of differentials d t,, of F each of which is

analytic on the boundary of 9) and such that N(df --

tends to

zero.

Given a differential df of class F, let 9)t be a compact subdomain lying in the interior of 9)1 which converges to 0 as u tends to infinity. We do not need to suppose that Define (4.13.1)

,, is a finite Riemann surface.

fµ (q) = - (df, 4 (0,

It is clear that df,,(q) is regular up to and including the boundary


138

(CHAP. IV

of W and that it belongs to F. We have f',

LF(p1, q)(LF(P2, q))-(/'(p1)) f'(p2)dA1dA2, (q)-f'(q) 2= J J D2--IItY 5U--9Rµ

and therefore (4.13.2)

f I (q) -f'(q) 12dA = -

f

got

an-m1A

JLp(p112)(f'(p1))/' (fi2)dA1dA2 2-9 F

since fLF (AL, q)(LI, (P2' q))-dA= J LF(q,

In view of the fact that

-J $LF(P1,fi2)(/'(p1))-f'(p2)dA1dA2= Si f'(p1)j2dA1, SR

Sut 912

we see that the right side of (4.13.2) tends to zero as ,u approaches

infinity; that is (4.13.3)

N(df - dfµ) -; 0.

4.14. A SPECIAL COMPLETE ORTHONORMAL SYSTEM

In the sequel we require a complete orthonormal system of differen-

tials in F with special properties, and we now construct this system by,a method which closely parallels that applied in the case of plane domains ([2a]). Assume that )J is orientable, let q be a point of V, C a uniformizer at q, and let Y. (q) = F,, (q, C) be the subclass of F which is composed of differentials dl which satisfy the conditions (4.14.1)

/()(q)=0, v=1,2,...,n-1; fclt(q)=I

where dCdV1

(q)

'0

Let-us consider first the case that 9 possesses' a boundary. In

§ 4.14]

COMPLETE ORTHONORMAL SYSTEM

139

this case, the class F,, (q) is non-empty for every n 1. In fact, let r,, s = 1, 2, , n, be n arbitrary points on U and consider the differential of the class F (4.14.2)

dT(p) = E x{ LF(p, r,)dp. :=1

The numbers x, can be chosen so that dT belongs to the class provided that the determinant (4 .14.3)

det :::

(L(q,

1

1

11

does not vanish. We want to show that we can choose the r, so that this determinant is not zero. Now, if this determinant vanishes for every choice of the r, F R, we would obviously have (4.14.4)

a aa

E1ak

k

k

T dq} - 0

(LF(q.

where the coefficients ak do not depend on r. Continuing this identity

over into OZ, we would then obtain (4.14.5)

E01ak k (LI, (q

k=

aS

r)

q

- 0.

aS

But this is clearly impossible since for r = q the left-hand side becomes infinite. We have thus proved that the determinant (4.14.3) does not vanish identically for arbitrary choice of the r{ and, hence, we can always construct differentials of the class F and of the form (4.14.2) which belong to

Thus, the class F,,(q) is non-empty for every n z 1. Let d be the and let greatest lower bound of N(df) for differentials of {d/} be a sequence of differentials of F,, (q) for which lim N(df,) = d.

The sequence {df,} plainly converges uniformly in any compact subregion to a differential d f (,a) a F,, (q). It then follows from the usual

reasoning that N (d f ( ,n)) = d. Moreover, if d j is any differential of F satisfying v = 1, 2, ..., n, f(') (q) = 0,

140


[CHAP. IV

then by the minimizing property (4.J4.6)

(df, df c,,,>) = 0.

It follows from (4.14.6) that the minimizing differential dl(,) is unique. Write (4.14.7)

dry =

df,n>

n = 1, 2, .. .

1/N (dfc ,

) Then {d(p,,} is an orthonormal system, and we now show that it is complete.

Let dl be an arbitrary differential of F, and let u = 1, 2, - . a,, = (dl, dT,), From Section 2.3 we know that the Fourier series (4.14.8)

go

dg = Z a,,dryc,

µ-l

is a differential of F. To show that the system is complete, we have

to prove that dl = dg. Let numbers a,() be determined in such a way that the differential ,-1

(4.14.9)

df, = Z a,9) dip,

"I

satisfies the conditions (4.14.9)'

fa ) (4) = f P) (4),

1,

u

$ Z 2.

Since (4.14.10)

faµ) (4) = E a,' 8)q ,) (4), ,-1

9;,A) (q) # 0,

the a,,,'> are uniquely determined. By (4.14.6), (4.14.8) and (4.14.9), (4% 14.11)

(d (f, -1), dry,) = a(-) - a, = 0, v = 1, 2, ...,

By (4.14.9) and (4.14.9)'

,-I

(4) = ft``) (q) -- f' aVl-") (1) = 0,

,u = 1, 2, ..., s -1.

-1

Letting s tend to infinity we obtain, since a uniformly convergent series of analytic functions can be differentiated term-by-term, ft") (q) = gcr) u = 1, 2, . . (q),

§ 4.14]

COMPLETE ORTHONORMAL SYSTEM

141

Thus / - g, and the existence of a complete orthonormal system is established.

In the case of a closed surface fit, we have only finitely many independent differentials and the selection of a complete orthonormal system becomes a problem of elementary algebra and can always be

solved. There are only finitely many classes F (q) which are non-

empty; but in each one we may solve the preceding minimum problem for the norm and construct a normalized differential dppn which is orthogonal to all differentials of the classes F.. (q) with m > n.

By this procedure, we can construct a particular complete orthonormal system of differentials which is useful in various applications. REFERENCES 1. N. ARONSZAJN, ,,La thdorie des noyaux reproduisants et ses applications, I," Proc. Cambridge Phzl. Soc. 39 (1943), 133-153. 2. S. BERGMAN, (a) "Partial differential equations, advanced topics," Brown University, Providence. R.I., 1941 (mimeographed). (b) "Sur les fonctions orthogonales de plusieurs variables complexes avec les applications a la thkone des fonctions analytiques," Mem. des Sci. Math., Vol. 106, Gauthier-Villars, Paris, 1947. (c) The kernel function and eonformal mapping, Math. Surveys, No. 5, Amer. Math. Soc., New York, 1960. (d) Ueber die Entwicklung der

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142

[CHAP. IV

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4 (1938), 473-485. 13. Z. NEHARI, "The kernel function and canonical conformal maps," Duke Math,

Jour., 16 (1949), 165-178. 14. M. SCHIFFER, (a) "The span of multiply connected domains," Duke Math. Jour.,

10 (1943), 209-216. (b) "The kernel function of an orthonormal system," Duke Math. Jour.. 13 (1946), 529-540. (c) "An application of orthonormal functions in the theory of conformal mapping." Amer. Jour. of Math., 70 (1948), 147-156. (d) "Various types of orthogonalization," Duke Math. Jour., 17 (1950), 329-366. 15. M. SCHIFFER and D. C. SPENCER, (a) "The coefficient problem for multiplyconnected domains," Annals of Math., 52 (1950), 362--402. (b) "Lectures on conformal mapping and extremal methods," Princeton University, 19491950 (mimeographed). 16. J. L. WALSH, Interpolation and approximation by rational functions in the com-

plex domain, Colloquium Publications, Vol. 20, Amer. Math. Soc., New York, 1935. 17. H. WEYL, Die Idee der Riemannschen Fldche, Teubner, Berlin, 1923. (Reprint, Chelsea, New York, 1947).

5. Surfaces Imbedded in a Given Surface 5.1. ONE SURFACE IMBEDDED IN ANOTHER

Suppose that t is imbedded in another finite Riemann surface 91. In other words, suppose that 971 is a subdomain of 91. Our purpose is to find identities and inequalities which connect the domain functionals of 9)1 (bilinear differentials, differentials of the first kind and their periods) with the corresponding functionals of R. The orientable covering of a non-orientable surface is an orientable surface with one symmetry. Therefore, as we saw in Chapter 4, the theory of the functionals of a non-orientable surface is merely that of an orientable surface in which the functionals satisfy a symmetry

condition. Moreover, if a non-orientable surface is imbedded in another, then its orientable covering is imbedded in that of the larger surface. Hence the imbedding of non-orientable surfaces can always be reduced to the imbedding of symmetrical orientable surfaces, and we may therefore suppose that the surfaces are orientable. Henceforth, unless it is explicitly stated to the contrary, we assume that all surfaces are orientable. If 9)1 is imbedded in OR, we shall assume unless it is stated to the contrary that the boundary components of 9)t are analytic curves on 91. Then the boundary uniformizers of S))2 are admissible uniformizers at the corresponding points of 91. This hypothesis is made for convenience in proving identities; it can afterwards be removed by a limiting process. We remark that, if 9)1 is closed, 91 is closed and V coincides with OR. We exclude this trivial case; then TZ will always

have a boundary. We shall say that the imbedding of 9)t in 91 is essential if each boundary component of Z which bounds on 91 also bounds on 91, and that it is proper if each boundary point of 971 is an interior point of 91. Let h be the genus of 9T1, m the number of boundaries, and let

G=2h-}-m--1, [] 43]

144

SURFACES IMBEDDED IN A GIVEN SURFACE

[CHAP. V

be a canonical homology basis for R in which K27i+1, - ' , K$n+,n--1 are the boundary cycles, K2. being homologous to the boundary component Ca of 2%. The cycles K1, , Ka,, are homologously in-

dependent on R, but this is not necessarily true of the cycles Ken+v

Ksn+,n-1, some of which may even bound on R. Let

1,2, ...,'rGo

be a canonical basis for 91. Then, on 91, G.

u=

(5.1.1) ,-1

If T (p, q) is the Green's function of T, and if (5.1.2)

q)

22

a2c (p, q)

apaq

then, as in Chapter 4, we have the canonical basis differentials ,,(q) = f2M(. q)dz, u = 1, 2, ..., Go.

(5.1.3)

Let (5.1.4)

PE,,, = (d

µ,

d.T,)st = - P (d °Zo, X,), ,u, v = 1, 2, ..., Go.

Corresponding to the cycles Kw ,u = 1, 2, - , G, we define &',(q) =

J

YM(p, q)dz =

K..

y(q), 1

and we write (5.1.6)

II,,,, = (d@',.,

G.

If dZ1 is the differential of 9)1 corresponding to the cycle K,,, and df is a differential of class M on 91, then G.

(df, dZ11) sot

= '- P (df, K,,)

Z a, P (d f , cf, ) ,-1

EaN,(df, dX,)et= (df, Za,,,d2',), = (df, d',,)e; = ,-1 ,-1

that is, (5.1ij

(df, dZ,,)It = (df, d3",,)lt, ,is = 1, 2, ..., G.

ONE SURFACE IMBEDDED IN ANOTHER

§ 5.11

145

In particular, 17,, = - P(dIW0, K,).

(5.1.8)

Let (d. ,,, dZ,)B= -P(d.1°E K,), u =1, 2,

(5.1.9)

G0; v-1, 2, . , G.

Then, using (5.1.7), Go

(5.1.10)

ep, = T a,QP,Q. Q-1

Also, starting from (5.1.6), we find Go

III _ I a,Q a PQ..

(5.1.11)

Thus the periods of T' and N,, over the cycles K, of 2 can be expressed in terms of the periods of the basis differentials on R and the coefficients of the transformation for the homology basis. Further, by (5.1.1) G.

I(K,,K,) = E

Q, a®1

Now let F. be any class of differentials on St. A clasu Fe of differentials on 0 will be called a corresponding class if each differential of Fst satisfies the period relations of FV. Specifically, if

F

FF(j1,

, jk), so that (df, d.Q)et

= 0, for each df a FF, then F. = F,v(i1, , it) is a corresponding class if df, considered as a differential on Ot, satisfies the period relations (5.1.13)

for F., namely (5.1.14)

(df, dZQ)u =

0, p = i1, ..., ii.

By (5.1.12), a corresponding class F., is symmetric whenever Fat is symmetric. Given a class FR in Yt, let 2' (p, q) be the bilinear differential for the class FM in R, and let L. (p, q) be the bilinear differential on 93t for some corresponding class Fe. We define (6.1.15)

4(p, q) = LF(p, q) -2F(p, q).

The double of 931 is composed of two surfaces 931 and sue, and the

146


(Caar. V

double of R of two surfaces 91 and R. It is convenient to take ft to be the subdomain of & which is conjugate to 1Y on the double of fit. The double of t is then obtained by identifying the boundaries of t z and 9R. With this convention, a point fi of It is uniquely defined and may be regarded either as a point of tt or of {. Meaning is thus attached to the symbols 1F(, q), 4(p, q). In order to simplify the notation, we set (5.1.16)

LF(p, q) = LM(p, q) + BF(p, q)

and (5.1.17)

2F (p, q) _2M(P, q) + RF(p, q)

where G

BF(p, q) _ .I a,,, Z,, (p) Z,(q)

(5.1.16)'

µ,,-1

and (5.1.17)'

-47(p, q) = Z ,,, X,,(p) -T,(q). ,,, ,-1

Comparing these formulas with (4.11.1), we see that a., and vanish if either ,a or v has a value different from 1T z = 1, 2, ,1, or jT, r = 1, 2, , k respectively. Also, by (4.11.5), (5.1.18)

ocµ, =

a,,, =

From (4.11.3) we have G

E aµ, r.= a,Q, e = i1, i2, ..., it,

(5.1.19)

N-1

where r',Q = (dZ,,, dZQ)O, and G,

(5.1.20)

E f,,, P,,e = a,Q, µ-1

Since 2F(p, q")

e = 91, ?2, ...' jk.

is of class Ft on R, it is also of class F. and

we have (5.1.14)'

(2F(p, q), ZQ(p))jR = 0, e = i1, is, ..., ii. Substituting from (5.1.17) and (5.1.17)', this becomes Ga

(5.1.21)

Q(q) = E fl,, e,, X, '(q), e = i1, ..., i=. N. +-1

§ 5.2] SEVERAL SURFACES IMBEDDED IN A GIVEN SURFACE

147

Comparing (5.1.21) with (5.1.5) we see that Ga

(5.1.22)

aQ, = E v'1

e

5.2. SEVERAL SURFACES IMBEDDED IN A GIVEN SURFACE

We now consider the case in which the set T2 imbedded in 91 consists of several components each of which is a finite Riemann surface. Assume, then, that R is the union of a finite number of domains,., v = 1, , k, each of which is a finite Riemann surface imbedded in R. We suppose that no two domains 17t,, have points in common. If the component EDt,, of l has genus h, and m, boundaries, we write

G, = 2h, + m, - 1, and we define the algebraic genus of 9 to be k

G=EG,. ,-i

Let Ki'), ..., KG (T)

and let be a canonical homology basis for Kll), ... Kr(,'), ..., KIk), ..., KGB

be denoted in order by (5.2.1)

K1,

KG.

We call (5.2.1) a canonical basis for R. Let dZF,') be the differential corresponding to the cycle K,(,v) on 9., and define dZ,(,') to be identically zero in SJR - 9R,. Then dZi3L), ..., dZGI ,

.100' dZi?

is a basis for the differentials of ,I which we denote by (5.2.2)

dZ1,

, dZG.

The formulas (5.1.1)-(5.1.12) are clearly valid under these more general circumstances.

The class FF on 9t is assumed given, and we suppose that F, is

148


CHAP. V

a corresponding class on WI,. We extend the definition of a differen-

tial d f, of WI, over R by setting it equal to zero outside the component W and we define FF to be the class of differentials

df,aF,. dl In other words, if is identically zero in R - WI and is equal to some (5.2.3)

=dfl-{-df$+....+dfk,

differential of F, on M. We say that F,a corresponds to the class FR.

If F = Fe corresponds to the given class Fa, we define the bilinear differential LF(fi, q) of this class on t to have the value zero for any two points ', q of unless both points lie in the closure of the

same domain 9, in which case LF (p, q) is equal to the bilinear differential Lp') of the class F, on M. The definition of LF (P, q) which has been given is the natural one, in view of the minimizing property of the singular bilinear differential in the case of a single finite Riemann surface. The norm of a differential on WI is the sum of the norms on the component surfaces. If q is a point of I,, the differential which minimizes the

norm among all differentials with the given singularity at q must clearly vanish on W,,, ,u ¢ v, where it is regular, and coincide with LF) on T4.

If if a F., we have the formulas (5.2.4) (6.2.5)

= 0, (df, LF(P, q))a _ -f'(q) (dl, LF(p, q))sc

Writing (5.2.6)

T,,, = (dZ,,, dZ,),

we see that the period matrix 11 r o, 11 is non-singular. For its deter-

minant is simply the product of the corresponding period determinants for each component of WI. Therefore the formulas (5.1.16)(5.1.20) are valid. In fact all formalism carries over immediately to

the case where WI is the union of a finite number of disjoint subdomains of R. 5.3. FUNDAMENTAL IDENTITIES

We assume that 9 is the union of a finite number of disjoint

FUNDAMENTAL IDENTITIES

§ 5.3]

149

surfaces imbedded in R, and we find formulas for the differences

I'P, - lip,, Zf, -- ,, lF = LF -3F. We begin by showing that for fixed q e R, IF (p, q) is a differential of class FF on R. Clearly 1F (p, q) is regular, and therefore of class M,

on D1. It remains to show that lp(p, q) satisfies the period relations

of F. Now (5.3.1)

(1M(p, q), Z()) ,v = - P(ZM(p, q), KP)

_ - P(LM(p, q), K,) + P(.'M(p, q), KP) = -ZP(q) + P(q) by (4.10.15) and (5.1.5). We note that (5.3.1) remains true if q is

replaced by q. For e = i., r

1, 2,

, 1,

=q), Ze(p))v ++(BF(p, q), Ze(p))u (IF(p, q), Ze(p))9k = (iM(p, - (e`F(p, q), using (5.1.16) and (5.1.17);

-

G

Ze (q) + We (q) + Z aP, .I',,#Z, (q) P..=1

by (5.1.16)', (5.1.17)' and (5.3.1);

-

Z' (q) + Q(q) + ZQ(q) using (5.1.19) and (5.1.22). Thus (5.3.2)

(1F(p, q), Ze(p))rot = 0,

-

1 #p, a 2, (q)l P..-i GO

- Q(q) = 0

e = il, ..., ii,

and IF (P, q) is in F.. The same argument can be applied to lF (p, but this is unnecessary since LF (p, q") and.PF (p, q") are in F. The fundamental identity is (5.3.3)

(lF(p, q), IF (P, r))st = -1F(3', q)

To prove this, we note first that (5.3:4)

(LF(p, q), LF(p, r))tt = LF(0, q)

for all q and r in R. For if q and r do not he in the same component of Ut, then both sides of (5.3.4) vanish, while if q and r lie jn Up, we have /,

\

/,

(LF(b,, q), LF(P,, r))8t = (Lp(p, q), L.-(P. r)jt" _ (LF) (p, q), 4.) CO, r)) D2 _ LIP) (r, q) = LF (PI, q)


150

[CHAP. V

by (4.12.3) if q is distinct from r and by taking the limit as r tends

to q if q = r. Also /, (.PF(Y, q), 3F (p, r))9t =.2PF(P, q)

(5.3.4)'

By (5.2.4), (lp(p, q), LF(P, r))9t = 0

so that (5.3.5) (2F(p, q), LF(p, r))9t = (LF(p, q), LF(P, r))9t = LF(f, q). Finally /, (lF(p,q), lF(C,r))% / /, /, r))9t- (2F(p, q), LF(p, r))%+(3F(p, (LF(p, q), lF(p, q),2F(p,/, r))9t

_-LF(9,q)+2F(9,q) _-lF(9,q) and this is (5.3.3). The identity (5.3.3) is also true if we replace r by 9: (5.3.6)

(IF (P, q), IF (P, r)) 9t

= - IF(r, q),

or if we replace q by q and r by y: (5.3.7)

(IF (P, q), IF (P, F)) It

IF (y,

q)

However, these identities are not immediate consequences of (5.3.3)

by analytic continuation. Using (5.2.4) and (5.2.5), we have (IF(p, q), lF(P, 9))9t

(IF (P, q), LF(p, r))9t-(LF(p, q),3F(P, r))9t+(.èF(t',/,q),- F(C, r))94

=-IF(r,q).

Similarly, by (5.2.5),

(l ($, q), lF(p, r))9t

= (1F(p, q), LF(p, #))9t-(LF(p, q),-TF(p, f))at+(2'P(P, q),2F(p,'))9t

= - IF (r, q) + (3F (q, 9)) -2'F (r, q) = - IF (r, q). Let

,(q) = ZF,(q) -'Y,,(q) In (5.3.3) and (5.3.7), take F = M and suppose that q is fixed. If we compute the periods with respect to r around a cycle K,,, we (5.3.8)

,

FUNDAMENT) L IDENTITIES

§ 5 3]

151

find by (5.3.1) (Im (p, q), J" (p))

(5.3.9)

(q)

and (5.3.10)

(ln1(p, 4}

Finally, writing (5.3.11)

y,U, = ITU, - T,+,

we compute the period of (5.3.9) with respect to q around a cycle K and obtain (since y,,,, is real and therefore symmetric) (.5.3.12)

y,,,.

(dale,

We have also (5.3.13)

(3F(p, q), LF(p, r))% =

-

/

/ (lp(p, q), LF(p, r))at = i,(r, q). (

The identity (5.3.13) shows that lF(r, q) is the projection of the singular differential -F(, q) into the space of the differentials of

class F. on M Other identities can easily be derived. For example, using (5.1.20) and (5.1.22), respectively, we obtain (5.3.14)

(BF(p, q), BF(C, r))et = BF(11, q),

(5.3.15)

(RF(p, q), RF(p, r))et = RF(f, q),

(5.3.16) (BF(p, q), RF(p, r))at = E

alalf'Qo)`

G

E a,., &,, (f') Z, (q) v. +=1

From (5.3.1) it follows that (5.3.17) (iM(p, q), BF(p, r)),t = Era,,.Z,,(9)`&.(q) -- BF(V q) 1,. +=1

while (5.3.2) leads to (5.3.18)

(la(p, q), BY(p, r))et = 0.

Finally (lm(-p, q), RF(p, r))et

= (LM(p, q), Rr(p, r))V - (2°M(p, q), RP(p, r))et = 0.


152

[Cxea. V

Taking r = q, q e 01, in (5.3.3) we obtain -(lF(p, q),1F(p, q))+ 5o. (5.3.20) l,(q, q)+(2F(p, q),2p(p, This inequality has a geometrical interpretation when F = S, where S is the class of differentials with single-valued integrals. In fact, let (5.3.21)

d' = -Y's (q, q) From Chapter 4 we know that 4' is the maximum external area of image domains of 91 under mappings by single-valued functions f

which have a pole at q such that df -2s (p, q) is regular there. If E denotes the corresponding maximum external area for 9t then (5.3.22) E - d' + (-C°s(p, q),2s(p, q))% got, since the maximum external area- for )l is not less than the corresponding external area of D1 defined by the mapping J, d f =2s (p, q).

But (5.3.22) is just the inequality (5.3.20). An analogous inequality is obtained by comparing internal areas.

Suppose that q is a point of D1, and let dfs (p) =

L9 (p, q) Zs (q, q)

dfs(p)

=Ys

(PI

q)

Ys (q, q)'

Since we know that NN(dfs) is minimal, we have No (dfs)

Nu (dfs)

-1

= Zs (q, q)

But

-N _st(dfs)

N (dfs) = Nyt (dfs) -Ngt--+st(dfs) Writing A

we therefore have (5.3.23)

= Zs (q, q) '

A S At - (Y8(I

q})'

`'s (q, q)'

(Ys(p, q'),Ys(p, 0) a_-a,

in analogy with (5.3.22). The inequality (5.3.23) may also be obtained from an identity. In corresponding notation, let (5.3.24) ki(p, q) = dtp(p) ---dfF(p)

§ 5.4] INEQUALITIES FOR QUADRATIC AND HERMITIAN FORMS

153

If q, r E D2, we have (kF(p. y), kF(p, q)) (

5 3 .

.

25

LF(q,

r)

Lip (q, q)LF(r, r`")

)

-I-2p

2F(q, r-) (q, q)LF(r,

)

2F(q, r) LF(q, q)9F(r, F) 2' (q, F)

PF(q,q)2F(r,P).

Taking r = q, we obtain in particular 1

(53261

q)

-

1

1

q)

(q, q)) a

(2F(p,

_ (kF(p, q), kF(P, q))v Z 0, and this gives a generalization of (5.3.23). We observe that (5.3.3) may be put into the form (5.3.27) lF(Y, q)=- (lF(p, q), lF(p,

(2F(p, q),YF(p, r))st-mn This formula will play an important role in the theory of variations of domains Tl. In fact, if Dl is near to 91 (in a sense to be specified

later), then lF(, q) = LF(9, q) -.'F(f, q) will be small and the first term in (5.3.27) will be of still higher order. Thus, we shall find (5.3.27)' LF(y, q)

q) ^' - (.7F(p, q),2F(p, r))S-R

and the right-hand side can be evaluated if the Y-differential of 91 is known. Thus, LF(p, q) can be determined approximately in terms of YF(p, q)5.4. INEQUALITIES FOR QUADRATIC AND HERMITIAN FORMS

We now apply the identities of Section 5.3 to obtain inequalities for quadratic and Hermitian forms with coefficient matrices II IF(gE q,) II, II LF(gK, J,) IL,

respectively. In this way we obtain necessary conditions expressed in the form of inequalities between qv ldratic and Hermitian forms in order that one or more surfaces can be imbedded in a given surface 9t. In the following section we derive from these inequalities necessary and sufficient conditions in order that a given surface X I can be conformally imbedded in another given surface 9t. A special case of these inequalities was' derived by Grunsky [2] in connection


154

[CHAP. V

with the conformal mapping of multiply-connected plane domains (see also [1]). We define (5.4.1)

F1, (P, q) = (2F (p, q), Y,- (p, r))st_art,

and correspondingly if r or q is/, replaced,/, by y or q". For example,

rF(r, q) = (2'F(Y, y))Ht-V Let A be a complex number and consider the norm over 92 (5.4.2)

E xlF(t', qp) -A E xpLF(-b, gp) N. `p-1 p=1

where the x,, are arbitrary complex numbers and the q,, are points of P. By (4.11.7), (4.9.2) and (5.3.27) wA find that (5.4.2) is equal to - I A 12 Z LF(gp, JA'

(5.4.3)

p,Z

2

r-l

g,)xpx,

W-1

N

- E [lF(q,, qp) + I'F(q, gp)]x,,x, p,'-1

Since the expression (5.4.3) is non-negative for all choices of the complex number A, we obtain the inequality N

2

E 4F (q,,, q,)x, x,

(5.4.4)

",'-1

N

N

E LF(q r-1

,

E [lF(gr, qp) + p,r-i

rp(q,,

q,)]X,9,-

We remark that, by the reproducing property, (5.4.5)

N

- E -vLF(gw

Nit E xpLF(P, q4))? 0.

p, r=1

p

(N

N

p, r-1 N

gp)) ? 0.

rF(q gp)xpx. =

p_1 N

E LF(gp, gr)x02, . E r-1

P. r-1

q )xp2,,

0,

§ 5 4] INEQUALITIES FOR QUADRATIC AND HERMITIAN FORMS

155

and it follows from (5.4.4) that the inequality N

(5.4.8)

I

N

N

2

E lF(gµ, g,)x,ix, S E LF(q,., q,)xµx, ' E IF(& ,a1

I. v-1

A. V-1

is true a fortiori. In particular, we observe from (5.4.5) and (5.4 8)

that N

(5.4.9)

E IF (q., q,)x,,R, !9 ID. ,,, v=1

Instead of the norm (5.4.2), we can take

N. ( E

(5.4.10)

A E x, LF(75,

µ-1

,'ml

Using the formula (5.3.7) we find that the value of this integral is equal to r

AV

J A 12NE LF(q,A, q,)x,4x, + 2 Rei 2 E IF(qu, U.,-1

(5.4.11)

l

N

- E [IF(gµ, q`,) + rr(q,., 4,v-1

Since the expression (5.4.11) is non-negative for all complex R, we have

E lr(q

,

(5.4.12)

N

N

S E LF(q., q,)x,,x, K, V-1

Now

E [lp(q,, q,) + r,(q,,,

K,-1

N

N

(5.4.13)

E rF(q,,, q,)x,.x,

E 90-11F C6 ,

qo)) z 0,

("-,L ,,.,-I and it follows from (5.4.5), (5.4.12) and (5.4.13) that N

E lp(q,,, ,-1

(5.4.14)

S 0.

For simplicity, write (5.4.15)

L

=

N

N

E LF(q,,,

E £°F(q,., Q.)x,,x.,

P. ,-1

/A, V-1

N

r = E rF(gP, P. V-1


156

[CHAP. V

and let N

1 = L' IF (Q,,, q,)xz, = L -..P.

(5.4.16)

N. I al

By (5.4.14) (5.4.17)

LS2:-- 0,

(5.4.18)

.P S L I,

and therefore

111=ILK-I21.

(5.4.19)

The inequality (5.4.12) becomes

12-ILI III +rILI =I11 {1l+-IL'I} +rI LI

_ - Ie ( {ILI-lY1}+rlLI so, and therefore

l2'12s(I2 -r)IL1.

(5.4.20)

Thus 2' z r and, if 2 I > q'', this formula provides an estimate for I L 1. From (5.4.13) and a formula analogous to (4.9.2) with respect to 91, we have

I' ° N

u1

A'YF(P, M) N. (µml E

(5.4.21)

N 1

K

Hence (5.4.20) may be written (5.4.22)

a

1 LIz

Z1YI.

NIy

N.( E x,,,2s

qµ))

µ=i Inequality (5.4.22) is thus a strengthened version of (5.4.18). Taking

N = 1, xl = 1, and dropping the subscript 1 on q, we have (5.4.23) L = Lip (q, q), 3 =2F(q, q), r = IF (q, q)' In this particular case inequality (5.4.22) takes the form (5'4 24)

1LF(q,4)

(` F(q,q))2 N,m(YF(p, q))

§ 5.4] INEQUALITIES FOR QUADRATIC AND HERMITIAN FORMS

157

We recognize that (5.4.24) is simply the inequality (5.3.26) written in a slightly different form. An inequality may also be obtained by use of the identity (5.3.12). Write (5.4.25)

Au, _ (d1,, d,3,)9j_U,

and consider the norm

N. µ-rlX

(5.4.26)

AE

Using formula (5.3.12) we see that this integral is equal to

+ 2 Re

A 12 E LM(qp,

,-1

-

(5.4.27)

N

-I

E [y,,, + A,,,)xµx,. ,=1

Since this expression is non-negative for all choices of A, we obtain

the inequality N

12

K. ,-1

(5.4.28)

N

N

S E LM

q",)xµx

it, ,-1

I [v,. + d,.Ylx,, ,

fp ,-1

Since

E A,,,xz, = N_,M ( E

z 0,

it, V-1

we have a fortiori N

(5.4.29)

2

N

N

E jr (gp)x,,x, S E LM(q,,, q,)xux, - E y,,xag,. I

P, ,-1

,b ,-1

p, ,-1

We conclude in particular that N

E

that is,

N

,4, v-1

N

(5.4.30)

S 0,

y,,,xpx, = E

N

E Il u,x,,x, S E I',,,x,,X,.

µ.,-1 v,,-1 Here both sums are non-negative.


158

[CHAP. V

We can even derive from (5.4.28) the better estimate N

E (y,. + A,,)xz, 5 0;

p, v-1

that is,

N

N

(5.4.31) - E y,,x,9, IA. =1

N

E Ap,xpx, = Nst_,r2 p, v-1

E xpaµ ( )} . (P-1

5.5. EXTENSION OF A LOCAL COMPLEX ANALYTIC IMBEDDING OF ONE SURFACE IN ANOTHER

We now apply the inequality (5.4.8) to obtain necessary and sufficient conditions in order that a local analytic mapping of a neighborhood in a given surface `7t* onto a neighborhood in another surface t can be extended over the whole of t* to give a one- one conformal mapping of 1't* onto a subdomain of 91. In other words,

we derive necessary and sufficient conditions in order that a local complex analytic imbedding be a complex analytic imbedding in the large. For plane domains these conditions have been stated when the surface R is the sphere (see [1]). We shall assume throughout this and the following sections of this chapter that the class F,, is symmetric, in which case we may assume in the preceding section the symmetry relations LF(p, q) = LF(q, P), -21F (P, q) =-FF(q, p)

The inequality (5.4.8) may then be written N

(5.5.2)

N

N

I

E lF(q

LF(q,, 4+)x,,9

, q+)x,Px,

N, +-1

£ lF(gµ, q,)x,,x,,.

,-1

Since

N

(5.5.3)

N

0, E -VF(qp, q.)x,,x, S 0,

E LF(gp, M, v-1

we have a fortiori N

2

E la(gp, q,)xµx, IS

E LF(gµ,4+)x +}, Q

or

N

(5.5.5)

E lp(gp, q+)xFx, I p, r-1

-- E LF(q, g,)xµx,. p,

Given an inequality such as (5.5.5) for a quadratic form, we can

EXTENSION OF A LOC4L IMBEDDING

§ 5 5]

159

pass immediately to an inequality for the corresponding bilinear form. In fact, if yl, y2, , yN are an independent set of complex numbers, we have N

N

1R, v=1

µ, V-1

4 2. IF (q, gv)x,.y:, = E IF (q,, q.) (x,.

y,,) (x,. -t- y,)

E lF(qu,gv)(x, -ya)(xy-yv)

1L, $,-I

Therefore by (5.5.5) N

4

E IF(q,,, q,)xµyv

µ, q=1

N

S - ELF (q,, qn) [ (xa -}- y,,) (x, -( Y,) + (x,. -y,,) (x, - y,)] ; 1", -1

that is N

I N

1 {`

(5.5.6)

11lF(q,,, q,)xy!

2

/,,,-1

q,) (x,,x9 -F

By a limiting process it is possible to pass from the inequalities (5.5.5) to inequalities between integrals. Let po be a point in the interior of ))1, and let zo be a uniformizer at pa. We denote the circle I zo I < a, a > 0, by go and, if p, q he in , we express the coordinates of P by z, those of q by C; that is z = zo(p), C = zo(q). Now

let a (z0) be a continuous complex-valued function of zo on the boundary a T4 of 9&. By a limiting process we obtain from (5.5.5) the inequality (5.5.7)

1f

fip. q)e(z)e(C)dzdC1 S

-f

Mo Me

W4 ano

In TZo we have the series developments (5.5.8)

lF(P, q) = E a,,,eC, -LF(p, 4) = E b1,z"c'',

where, because of the symmetry assumptions (5.5.1), (5.5.9) a,,, = a,,,, b,,, = 5,,. Taking (5.5.10)

1E

a,, e (z) = 2,ri v-o

160


CHAP V

in (5.5.7), we obtain 'V

IV

(5.5.11)

1

is, ri 0

i

SE o

The inequalities (5.5.11) are valid for arbitrary choice of the finite set ao, ai, - , aN. Thus inequalities for the bilinear differentials are easily transformed into similar inequalitites for the coefficients in their local developments. Let us consider now an orientable surface % with boundary C. In saying that the surface 9 is complex analytically imbedded in the surface R, we mean that there is a one-one conformal mapping of 9t onto a subdomain 9l2 of R. In order that such an imbedding be possible at all, we must assume that a topological imbedding of 91 into 91 exists. If 1t' is the image of T in 91 under the topological mapping, there is a correspondence between the cycles of 9t and 9)'t'. By introducing on fit' as uniformizers

the parameters at the corresponding points of R, we make 9)t' into a Riemann surface analytically imbedded in R. Given a class Fjj, let F., be a corresponding class on 9)r in the sense of Section 5.1. We then choose F. to be that class of differentials on 91 which have vanishing periods on those cycles of 9t which correspond to the cycles

associated with the class Fv,. The question now anises whether this topological imbedding can be realized analytically. Of course, different topological imbeddings give rise to different analytic imbeddings, provided the latter exist. Consider, for example, the case of two circular rings R and W in the complex plane. The ring 91 may be topologically imbedded in 9t in two different ways according as the essential cycle of 91 bounds in 91 or not. The first

type of topological imbedding is always realizable analytically while the second is possible only if the moduli of 9`t and 9t satisfy a suitable inequality. The choice of a corresponding class Fve depends on the topological type of the imbedding. For example, in the case of the rings 9t and #R, we have two classes of differentials in each domain - namely the classes S and M. In the first type of imbedding, the classes M and S on 9 correspond both to the class M and to the class S on DR while, in the second type of imbedding, the class.S on % corresponds

only to the class S on R.

EXTENSION OF A LOCAL IMBEDDING

§ 5.53

161

If there is a conformal mapping of % into 91, let the points n x of T correspond to the points p, q respectively of R. Then, ii Ag(n, x) is the bilinear differential of 9t, we have AF(n, x)dndx = LF(p, q)dpdq

(5.5.12)

where LF(p, q) is the bilinear differential of V. Again let lg(p, q)dpdq = [LF(p, q) -2F(p, q)]dpdq AF(n, x)dndx --.2PF(1', q)dpdq.

Writing

dF(a, x) = AF(n, x) --TF(p, q) dpdx' the inequality (5.5.5) becomes (5.5.14)

N

N

Z dg(n,,, n.)x,x,

(5.5.15)

C-E

a, .=1

The inequalities (5.5.15) are obtained from (5.5.5), the latter being

derived from (5.5.2) by neglecting certain non-negative terms. If we do not neglect these terms, we obtain instead of (5.5.15) N

(5.5.16)

_

n.)x,,x,12:!g

E' N dg(v,, n.)x,,z.,

E AF(np,

E dg(sr,,,

where (5.5.17)

dp dq

dF(n, x) = AF(n, x)

q)

dndx'

The inequalities (5.5.16) are necessary conditions in order that 91 can be complex analytically imbedded in R. Now assume that the surface 91 is locally imbedded complex analytically in R. That is to say, assume that a neighborhood of a point no of 91 is mapped one-one and conformally onto a neighborhood in R. Then for n, x in a neighborhood of no we have the local series developments with z = z(), _ C(m): 00

dF (n, x) = E t,,,zM r, - dg (5.5.18)

K, -0

- AF (n, x) = E bp, ez" µ, V-0

£ CM. z'` '',


162

[CHAP. V

and (5.5.15), (5.5.18) may be transformed respectively into the inequalitites (5.5.1 c)

I

E Cx, X,,

E bN,xµx,,

N, -0 N

l

(5.5.20)

!I

N

N

s

E CN,xx,.

E C,,,xNxv I S E b,,vxµxv

I

N,+-0

N, ,e0

N,,=0

We investigate the circumstances under which this local imbedding can be extended to a complex analytic imbedding in the large, and we base our considerations on the following theorem: THEOREM 5.5.1. Let

0

V (n, x) = E V"e4`',

v,,, = V'-O'

N. , -0

be an analytic bilinear differential defined for n, x in a neighborhood of the point no of 92. I/ the kernel Ar (n, x) of % has the local development m

- Ar(n, ic) = E b,,,zN O

(6.5.22)

N, -o

and it for every finite se) of complex numbers

xo,

xl,

, xN the

inequalities i

E v,,,xx`

(5.5.23)

L'V-O

f

E b,,,xx", N, -0

are satisfied, then (5.5.21) is the local development of a symmetric analytic bilinear differential 17(a, x), of the class F on %, which is everywhere regular.

Similarly, if (5.5.24)

W (X' x) = E w,,40 cv, w,,, = *,,,, N,I-0

is defined for n, x near no and if N

(5.5.25)

0

E w,,,x,,z, is, +-0

N

E b,, xO9, is, V-0

, xN, than W can be extended over % and defines an analytic Hermitian bilinear differential of the class F in its dependence on n. for every finite set of complex numbers xo, x1,

PROOF: Let {dV,} be a complete orthonormal system for the class

F of differentials on Q and, for n near no, let

§ 5.61


(er) =V -P `1 ,

(5.5.26)

163

,8,,, 7" 0.

,

The existence of such a system has been proved in Section 4.14. For formal reasons it is convenient to define 9,,, = 0 for µ > it. The j8,,, then form an infinite triangular matrix with zeros under the principal diagonal.

We have by (4.9.6)

x) _ '-0

(5.5.27)

(;,(x))

,

and hence by (5.5.22), (5.5.26) and (5.5.27) 00

(5.5.28)

b,,, = E Peg e-0

Since fu,. vanishes for 0 > ,u, the sum in (5.5.28) involves only finitely

many terms. We remark next that the infinite triangular matrix 1' 15,,, 11 has an inverse matrix 11 y,,, 11 of the same form and that each

element of the inverse matrix can be calculated by finite algebraic operations. Let to,

(5.5.29)

QD

= E VQaVQ,,Ya

t,,, = 4,0,

0. a-0

define a matrix associated with the coefficient matrixv,,,

of the

given local development (5.5.21). Using (5.5.26) we obtain by formal

calculations with power series 0

(5.5.30)

ao

V(n, ,) = E VQ,xer = 1E e, a-0

is, ,-0

.(x)

We know nothing about the convergence of the last sum and the equality is to be understood in the sense that formal operations on both sides of the equation lead to the same coefficient for each term

zzCa. On the other hand, it should be observed that the numbers t,,, exist since the defining sums (5.5.29) are finite. Let a,, it = 1, 2, . , N, be arbitrary complex numbers and set N

al, =

(5.5.31)

Then by (5.5.32)

ypyQr.

,-0

(5.5.29) N Fr

P. -0

N

t,,,aµa, _ I vQaaQaa. Q, a-0


164

[CHAP. V

Using the inequality (5.5.23) and the identity (5.5.28), we obtain N

(5.5.33)

AT

E t,,ra,ay

I

I

,y vm0

SE E+ Qpxp Q°0 0-0

12==

E I ae

12

Q-0

for arbitrary choice of the a,,. We now extend (5.5.33) from quadratic forms to bilinear forms. We clearly have N

N

N

E t,,,.a,,a, + I R. v-0

(5.5.34)

2 1 tp,apb, p, v-0

P. v=0

N

SZ(Ia.I2+Ibel2±2Re{aQbQ}) e-0

from which we infer that (5.5.35)

I

Et,,a,,b,S

EN

2

1,9.0

e-o

for any choice of the finite sets al, a2, Choose in particular

, aN and bl, b2,

, bN.

b, = yp(x).

a,, =

(5.5.36)

Q..o

Then (5.5.35) becomes

1

(5.5.37)

E tµyk(n)y,,(x) Ip.v-0

E I'p, (n) I4 + E' I(x) I'

.

0-0

2 p-0

If n and x are interior points of W we may let N tend to infinity and we obtain by (5.5.27) (5 53$)

I

S

Z t ,w( cc

Ip.v-0

1

2 1

ac)

+ AF(x. x)}

Let us return to the formal identity (5.5.30) which, by (5.5.38), has now become a valid equation connecting two series uniformly convergent in a neighborhood of the point no. Moreover, the second sum in' (5.5.30) is regular analytic over the whole of 91 and is obviously of the class F. Thus the extension of the local development (5.5.21) over 9 has been effected. The second part of the theorem is proved in an analogous way. Let go

(5.5.39)

Sp = E WQdYQ,-(Ya,)-, Q, 01-0

Spv _ (s,),


§ 5.5]

165

be the matrix associated with the coefficients wi,. Again introducing

numbers a, and a, related by (5.5.31), we find N

N

E

(5.5.40)

E wQ,aQa,.

kr +-0

Q. 0-0

By (5.5.25) and (5.5.28) we obtain N

N

0< E

(5.5.41)

1 2_

µ,r-0

Q-0

Hence N

E sa,,5, S

(5.5.42)

/b r-0

2

aQ1 +I bQI$)Q-E 0

Thus (5.5.43) 1A,

2

V"0-0

Ofr , 7G) + AF(9L, x)}.

For a, x in a neighborhood of no we have by construction (5.5.44)

E

,t, r-0

W(n, Q, a-0

By means of this identity we are able to extend W (a, x) over the whole of 91 and it defines a Hermitian bilinear differential which, in its dependence on n, is of the class F. This completes the proof of our theorem. Let us apply the above theorem to a given local complex analytic imbedding of the surface 9 in the surface R. Assume that a neighborhood of the point no of 91 is mapped one-one and analytically onto a neighborhood of a point p0 of 91. In other words, if z is a local coordinate near the point no of 9 and if w is a local coordinate near po on 91, we suppose that w is an analytic function of z with a non-

vanishing derivative in . If n and x are two points of % lying in this neighborhood of no which have coordinates z and C respectively, we then have the local developments (5.5.18). Now suppose that the

inequalities (5.5.19) are fulfilled for arbitrary choice of the finite set x1, x2, , xN. From Theorem 5.5.1 we conclude that the bilinear differential dF(n, x) given by (5.5.14) is regular everywhere on 91. We now extend the local mapping by analytic continuation, and we assume first that the point x is fixed, me 3. From (5.5.14) (5.5.45) dW(p) = dQ(x)

166


[CHAP. V

where (5.5.46)

dW(p)

=

dx(p, q) dp

and where dQ(a) is everywhere regular on 9`t. Let o be a subdomain of ) containing x in its interior but with ro in its exterior. Integrating

(5.5.45) along a path beginning at the point no and avoiding the o, we obtain

domain

(5.5.47)

W (P) - W (po) = 0) - Q (-no) This relation may be used to define p (n) outside . Moreover, p (a) defined in this way will be regular analytic except at points where W' (p) vanishes. But if (5.6.48)

W'(p) =

dx2p(p, q)

vanishes at a point p of 9, then by varying the point x inside JU we can make YF (p, q) different from zero at the point in question. Thus the mapping p(x) defined by continuation will be everywhere regular so long as the point p remains in the closure of 91. In this mapping two distinct points cic, x of 9t go into distinct points p, q of OR. For assume that p(n) = q(x), a x. Then3F(p, q)

becomes infinite in (5.5.14), which contradicts the regularity of dF(x, x), unless

dp or dq vanishes. But if LP dv dx do

vanishes, say at

a point ni, then there are two points n', x" in a neighborhood of a1 where LP is 'different from zero and which map into the same

point of 1, contradicting the regularity of dF(r, x). Thus distinct points of 91 go into distinct points of 9R and LP is regular analytic

and non-vanishing. But then we conclude from (5.5.14) and the regularity of dF(;c, x) that distinct points of OR correspond to distinct points of 91, and therefore the mapping p (x) is single-valued on R.

Hence the correspondence p(() defined by continuation is one-one

and analytic so long as the point p remains in the closure of N. In order to prove that the point p remains on 1 we must suppose that the stronger inequalities (5.5.20) are valid for arbitrary finite


§ 5.5]

167

sets x1, ' , X. Since N

E b,,.x,x, Z 0,

(5.5.49)

Is. v-o

by (5.5.19), we have from (5.5.20) N

E C"x,,z, Z 0.

(5.5.50)

µ, vm0

Let

(5.5.51) DF(n, x) = dF(r,

AF (7r, x) _

q)

ddxd

By (5.5.18)

DF(n, x) = E (by, -

(5.5.52)

a, ti-o

Moreover N

N

0< E (b,,,, - C,,,,)x,,x, S E bµyxµxv.

(5.5.53)

Thus the condition (5.5.25) is satisfied for DF(n, x), and we have from Theorem 5.5.1 that DF (n, x) can be extended analytically over the whole of S91. If an interior point n maps into a boundary point fi(n) of 91, we see from (5.5.51) and the non-vanishing of dP

that DF(n, v) is infinite, a contradiction. Thus the mapping p(n) carries % onto a subdomain of 9t, and we have the following theorem: THEOREM 5.5.2. A local complex analytic imbedding P (7c) of 9t into tR defines an analytic imbedding of the whole of 91 into 91 if and only if 2N

(5.5.54)

I

E d4(x,,, x,)x,,x,I

E

µ, vat

for every N-tuple P"_1 of points x1, x2,

E dF(x,,, x.)x,,xv /i, v=1

, xN lying in some neighborhood

where p (n) is defined and for arbitrary choice of the finite set x1, x2,

,xN

(x,, complex). Here (5.5.55)

d7 dF (n, x) = AF(n, x) -.F(, q) dp do dx.

We may also express a necessary and sufficient condition for an analytic imbedding of 91 into R in terms of the coefficients of the series development of the map function p (n) at a given point no e 91 and, in terms of a given local parameter. We have

168


[CHAP. V

THEo1EM 5.5.3. Let p(a) map a neighborhood of no a 9't into a neigh-

borhood of P. = p(no) a OR. Using a local parameter z(n) around no we may calculate the series developments (5.5.18). Then, a necessary and sufficient condition that p (n) can be extended over % to give an analytic imbedding of T into 9R is the validity of the inequalities (5.5.20)

for every N-tuple of complex numbers x,. Let us proceed next to the case that % is a closed surface. For the sake of simplicity, we restrict ourselves to the consideration of the class M. We are sure that the kernels Am (a, x) and AM(n, x) exist in % and obtain again (5.5.20) as a necessary condition for an analytic imbedding. But now we can make even a more definite statement. 9't must be mapped into 91 by the function p(n) and we must neces-

sarily have the equality (5.5.56)

AM(n, ic)dndx =2M(p, q)dpdd

since there is only one differential 2M on any surface. Similarly (5.5.57)

AM(n, x)dndx = 2M(ifi, q)dpdq.

Thus, we have identically on 91 (5.5.58)

di(n, x) = di(n, x) = 0

and instead of the inequalities (5.5.20), we may write the equalities (5.5.59)

cop=0,

Cµ,=0.

These conditions are, conversely, sufficient in order that 9 be analytically imbedded into 9R as can be derived from (5.5.57) in just the same way as before. 5.6. APPLICATIONS TO SCHLICHT FUNCTIONS

If 9 is the w-sphere and 9t is a subdomain thereof, a function p(n) which gives an imbedding of 9 in fR is called "schlicht." In particular, p (n) may be the identity mapping. Or we may take 91 to be a disc J w I < a, in which case the functions p (n) become bounded schlicht functions. We show first how the coefficient problem for schlicht functions in multiply-connected domains of the complex plane can be treated. In this case, we choose SJR to be the whole z-sphere and then the com-

plex variable z is a valid uniformizer at each point of 9t except oo.

§ 5.6]

APPLICATIONS TO SCHLICHT FUNCTIONS

189

The &-kernels have the simple forms (5.6.1)

1

X (Z' C) =

0. E'

Let % be a domain in the z-plane which contains the origin and has

finite connectivity. Let w = ip(z) be regular analytic around the point z = 0. Let us assume further that ip(0) = 0. We want to derive from Theorem 5.5.2 the conditions that w can be extended to a schlicht mapping of S)l into 91. We assume that the class F is S. In the particular case under consideration, S is a symmetric class.

By (5.5.14) and (5.6.1) we have

-

"(z)P/(_) n[p(z) --T(C)]a We have to develop this expression into a power series around the d(z, ) = A(z, C)

(5.6.2)

origin and then insert the coefficients of the development into the inequalities (5.5.19). For this it is convenient to introduce certain sets of auxiliary functions which will provide an interpretation of the inequalities obtained. Let A'(z) and B,(z) be defined by the generating functions (5.6.3) A(z, 4) = Z (v + 1)A,',+1(z)6", A(z, ) = Z (v + V-0

V-0

Clearly, the differentials A' (z) and B' (z) are defined over the whole of 91. In fact, denoting by C the boundary of fit, we have (v

(v

1

+

(' A(z, C)

f

dC +

1)(B.+1(z)) = 21 f (A(z, Z))

v -}- I

dc.

We see from (5.6.3)' that B; (z) is regular analytic in 91 while A; (z)

has a pole of order v + 1 at the origin. Set, therefore, Av(x) =

r

}

+ o(a + 1)a µ+x] 14-o


170

[CHAP. V

If z tends to the boundary C of %, we have A(z, C)dz = (A(z, g)dz)-

and hence, comparing coefficients in (5.6.3) we obtain (5.6.5) A,'+,(z)dz = (B,+1(z)dz)for z on the boundary C. Thus, with any domain % in the complex plane we may associate a sequence of analytic functions A,, BY which are related 1* (5.6.5). Since we are dealing with the class S, the functions A, and B, are single-valued on 91. On C we have (5.6.6) Av}1(z) = (B,+x(z))- + constant, and the functions A, and B, are easily seen to be determined except for an additive constant. By (5.6.3) and (5.6.4) we have the series developments A (z, ff b)

=

[-(

1

pp b) 9

(5.6.7)

+ µ, Y-1

1Cr 1

A (z,

where (5.6.7)'

by the symmetry property of the A-kernels. Having defined sequences of functions related to a given domain s91,

let us now define sequences of functions related to a given function w(z). We consider the generating function (5.6.8)

log

z-

E y,,,z"C'. µ,,-0

We have, consequently, log

CI

'/(/C) -- log T(Z)

(z)

z

_ -0-1 E V,- -}- E / 00

K,Y-0

On the other hand, for fixed z # 0 the function log l

I

_!P(z)] is

analytic in C around the origin and may, therefore, be developed into a power series

§ 5.6.]

(5.6.9)

APPLICATIONS TO SCHLICHT FUNCTIONS

log

i = E F,. f

[1

I

171

1 a".

It is easily seen that F (t) is a polynomial of degree µ in its argument t.

Comparing (5.6.8)' and (5.6.9) we obtain

F. (_(17) ) =

(5.6.10)

-

-

E y",x". K=u

For v = 0, we define Fo identically zero. Thus - vF, is a polynomial

of degree v in 1/4p(z) whose principal part at the origin is exactly 11z''. It is uniquely determined by c (z) except for an additive constant.

It is called the v-th Faber polynomial of the function 97 (x), and it plays an important role in conformal mapping (see [5]). The above definition of F, also provides a simple construction for it. Differentiating (5.6.8) with respect to z and C, we obtain (5.6.11)

4''(0

µv}'",z" '

ac(z - )2

9p(C)]2

1

k.

From (5.6.2), (5.6.7) and (5.6.11) we obtain (5.6.12)

d(z, C) =

1

E ,uv(a,,,a, -1

Now we are in a position to give necessary and sufficient conditions that 9' (z) be schlicht, in terms of inequalities which depend, on

the coefficients at,,, and (5.5.19) we have N

(5.6.13)

E

determined by the domain 1. From II

N

The inequalities (5.6.13) are Grunsky's necessary and sufficient conditions that 97(z) be schlicht ([2]). Since the yN, are easily expressed in terms of the coefficients in the Taylor's expansion of 9'(z) around the origin, the inequalities are conditions on the coefficients of 9'(z). Unfortunately, these conditions appear in such an implicit form that

it is difficult to extract much information from them concerning the possible values of these coefficients. As another example, we consider bounded schlicht functions c in a plane domain M which satisfy, say, the inequality I p S 1.

172


[CHAP. V

In this case 92 is the unit disc and we have

(5.6.14) (x, ) = x(z

1

--3(z, ) _ all

c)$,

1

z

z-

Since.'(z, ) does not enter into the inequalities (5.5.19), we obtain the same conditions (5.6.13) as before if we apply (5.5.19) alone. We therefore apply (5.5.20). We have to introduce a new sequence of functions connected with 97(z). Starting from (5.6.9), we find

log [1- T(zOW)-] = E -o 00

Fn(T( )) ='-0 E Then

log [1- c'(z)(q (C)) j = E AMIZA F.

(5.6.16)

OD

Differentiating this identity with respect to z and f, we obtain cc

4" (z)(4', (0) -

E 1`vA

,-1

11-4'(z)(w(0) '2

1 "-1.

From (5.5.17), (5.5.18) and (5.6.17), we compute

cl 1, .

(5.6,18)

i

v

+

Thus we obtain finally the following necessary and sufficient conditions for the coefficients of bounded schlicht functions:

:S

H

(5.6.19)

E

/t,,-1

E f,,,xvx - E (pp. +

µ,,-1

µ,,-1

In particular, let % be the unit disc with center at the origin. In

this case we find easily a,,, = 0, f4,,, = 8,,,/v (where 8,,, = 1 if p = v, 0 otherwise). Thus (5.6.19) simplifies to N

(5.6.19), I

E Ywx,,x,

Mr,-1

12:!9 N

E

1 I x,, I2.

M-1 14

AV

AV

E 1 I x,, : + E

\ i-1 /

AN,x,,x, .

µ,,-1

I

EXTREMAL MAPPINGS

§ 5.7]

173

Taking, for example, N = I we have the condition (5.6.19)"

I a2

- a3al 12 S I al I°(1 - I a112)

for the first three coefficients in the development

p(z) = E az'`. µa1

It is worthwhile to remark that the inequalities (5.5.15) can be applied in a different way to obtain information about schlicht functions in plane domains. For example let % be the unit disc; then A(z, C) and A(z, ) are given by (5.6.14). We let (5.6.20)

1

U(x, C) _

a { [T(z) - 49(c)]2 [z - C]2 Then U(z, C) is regular in % if q, (z) is schlicht there. For z = we find (5.6.21)

U(z, z) =

6,n L q,'(z) )

q,'(z)

2

67C

{ ,

z}

where {qp, z} is the Schwarz differential parameter. In view of (5.6.2) and the fact that A(z, ) = 1/2(z - C)2, we may

write (5.5.15) in the form N

(5.6.22)

I

I'U' E U (x,,, z,)xx,

S

1

N

,-x In the special case N = 1 we derive from (5.6.21) and (5.6.22) the estimate (5.6.23)

I{Q',z}I

S(I-IzI2)2

The example of the function z/ (1 - x)2 shows that the estimate (5.6.23) is best possible. 5.7. EXTREMAL MAPPINGS

In Section 5.5 we established sets of inequalities for mappings of a domain J2 into a domain 91. We now study the class of extremal mappings for which some inequality becomes an equality, and we

are thus led to examine the process by which we passed from identities to inequalities.

174


[CHAP. V

The inequality (5.5.5) was obtained from (5.5.2) which, in turn, was obtained from (5.4.4). In passing from (5.5.2) to (5.5.5) we neglected the term N

E 3F(q,, gr)xµxr

p, r=1

while, in passing from (5.4.4) to (5.5.2), we neglected N

E r (q,4, Mxxr

1+, v-1

Here (5.7.1)

r. (q,, 4r) = (-Tg(P, q,4), 2F(P, q.)) It-a.

If there is equality in (5.5.5), then necessarily (5.7.2)

p F(qp, Z,)xp:Gv = NLr rF(gp, !.. .L N

p, s-1

,4. v-1

But then, since the left- side of (5.7.2) is non-positive by (5.5.3), the right non-negative by (5.4.6), each `side must be zero. This implies,

in particular, because the right side of (5.7.2) is zero, N

E2'F(l, 4,4)x,4 = 0

(5.7.3)

,4-1

at each interior point P of 91- 9)t. By analytic continuation the sum in (5.7.3) is therefore identically zero in R if there exists an interior point of 91 -- 9)2. On the other hand, .2F(p, qp) becomes infinite if P - q$ and hence it is impossible that (5.7.3) should hold throughout R. We have therefore proved that, in the case of an extremal mapping,

the set 9t -- R has no interior points. In particular, we conclude from (5.7.1) that

rF(P,q)=0

(5.7.4)

for an extremal image domain Tt. From the identity N

(5.7.5)

-E

N

Qr)xpx, = N9

l

E xpYF

(P,

q11))

and the fact that the left side vanishes for an extremal mapping,

EXTREMAL MAPPINGS

§ 5.7]

176

we have N

E 2F(p, gp)zp = 0

(5.7.6)

pat

for each p of R.

If there is equality in (5.5.5) for some particular set x1, xs, -, we see moreover that the integral (5.4.2) must vanish for some value of the complex number A. For this particular value we therefore have N

N

E

(5.7.7)

A E 9, LF(p, q,.)-

A-1

p-1

Take the scalar product of (5.7.7) and lv(p, q) with respect top over V. By (5.3.3), (5.4.1) and the reproducing property of LF(p, q) we obtain N

(5.7.8)

N

E xp[lF(q, qp) + rF(q, q14)] = A E zp(lF(q, q,.))-.p-1

p-1

In the extremal case now being considered, (5.7.4) and (5.7.6) hold. Hence N

N

E x,,Lg(q, qp) = A E xIF(q, q,,)

(5.7.9)

p-i

p-1

for q e V. Comparing (5.7.7) with (5.7.9), we see that I A I = 1, and

we therefore write (5.7.7) and (5.7.9) in the common form N

N

E Al xpLF(q, qp) = E (A1)-xpl,(q, qp).

(5.7.10)

µ-1

p-1

That is, setting (11)-x = yp and replacing q by p, we have N

N

(5.7.11)

E YpLF(p, qp) = E yplF(p, qp)

p-1

p-1

Using (5.5.12) and (5.5.13) we may express (5.7.11) as follows: N

E yp2F (p, qp)

(5.7.12)

p=1

= E [y14A(v, xp) d7r p=1

(dx1 "Y

p

xp) (d x) j Let x tend to a boundary point of %, and use a boundary unifor-

176


[Cxap. V

and hence, in view of

mizer at n. Then AF (n, x,,) = (AF (n, (5.7.12), N

(5.7.13)

Re L E

dp 4a)

=0

for n on the boundary of 91. This equation determines at each point a E C the direction of the tangent vector d4 of the image in 91 of the boundary C of 91. Thus (5.7.13) is a differential equation for the boundary of 911 imbedded in R. Using a boundary uniformizer, we may integrate (5.7.13) and we obtain N

(5.7.14)

Re

£' y,,.FF(p, qp) = constant

where (5.7.14)'

ap

Hr (p, q) = 2F(fi, q)

The boundary curves of 91 are, therefore, the level curves of a harmonic function on 91 which has poles at the points q,,. These level curves are generalizations of the isothermal curves in the plane which are defined as level curves of rational functions. Since 91- M has no interior points, the boundary curves (5.7.14)

of $t are piecewise analytic slits lying on W. We assumed at the beginning of this chapter that the boundary uniformizers of a domain 91 imbedded in 91 are admissible uniformizers at the corresponding points of OR. This restriction may now be removed in the main for-

mulas and identities by a limiting process. For if 9R is any finite Riemann surface imbedded in 91, it can be approximated by a sequence where M,, tends to P of finite Riemann surfaces 971,,, 91,, C

as n becomes infinite. Since all functionals considered are derived from the Green's function and since the Green's function of 971 together with its derivatives tends uniformly to the Green's function of 9)1 in any compact region interior to 0, we see that the functionals of Dl converge to the corresponding functionals of 9)1 in any compact

subdomain interior to 0. We shall not carry out the details of the approximation process, but we observe in particular that the inequality

(5.4.8) is valid for any finite Riemann surface imbedded in R. This remark is necessary since we found that the extremal domains 971

EXTREMAL MAPPINGS

§ 6.7]

177

are slit domains in 91 with boundary uniformizers which must be singular uniformizers of t at a finite number of points. We have shown that every domain Jt C R which is obtained from 92 by an extremal mapping into OR is a slit domain whose boundary

slits satisfy a differential equation (5.7.13). We now remark that a partial converse of this statement is valid; namely if 9 is mapped on a slit subdomain `t of R with differential equation (5.7.13) for the boundary slits, where F,, is such that Fa = S is a corresponding class, then the mapping solves somes extremism problem for the inequalities (5.5.16). In fact, suppose that 9t is a slit domain whose boundary satisfies (5.7.13). Consider the expression N

N

(5.7.15) E Y,[YF(p, qµ) - LF(p, qk)] + I Yµ

qµ) = d

P-1

K-1

It represents a differential on S2, and for p on the boundary of 1t we have, in view of (5.7.13) and the behavior of the LF-kernels there, (5.7.15)'

Re {A (P)dp} = 0.

Moreover, d (p) is obviously regular on 0. Thus, since F., = S, d (t) vanishes identically. Then N

(5.7.16)

N

y 1F(', q,.) + EYNLF(p, qµ) = 0.

Taking the norm over TZ of (5.7.16) we obtain by (5.3.3) (5.7.17)

2Rei E YµY+IF(4µ,

,E y/45,[lF(q,, gµ)+LF(gw

=0.

of the first sum in (5.7.16) On the other hand, the norm over must equal that of the second, and hence we have N

N

(5.7.17)'

q,)

E y47 JF(q,, qq) _ £ V-1

V-1

Thus, instead of (5.7.17), we may write (

[Re j

12

N

E

q,)

rU. V-1

(5.7.18) N

E Y/4YVLF(gµ, M. -1

j

N

E YpY+IF(gy, q,.) µ. +-1

178


[CHAP. V

Comparing this result with (5.4.8), we see that the domain 91 has extremal character in 91 with respect to that inequality. Referring back to our domain 9, we can express this property as an extremal property with respect to the inequality (5.5.16). Thus the converse of the above result is proved; every domain bounded by slits satisfying the differential equation (5.7.13) for suitable F results from an extremal mapping. 5.8. NON-SCHLICHT MAPPINGS

So far we have dealt with the problem of the schlicht imbedding of a domain 9 into another domain 91. The question arises whether similar criteria might be developed for the case of a non-schlicht mapping of 9t into R. In order to show the use which can be made of our L-differentials let us consider the following special problem. We consider an orientable surface 91 with boundary and distinguish on it a point xo. We choose a fixed uniformizer z at xo and consider all functions on 9t which have near xa a series development (5.8.1)

j (z) = 1 -{- alz -}- a2z2 + ...+ anxn -f- .. .

and have positive real parts everywhere on W. We want to give necessary conditions for such functions in terms of their coefficients a,. Clearly, this is a special case of the general subordination problem

with 9 the half-plane Re w > 0. We derive our conditions on t (n) in the following form. We select N points xi on % and N complex numbers x.. We form the differential N

H'(-c) = Exs Am (a, xr)

(5.8.2)

S-1

and the integral

(5.8.3)

J

I H'(=) 12 Re {f (n)}dA. = J,.

Because of our assumption on f (n), the number J, will be nonnegative. On the other hand, we can easily evaluate the integral. Using (4.10.8)' and (5.8.2), we have

NON-SCHLICHT MAPPINGS

§ 5.8]

N

_

_

r I, ' E xixk rAM(n, xi)(AM(r, xk))

1

i, k-i

JJ

(5.8.4)

179

cL [f(v)+(1(n)) ]dA

9t

L xixk1M(xk, xi)[/(xk) + (f(x,))-] Z 0.

We thus derive the result: THEOREM 5.8.1. If 1(n) is an analytic function on % with positive real part, the kernel - AM (n, x) [f (n) + (/ (m)) -] is positive-definite.

Here AM (a, x) may be replaced by Av (n, x) for an arbitrary class F, since f (n) is single-valued on SJI. Let

ÀM(n, x)[f(n) + (f(x)) ] = E C'? _r is, ,-0

be the series development of the above kernel near xo in terms of the local uniformizer z(n) = z, z(x) = C. If we assume that Am (n, x) is known, the coefficients cµ, can easily be calculated in terms of the coefficients a, of the function f (n) considered. From the inequalities (5.8.4) we derive, by the method of Section 5.5, the following inequality

in terms of the c,,,: N

(5.8.5)

£

N. -O

zt, 0

for every choice of the complex N-vector x,,, ,u = 1, 2, - . N.

The inequalities (5.8.5) impose an infinity of conditions on the coefficients of a function f (n) on % with positive real part. They are closely related to analogous inequalities established by Caratheodory

and Toeplitz in the case when the domain R is a circle. It is easy to generalize the above method to a wider class of domains R. Suppose there exists on 91 a function 0(p) which has on R a positive real part. In this case, we may give necessary conditions for a function J (x) on % to map % (non-schlicht) into 91. For this purpose we would consider the integral (5.8.6)

f I H'(n) 12 Re {O(f(n))}dA,, = Tf,O.

This integral must again be positive and treating it in the above

180


(CHAP. V

way, we can again derive an infinity of inequalities for the coefficients a., of the function /(rc). REFERENCES 1. S. BERGMAN and M. SCHIFFER, "Kernel functions and conformal mapping," Compositio Math., 8 (1951), 205--249. 2. H. GauNsI Y, ,Koeflizientenbedingungen fur schlicht abbildende meromorphe Funktionen," Malls. Zeit., 45 (1939), 29-61. 3. J. A. JENKINS and D. C. SPENCER, "Hyperelliptic trajectories," Annals of Math.,

53 (1951), 4-35. 4. A. C. SCHAEFFER and D. C. SPENCER, Coefficient regions for schlicht functions,

Colloquium Publications, Vol. 35, Amer. Math. Soc., New York, 1950. 5. M. SCHIFFER, "Faber polynomials in the theory of univalent functions," Bull. Amer. Math. Soc., 54 (1948), 503-517. 6. M. SCHIFFER and D. C. SPENCER, "On the conformal mapping of one Riemann

surface into another," Ann. Sci. Fenn., A. I, 94 (1951).

6. Integral Operators 6.1. DEFINITION OF THE OPERATORS T, t AND S We continue to suppose that D1 is a subdomain of a finite Riemann surface 91, and that the classes FF and FF of differentials are related

as described at the beginning of the preceding chapter. Moreover, for simplicity, we suppose that T is orientable and has a boundary. In this chapter we shall be concerned with integral operators T which

transform a differential f' of FF into a differential T l' on N. We shall again denote the bilinear differentials of 1)l and 91 by LF(p, q) and 3F(p, q), respectively, and we introduce the integrals Yfr(p, q), SF(p, q), E(q, p) aTF(p, g) ap

g)

= LF(b, q),

ap

_2',(p. q),

(6.1.1)

a F (g, p) ap

= . . 2 ' ( q,

).

We write (6.1.2)

lF(p, q) = LF(p, q) -2'F(p, q) (as in Chapter 5). Scalar products and norms over fit, $t will be distinguished, wherever necessary, by attaching subscripts $)2, N. From the point of view of the comparison of the two domains 931 and 9, one is led quite naturally to the following question. Scalar multiplication of any differential of Fu with the kernels - LF (p, 4) and LF (p, q) leads on the one hand to reproduction of the differential and zero on the other. What then will be the outcome of the same multiplication with the kernels (p, q") and L°F (p, q) ? In each case the result will be a differential on 91 which is a linear functional of the given differential on 92. To form these linear functionals, define (6.1.3)

Ts(q) = WF(q, p), /'(p))se, [181]

182

INTEGRAL OPERATORS

(6.1.4)

(f'(p), _T1, (p, q)) sot = (Yip (q,

(6.1.5)

Sf(q) = Tf(q) + Tf(q).

[CHAP. VI

The differential c is regular analytic on the whole of J1, while T and S are regular analytic on D and on R - TZ but, as we shall see, are discontinuous across the boundary of St. If 2'F (q, p) is symmetric, that is if 2'F(q, p) = 2F(p, q), then (6.1.6)

JF(q, fi) = 2F(fi, q) = (-TF(q,

p))-.

In this important case we see that, formally at least, (6.1.7)

Tf(q) _ (Tf(q))-, so D is obtained from T by applying the "N-operation" described in Chapter 2. It will often be convenient to extend the definition of a differential of FM over the double Z of R by defining f' to be Then identically zero in R - D% and setting (6.1.8)

Sf(q) _ Computing periods with respect to q in (6.1.4), we see that Tf satisfies (-2F(q,p),'f'(p)),.

the period conditions for a differential of class F. By (6.2.10) below, NII(fif) < oo, so T f is in F%, and we may therefore think of 1` as an operator which projects each differential of FM into a corresponding differential of F%. If q is a point of 90l, the integral on the right of (6.1.3) is to be interpreted in the sense of (4.9.8). However by (4.11.6) (6.1.9)

(LF(q, p), f'(p)) = (LF(p, q), f'(p))V = 0 since f' belongs to the class F. Thus, if q is a point of X12, (6.1.3)'

p), /'(P)) t and it follows that Tf is regular analytic throughout the closure of 1 and by (5.3.2) is in F. when considered as a differential on V. From (6.1.3) we see that Tf satisfies the period conditions for a Tf(q)

(IF (q,

differential of F. on R - M. Let q be an interior point of 91 lying on the boundary C of '!t, and suppose that /' is analytic on an arc of C containing q. Then the difference between the values of Tf (q) on the two edges of C is equal to (f' (q)) -. Let C be a uniformizer at q which is also a boun-

dary uniformizer for M. Let f in 91 correspond to the uniformizer

§ 6.1]

THE OPERATORS T, Z' AND S

183

circle I C I < a about q and write Jet' = 93t t1 f. Choose q, and q,,

to be points in f, in 931 and 9 - 9 respectively. We wish to consider the difference T, (qj) - Tf (qe) as q, and qe approach q. Let tt = (q, ) and t, = C (qe) and, for p e 9}2', z = C(p). By (4.11.1) and (4.1.5), we have for 5 e ', q' = q{ or q

2F (q',1) = &M (q', P) + regular terms 1

= x[C(q,)

- z]2 + regular terms.

Thus

51 (q', p) = 7S(x - C(q')) + f (q', P)

where e (q', p) is continuous at q. Also, PF(q', p) is continuous at q for P E 91- M Z-. Then Tf(g2) - Tf(g6) = (2'F(ga, p) -YF(ge, P), f'(P))%Z-vr

- 1 f [e (qa, P) -'(q p)) +2-

J

Lx

1 tt

x

1J

(/'(p))-dti

(f'(fi))-dz

where the scalar product over 932' has been evaluated according to (4.9.8). As q, and qe tend to q, the first two terms tend to zero since the integrands are continuous at q. The discontinuity of the last integral

can be evaluated by Cauchy's theorem to give (f'(q))-. In fact, the boundary of 01', considered in the r-plane, consists of a semicircle s and a segment y of the real axis. The integral along s also vanishes at q. In the integral along y, dz may be replaced by dz. The conjugate of this integral is 1

27ei

f

- ti -- x - toJ f' (z)dz. 1

1

Y

If we complete the path along s, we add an integral which vanishes at q, and the result may be evaluated by Cauchy's theorem, giving

INTEGRAL OPERATORS

194

(Cswr. vI

I' at the point 1. As q, and q, tend to q, this yields f(q) where f' (q) = d//d, C a boundary uniformizer at q. Thus for qi = qE = q, we have (6.1.10)

Tf(gt)-T,(qe) _


Sf1+r, = Sf1 -I- S,,2.

On the other hand,

S,,, = AT, + 2f.

(6.1.12)

Taking in particular A = - i, we have S-.f = i(Tf - 1'`f).

(6.1.13)

Also, assuming symmetry of Fvt, (T,1, f:) =

J ,a

(J2p(q, P)(f$(q))-dA,) (fo(p))-dA. ,a

(T,,,/). The interchange may be justified if we replace -TF by 1F. Similarly (but without requiring symmetry)

(f2'q

, fi)(fa(q))- dAtl fi(p)dA, ('1,,, fa) = f a Wt Finally (with symmetry), we find, by adding (6.1.14) and (6.1.15) and taking real parts,

(6.1.15)

(6.1.16)

Re {(Sf1, f=)} = Re {(S,,, fi)}.

Thus 1' is a self-adj oint operator in the Hilbert space with Dirichlet metric. On the other hand, T, and hence S, are self-adjoint operators

only if we consider a Hilbert space in which the metric is based upon the scalar product (6.1.17)

(j', g'] = Re {(f'. g')}.

SCALAR PRODUCTS OF TRANSFORMS

§ 6.2]

185

6.2. SCALAR PRODUCTS OF TRANSFORMS

The operators defined in Section 6.1 transform the Hilbert space F., which consists of differentials analytic in fit, into linear spaces of piecewise analytic differentials on R. In these spaces, the metric

is again defined by Dirichlet products over N. In the case of differentials which are discontinuous across the boundary of Dt,

the scalar product over 91 is the sum of the scalar products

over )t and the components of % - V. In this section we shall investigate two preliminary questions: first, to find certain metric relations among these spaces, such as orthogonality; and second, to express the metrics over OR of the transformed differentials

in terms of the metrics over W of the original differentials. We show first that (T1,,1''r,)at = 0

(6.2.1)

where f;, f= are any two differentials of the class F,vt on Wt. In fact,

by (6.1.3)' and (6.1.4),

(T1,,1`; )tt=--J f f l,(, p1)(f p1))-2'F(p2, q)(f2'(p2))-dA,,dAEsdAc

+f f J

2'F(q,p1)(/ (p1)) = 's

Since

- (lF(q, pl), 2F(q, fi2))t1t = (2'F(q, C1), 2°F(q, p2))sot

by (6.1.9), we have

f

{f.'F(q,

l p1)(.e`°F(q,ia))-dA}} (f'(p1))`(f2(p2))-dAX,dA

U in It

But .?F(q, p) is orthogonal to all differentials of the class F on 9t; hence

z (2F (q, Pi), 2F(q,Y2))8t = 0,

and this proves (6.2.1). Thus we have proved the following lemma: LEMMA 6.2.1. The T-transforms and P-transforms o t any two differentials on Wt are orthogonal with respect to the scalar product over 9.

Next, by (6.1.3)', (5.3.27) and (5.4.1):

INTEGRAL OPERATORS

186

T,)

= f ITIfartf lF(q, pi)(lF(q,

+J

{$J

2F(l,

[CRAP. VI

pa))-(fi(pi))-fs(P2)dAz,dA,j dAt

61)(2F(q, fia))tz(pz)dAi dA,JdAC

tm

W-2n

-' f f [lF((2, P1) + TF( 2, P01 (fi(pl))-fa{p2}dAdA: 9R 99

+ f f rF(y2,

- f f li

21

dASt

p1)(fi(p1))-fx(p2)dA,1dA$,

This gives: LEMMA 6.2.2. The scalar product over R of two T-transforms can be

expressed as a Hermitian form over ) with kernel 1: (6.2.2)

(Tf ,

T1,)8t = - f f lF(fi2, p1)(fi{pa)))-fz(p2)dAX1dAZ, Stmt

Even more simply we have, for the D-transforms, (2'11, r1,)8t=f (6.2.3)

j

-

f2n »tf

--WF(q,pl)(2'F(q,p2)) 1 (pl)(ff(p2))-dA-I dAa=}dAC

f f -rF(p8,

x)

dA,dA,,,.

We are now able to express the product of two S-transforms by integrals extended over 't. By (6.2.1), (6.2.2) and (6.2.3), (Sf1, S11)81 = (T11, T1,)81 + (i,1, 2*1,)81

_

-f f

lF(

2,

(6.2.4)

-- f f . 2 ' P ( 2 , I'1) fi (p1) (fa (p2)) -dA,1dA=1 a Wt

SCALAR PRODUCTS OF TRANSFORMS

§ 6.2]

f f LF(p1, fi2)(f1(L'1))

187

1t(p2)dA21dAz2

RE

+f

f

2) -- LF( 2, Pt)]

1z(p2)dAdAss

+m ant

f f [YF(P1> fi2) -YF(fi2> Pi)] (fi(p1))-fa(P2)dA,1dA,n

(6.2.4)

+f

f

-f

f (-VF(YV

'(P.)) 'C 2)dAXIdA Z2 )(f'(P2))-dA,1dA,2.

OR say

Now (6.2.5)

- ff L,(p1, fi3)(ff(p1))- f2(p2)dA.1dA,11 = (t in In

Further, by (4.1.6), (5.1.16), (5.1.16)' and (5.1.18),

f f [La(p1,

a)

-

LF(

2,

C1)l (f1(Y1))-fa(Pa)dA,1dA,lz

(6.2.6)

= 2i Z Im {a,,,}(dZ,,, df1)u((dZ,, df2)n)- = 0 since all terms (dZ,l, 0"-'L df{)n occurring in the above sum vanish for differentials of class F on V. Similarly

(6.2.7)

ff

s)

` 2i E Im {fl,,,} (d. /a, r-1

p1)] (/;(p1)) ,,, df1)sx

((di' d/2)vc)

The last two terms in (6.2.4) combine to give

2i Im Iff

fs(-bs)dA,1dA% .

Substituting thesest's results into (6.2.4)1 we obtain finally

INTEGRAL OPERATORS

188

[CHAP VI

(S11, S,,)gt = (T11, Tf2)a1 + (2'1l, Tfa)01

_

f1} - 2i

E Im {#,,v} (d-*',, d/1)9( (d-wv, dfE)m)

P, v=1

(6.2.8)

/, + 2i Im Iff YF(p1, YZ)(fl(P1))-f2(P2)dAzIdAz,

In particular, taking f1 = f2 = f, we have N1('1)

= N (df) - 2i E Im

(6.2.9)

t', n -1

(d°-,,,,

This formula shows that the norm of the S-transform is, up to a bilinear combination of terms (d 9t,,, d f )9R equal to the norm of d f .

We observe in particular that (6.2.10)

NS(T1) < oo.

Formula (6.2.9) becomes particularly elegant if YF(p, q)=58F(q,p).

In this case of symmetry, we have Im Pµ, = 0, so

N(df) = NN(df)

(6.2.11)

since df = 0 in R - fl?. Thus: LEMMA 6.2.3. If the class F., is symmetric, S, is a norm-preserving transformation.

Let A be any differential of the class F., df any differential of E. We shall prove the following lemma: LEMMA 6.2.4. The discontinuous T-transforms in R are orthogonal to all analytic differentials of F, and the scalar product 61 a Z'-trans form

in the metric o t R is a scalar product over D: (6.2.12)

(T,, dh), = 0,

('r, dh)

(dl, dh)9g.

In fact, (T,, dh)Ot

= f I J2(q. Wt

and

Ot

p)(h'(q))-dACI

(fo(p))-dA1

=0

THE ITERATED OPERATORS

§ 6.31 (

f, dh), = j, j

189

fYF(q, fi)(h'(R'))-dACI f'(P)dA.

= - f f'(p)(h'(b))-dA:

(dl, dh)m.

12

We point out that from the point of view of the Dirichlet metric in 91 the operators Sf = Tf + T. and S_sf = i (Tf - 2f f) are indistinguishable. We clearly have by (6.2.1) (6.2.13)

(Sfl, S,z) = (S_v1, S_s12)

We might equally well have defined Sf = T f - Df. Merely for definiteness we adopt the definition (6.1.5). 6.3. THE ITERATED OPERATORS

In Section 6.], various operators were defined for the classes F. which are one of our main interests in this book. However, since these operators carry differentials of t into differentials analytic on )I1 and on flR - 1t, it appears desirable to extend the domain of the operators T, 1', and S to such differentials on St. We extend each differential in F. into a piecewise analytic differential on 91 by defining it to be identically zero in 9t - V. Instead of formulas (6.1.3) and (6.1.4) we shall now use the following extended definitions, viz. (6.3.1) T,,(q) = (2F (q, P), f'(j))st, and (6.3.2)

Df(4)

R'))aj,

valid for all differentials piecewise analytic on R. Let Ex be the class of differentials which are regular in the interior of T't and in the interior of each component of R - fit, but which are not necessarily continuous across the boundary C of V. We suppose that the differentials of Evt have finite norms over 91. Particular subclasses Da of ER may be distinguished by supposing that the differentials of D,, form a corresponding class on St and on each component of 9R - V. In particular, if all the corresponding

classes are the class M. then D1 = E,,. We observe that all the

INTEGRAL OPERATORS

190

[CHAP. VI

results of Sections 6.1 and 6.2 remain valid under these more general circumstances. We remark that the class F,, is contained in every D,,; in particular the 2-transform of an arbitrary d/ e EVt also lies in every Dot. However,

a class F. will not lie in very D,, if F. contains a proper subclass which is also a corresponding class. The operators T and D transform

a class D. into itself, hence the importance of these classes in the theory of these operators. Now we can iterate our operations, and form operator products on differentials of D.: (T2(q))f = (YF(q, p), T1(/))rx,

(6.3.3) (6.3.4)

(D2(q))1 = (-eF(q, ), (fi1(p)) )K, (TT (q))1= (YF(q, P1(p))1R, (DT(q)), = (2F(q,.), (T,(p))-)9z

(6.3.5) (6.3.6)

and, therefore, we have also defined: (6.3.7)

S2= (T-I-r)2=T2+TP-f-I'T4 2rt2.

We shall prove, with no assumption of symmetry, that (6.3.8)

(SZ(q))1 = f'(q),

or, symbolically, (6.3.8)'

S2 = I,

where I stands for the identity transformation. Thus we have: LEMMA 6.3.1. S is an involutory transformation of D% into itself.

In particular, if f e F., this means that (S2 (q) )1 is equal to 1'(q) for q in l and equal to zero for q in 91 - V. This elegant result justifies our extension of the operators to the fit-domain, and forms the basis of significant further developments. Proof of (6.3.8)' turns essentially upon the formula (6.3.9) T2 = I -}- T which we now proceed to establish, by rather lengthy calculation. We assume at first that df is a regular analytic differential in the closure of t and also in the closure of R - SJ,)l. Then df can be written

as the sum of differentials df,, each of which is regular analytic in


§ 6 31

191

LUt and in each component of R - 9, and different from zero in one only of these domains. Since all the operators are linear it is sufficient to prove all necessary identities for differentials of the type dl t. Finally we shall be able to remove the restriction of analyticity

in the closed regions by a limiting process of the usual form. We prove first: LEMMA 6.3.2. For each differential .' (p) on T such that the class and for each F., is single-valued on the cycle '',, belonging to d f e F,., we have

T,)st = (f', ff,,)gr In fact, by (5.1.17), (5.1.17)' and (5.1.4) we have

(6.3.10)

(-°

(6.3.11) (2F(p, q),

,s(

'

))tJt= (-L

gl l p))St+ i &-p"'ft (q). e,Qs1

Using (5.1.20) and the fact that 2'M is orthogonal to all differentials

on %, we thus obtain (6.3.12)

q),

JA

Now we apply formula (6.3.12) and we derive the relation

(Z", Tf) =

f at

(6.3.13)

j f (°r"°F(pl, P2))-f'(P2)dAx } dA at

= fi' () (5.&v P2)(°- ,A(PO))-dAdAza = f/'(P2)((P2))-dA:,

This proves the lemma. Next, (6.3. 14

(T2(q))f = (.2F(q, p].), Tf(pl))8t

= (2F(pl, q),

Go

2i E Im p. v-i

d

)+x

v(q)

using (4.11.1) and Lemma 6.3.2. Further, 2'F (pl, q) - 2s (pl, q) is a differential of class Fqt, so by (6.2.12) we have (.2'p (p1, q) -.rs(p1, q), Tf(p1))et = 0

INTEGRAL OPERATORS

192

[CHAP. VI

or

(6.3.16)

q), T,(pi))st = (2108C61, q), T,(pi))st

Suppose first that f' vanishes outside 9t and is regular in the closure of 931, and that 932 is properly imbedded in R. Let C be the boundary of 932 oriented in the positive sense with respect to 931,

and let - C denote the oppositely oriented boundary. When the integration is over C we assume that the boundary values of the integrand are derived from 931; when the integration is over - C, we assume that the boundary values are derived from 3t - V. The boundary of 91 will be denoted by B. Let q be an interior point of 91t or of 9't - Tt and let f correspond to the circle 1 ! < a at q. Integrating by parts we obtain (3s(p1,q), Tf(pl))at=(2s(pi,q), T,(P1))V+(2s(p1,q), Tf(pl))st-sR

II ss(Pl, q) (Tf(pl))-dzi c

(6.3.16)

2i J Jrs(pv q) (T,(pi))-dxl

-c+s

+ 2z

us(pi) q) (Tf(pi))-d21 al

where the variable zl in the boundary integrals is a boundary uniformizer. Now (6.3.17)

2i

f

q) (T,(pi))-ddi = o(1) as a -* 0,

of

since T.(pl) is regular near q. By (6.1.10), we have (6.3.18) -2.. f ss(P q)(T,( 1))-dx1=-2i5Hs(1> . q)f'(pi)dxi=f'(q) c-c

c

by Cauchy's theorem, using the fact that f' (q) = 0 if q is in fR - 93't. In the integral over B, pl is a boundary point of 91, so that pl = i Then (f'(pa), YF(fi1, P2))st (T>(pi)) _ (f'(ps), F(t 1, pa))st = _ (f'(pa), (.'F(a, pi)) )st


§ 6.31

193

Thus

-2i f

Pa)dA,] A,

4)(Tr(pi)) a

a

at

f

f Ls

at

a

= f' (ps) [ 2i

= $t2) ar-OF{

t z,

'r)dzi] dA y

z, 4)dA,

by Cauchy's theorem;

= f 2' (q, fi2)f'(12)dA,2 + 2iK, +-1 Im {#,.n}(df, dXp)%Z'(q) G.

using (5.1.17) and (5.1.17)'. That is,

-

2i

f Es(PI, 4)(Tr(p1))-dz,. B

(6.3.19)

G

= 2r(4) + 21 Z Im {#,,,} (df, dZ,)st X,(4) M, V-1

Combining (6.3.14) through (6.3.19) we obtain (6.3.9). If Ut is not properly imbedded in fit, we let Mtt,, be a sequence of surfaces which approach X12 as p tends to infinity, Yt,, C 9,,+1 C V.

If q is a fixed interior point of D2 or of R - M, it is clear that the functionals T,,, P,, for the surface 9't,, approach the corresponding

functionals T, 2'f for the surface 9 and therefore (6.3.9) is valid for

Tt.

To remove the restriction that d t is regular in the closure of `t, we use the fact, proved in Section 4.13, that we can approximate dl, in the sense of the norm over 7t, by differentials 4,, regular in the closure of V. If q is a point of OTt, we have by (6.1.3)' (6.3.24)

{Tr(4) = -(IF(q, T,,(4) =

f' ('))Wt,

(lg(4, ),

Hence, by the inequality of Schwarz,

[Cu". VX

INTEGRAL OPERATORS

194

TI(q) - T.(q) 12

< f I IF(q,

f I IF (q, P) I2dAa

dAt }2

j2dAZ.

Since the right side tends to zero by (4.13.3), we see that T. (q) -* Tf(q)

(6.3.21)

uniformly in the closure of 0. Hence, in particular, f (Z,.(q) -T.(q) I2dAc - 0

(6.3.22)

In

as n tends to infinity.

If q is in the interior of 91 - 1, then T1(q) = (.'F (q, P)' /' P))IR,

(6.3.23)

IT. (q) = (2F (q, where . 'F (q, ) as a function of p is bounded. Hence (6.3.21) holds

uniformly for q in any compact domain interior to 91 - 0. Suppose that q is an interior point of 1, in which case

f

(T2(q))r-- (T2(q))* = - lp(q, p)(T,(p) -T.(p))-dA. a

+ 5p(q, K-V

By the Schwarz inequality

I flp(q, P)(T1(p) - T.(p))-dA,

a

f 1I1, (q,:0)12dA.}

a

f J1

and the right side tends to zero as n approaches infinity by (6.3.22) Let 3 denote a compact region interior to 91 - ?2. Given e > 0, we


§ 6.8] can choose

195

such that

f I PF (4,

1 adz < 62.

By the Schwarz inequality

f YF(4, P)(Tf(p) -- T.(p))-dA. f I rF(q, p) 12dA.

f I T,(p) - T..(p) I2 dA.} M-M-s `-ant e where, by the triangle inequality,

{ ft T,(p)-T.(p)I2dA.}i_{ f IT,(p)I2dA,}i+{ ft T,, (p) I2dA. {N9t(T,)}1 + By (6.2.9)

Ns(Tr) S NN(df) + 2 E I Im Since I

we have

(dZ, dl) v

(d. dt),I

M. .-1

(df°,., df)v l

{NN(d.° ,,)}i {NN (d f)}},

NN(T,) S ANN(df), where

A = 1 + 2 1 Im

{No(d.,,)}i{NM(d.',.)}i

is a number depending only on 91. Similarly Nst(T.) S ANN(df.). Since NN (d f.) tends to NN (d f ), we have }2:!g

{ f I T,(p) - Tn(P) 12 dA,

B,

where B is independent of n. Thus

f2F(q, P)(T>(P) -'ri(p))-dA, < Be.

INTEGRAL OPERATORS

[CHAP. VI

196

For fixed

,

f.p(q, p)(T1(p) - T.(p))-dA, --> 0 a

by the uniform convergence. Since e is arbitrary, it follows that (T2(q)),,. -* (T2(q)),,

for each q in the interior of 9)t. Suppose that q is a fixed interior point of 91 - 1'l. Then

f 2F (q, p) (TT (p) - T. (p)) -dA, -+- 0 9 as n approaches infinity, and it remains to show that

f2

(Tf(p) - T.(p))-dA, -4- 0.

Let J be a compact subdomain lying in the interior of 91 - TZ and containing the point q in its interior. By choice of we again have

f `PF(q, p)(Ts(p) - T.(p))-dA, I < Be. Let fb be the circle I C I < b at q. Then

f

p)(T,(p) --

e

f 2F(q, p)(T,(p) - T.(p))-dA,

-

l f F(X, p)(T,(p) - T.(p))`dR, 2i

where the integral over fb has been evalpated according to (4.9.8). Here the right side tends to zero by the uniform convergence. It follows that (T'(q))n -- (T2(q))i,

for each q in the interior of R -

R.


§ 6.33

197

It is obvious that 2''.(q) --- Tf(q)

for each q in the interior of R. Further, f;, (q) tends to 1'(q) for each q in R which is not on the boundary C of TZ. Since (6.3 9) is valid for f,,, it remains valid in the limit for each q in the interior of M or in the interior of t -- Tt. This removes the restriction that f' is regular in the closure of U. Finally, the same argument applies to the case where f' is difierent from zero in one of the components of 91 - 9, and (6.3.9) is therefore proved in general.

Next, we have the much easier identity

(T2(q))f = -Tf(q), or T2 =

(6.3.24)

This follows from (4y.11.7), since

T.

e F. In fact,

(T'2(q))f = (q,(p), 2F(E', q))81

Finally, it can be shown that (6.3.25) T1 = 2`T = 0. The first relation follows from (4.11.6). For the second we have (TT (q))f = (-PF(q, fi2), (Tf(p2)" )It

=

f Ut

=

f

3F (q,

2)

[f (2F (p2, ps)) -f'

dAdAz2

sx

[fq, 2)(YF(p2, pi))-dAs.1 dAi'i = 0,

since TF(p2, p1) is orthogonal to 2'F(q,Y2) by (4.11.6). Collecting these results, namely (6.3.7), (6.3.9), (6.3.24) and (6.3.25), we have

S2= (T + P)2=T2+TT -{-TT+T2

=I+.T+0+o D=I.

We summarize the relations between the operators T, D and S in the following theorem: TiEoREM 6.3.1. The operators T, 2f and S satisfy

T2=I+D, 2'2=-T, TP=DT=O, S2=I.

INTEGRAL OPERATORS

198

(CHAP VI

6.4. SPACES OF PIECEWISE ANALYTIC DIFFERENTIALS

The subclass of DW composed of differentials which are orthogonal to the differentials of FR will be denoted by OR. By (6.2.12) we see that T,, belongs to OR.

We show first that

DM=F,x+ On.

(6.4.1)

In fact, let a differential dh of Da be given We determine the orthogonal projection of dh into the space FR, that is, we determine the differential df of F, for which N (dh-df) is a minimum. The existence of such a differential dl is guaranteed by the completeness

of the Hilbert space. The difference dh - dl is clearly orthogonal to each differential dg in FR. This is geometrically obvious, and also

follows from the usual analytic argument, depending upon the minimum property of dl. Thus, if dg E FR, and e is an arbitrary complex number,

NR(dh-df -edg) =NN(dh-df)-2 Re {e(dh-df, dg)%} + 1 e 12 Not (dg) Z NN(dh-df), whence

(dh - df, dg)% = 0.

Thus dh - dl belongs to OR. It follows that D. C OR + FS. But it is obvious that DR D OR + F., so (6.4.1) is true. By T (FR) we mean the class of differentials Tf, dl e Fez, and T(FR) = 0 means that each differential of T(FR) is zero. We now prove that (6.4.2) T(F) = 0, P(FF) = FF, T(Olt) C Ost, T(OR) = 0. The first relation is a consequence of (4.11.6). By (4.11.7), we have

Df(4) = (/'(p), 2'F(p, 4))s1 = -f' (q), which proves the second relation. To prove the third, suppose that dg e Olt, d Fs. Then .

To) =

J

f, a}L J

= f9

) C5 ('((i-9F(p2,

$r))-

dA=s

p1))-f'(12)dA51]dA11 =

0,

THE VANISHING OF A DIFFERENTIAL

§ 6.5]

199

again by (4.11.6). Finally, for dg a O., we have DD(q) = (g'($), 3F(P, Mat = 0 since 2'F(P, q), as a differential in p, belongs to F.. This proves the fourth relation. Combining (6.4.1) and (6.4.2), we have (6.4.3)

T(Dot) C O01, D(DM) = Fat, T1(D) C T(Oot) C OR.

Now we are able to characterize the operators T and TT' by their projection properties in the space D.. In fact, let df be an arbitrary differential of D%; from (6.3.9), we have (6.4.4)

dl = Tf - Df.

Using the second and third relations (6.4.3), we see that (6.4.4) is just an explicit representation of the orthogonal projections of df into F91 and Ow. Thus T2 and - T appear as the projection operators

of an element of DR onto the spaces Os and Fe, respectively. In particular, if dl e O., then Tf = 0 so (6.4.4) becomes 7' f2 = df

or

T2(06t) = OR.

But by (6.4.2), T2(09t) = T(T (O31)) C T(Ost) C O.

Thus (6.4.2)'

T(O,)=Olt

and hence (6.4.3)'

T(Dst) = O.

6.5. CONDITIONS FOR THE VANISHING OF A DIFFERENTIAL

We now consider the subclass On of F. consisting of elements which are orthogonal to all elements of Fft. We remark that the class

F,, is non-trivial, since M has a boundary. Since 'F (P, q) belongs to F%, every differential df of the class O. satisfies (6.5.1)

fif(q)

0, q e R.

INTEGRAL OPERATORS

200

[CHAP. VI

In fact, by (6.3.2), (6.5.1)'

,(q) _ (f'(p),

4))rs = 0.

For d f e F,,t we write (6.5.2)

q), f'(p))'%.

J,(q)

It is clear that J,(q) is an analytic function of q in 91 - 9t. For

qe 2, (lF(p, q), f'(P))vt

Jf(q)

so that Jf(q) is also regular in the closure of 9?. For dl e O. and q on the boundary of R, so that q = q, we have Jf(q) = 0 by (6.5.1)'. By analytic continuation, we then have (6.5.2)'

Jf(q) = 0, q e 01 - 2, df a On.

Thus the elements of O,, satisfy the conditions (6.5.1) and (6.5.2)'. We prove next: LEMMA 6.5.1. It dl is in Owt, then (T2 (q)) f is regular in the closure

o/Tt. For every d f e F., we have (6.5.3)

(-29F(q, p), f'(p))m = - (lF(q, and hence this expression is regular in q throughout the closure of V.

In particular Tf(q), considered for q in 9N, is in F.; thus (IF(q, p), Tf(p))9t is regular in q in the closure of 't. If d f e On, then for q e slt - 2

we have by (6.5.2)' Tf(q) = (2F(q, p), f'(p))at = (2F(q, p) -.F(p, q), f'(p))at (8.5.4)

= 2i £'

(q) (d. ,, df)at

R. 9 -1

using (5.1.17), (5.1.17)' and (5.1.18). Moreover

(655)

(Q'z?(q,p),

= (-WF(q, p), Q(P))jt (LF(q, p),

(p))a + (lF(q, p), Ze(p))0 and these terms are regular in the closure of V. This is obvious in the case of the last term. Using (4.10.8) and the formulas of .

§ 6.5]


20i

Section 5.1, the first term is G

( Pve ffK (q) E #,,, p. v=1

while the second is G

Z

Z,.(q)

L,v=1

and these expressions are regular in the closure of 9l. Using (6.5.4) and (6.5.5), together with our remarks concerning regularity, we

see that (3F (q, P), T1 ))st-m

is regular in the closure of 9)t. Thus (T2(q))f = (.F(q,P), Tf(p))M + (3F (q, p)' Tf(p))%_got is regular in the closure of 1J1. This proves the lemma. By (6.3.9), we have (6.5.6)

(T2(q))f = f'(q) + l'f(q) However, for df e O0, Zf - 0 by (6.5.1), so (6.5.6) implies (6.5.6)'

(T2(q))f = f'(q) Combining (6.5.6)' with the preceding lemma, we have LEMMA 6.5.2. Each differential of the class On is regular analytic in the closure of 9)t. Suppose that 9R is properly imbedded in R. Since

Jf(q) - Tf(q) = 2i E Im

df) ot T,(q)

is. vat

is continuous as q crosses the boundary of 9t, and d t is regular on the boundary of 9)1, it follows from (6.1.10) that

Jf(g8) -Jf(g.) = Tf(q) -Tf(q.) = (f'(q))-, where /'(q) = df/dC, C a boundary uniformizer at q. But by (6.5.2)', Jf(ge) = 0. Thus 1. (q) has in 9t the boundary values (/'(q))-. Therefore the differentials f' (p) + If (p) and (f' (p) - Jf (fi)) li, which are regular everywhere in the closure of 9)t, are real on the boundary and we have

INTEGRAL OPERATORS

202

(CHAP. VT

a

a,. Teal,

(6.5.7)

v-1 G

Jf(p))/i = E b,, Z,, (P), b, real.

(6.5.8)

P-1

Combining these equations we obtain G

If Fm is the class S, we conclude immediately that

0 by

Lemma 4.8.1. 0 If Fm is not the class 8, then we can conclude that if the corresponding classes and the imbedding are such that it is

possible for a differential of F. to have non-vanishing periods on any cycle of t not included in the period restrictions for F.. In fact, let

Fe = Fm (i1, i9, , i=) where 0 S I < G. Given any cycle K, of Tt, v 0 i1, , it, there is an integral W, of the first kind such that dW, a F9t and P(dW K,.) (6.5.10) (dW,, dZ,.)

On the other hand, each dW, is in Fvt and therefore orthogonal to

die O. By (6.5.9) and (6.5.10) we have G

(6.5.11) (dW,, d1 )m = E e 81N = c, = 0, 0-1

so T =Zc;T Zfr

(6.5.12)

v

i1, ia, ..., i{,

)

But df e F., so we have 2

(6.5.13)

(dl, dZ,,)m = E c'..F',{ = 0,

v = 1, 2, .. ,1.

T-1

Since the matrix I'{s4 11 is non-singular because of the proper imbedding hypothesis, it follows that ci, = 0, T = 1, 2, - ,1. Hence

For the existence of the differentials dW, e F we must suppose that the class F. is the minimal corresponding class on XR, i.e., that FIR contains no proper subclass (with period restrictions on additional

§ 65:

THE VANISHING OF A DIFFEREATIAL

203

cycles K,.) which is also a corresponding class. In addition we suppose

that the imbedding is essential (see Section 5.1). Then we have the following

LEMMA 6.5.3. Let 1R be properly and essentially imbedded in R, and let Fa be the minimal corresponding class on fit. Then the only element o l FE which lies in 0. is the zero di f f ei ential.

We remark that the conclusion f' (p) = 0 remains valid under the weaker assumption that any boundary component C,, of t which is homologous to zero on 91 but not on TZ be a cycle K4 associated with the class F. However, some hypothesis of this nature is essential. For consider the case in which F. = M, and suppose that there is a boundary component C,, of D1, which is homologous to zero on JI but not on V. Let ZZ,,, F, be the differential of the first kind on 0 corresponding to the boundary cycle C. Then

(6.5.14) l(q) = (Zzn+, (p), 2M (p, q))sot=-(P(-TM(p, q), 0 since C,, is homologous to zero on R. Similarly, for q e OR - 9A, (6.5.14)' J(q)=(2M(t", q), Z;h+,,(P))U = - P(2M(p, q), C,) = 0.

Hence, for the particular differential Z;,a+,(p) of class M on fit, 2'(q) is zero for q e 91 and J(q) is zero for q e JR - 7t, but Zi +µ(p) is not identically zero. We observe that the equation (6.5.1) characterizes the class 0,., that is, if rf(q) vanishes identically for df a F., then df e On. For let dh be any differential of class F.. Then by (6.2.12), (df, dh),n = - (2'f, dh)vt = 0. (6.5.15) The same conclusion follows if we know merely that 2'f vanishes in SJR - R, since T. is analytic in #R. Thus by Lemma 6.5.3 we have LEMMA 6.15.4. Under the hypotheses of the preceding lemma, it 2, vanishes in DR - t and df a FV, then df vanishes identically.

Let us make an application of this lemma to properties of the norm of an operator. If F,, is symmetric, we have by (8.2.11) (6.5.16)

NN(Sf) = Net(Tf) + NR(1'f) = NN(df).

From this identity we conclude that N,(df) NN(T,), NO(dl) k Nvt(Tf) (6.6.17)

INTEGRfL. OPERATORS

204

[CH ?. VI

If equality occurs in either relation, then T, = 0 in R - V. This implies LEMMA 6.5.5. Let 1l and FM satisfy the conditions of Lemma 6.5.3,

and let F,, be symmetric. Then for any non-zero differential of F., we have the proper inequalities NV(df) > NN(Tf), NV(df) > Ngn('f). Let us derive a similar result for the S-operator. We make the

same assumptions that were made in the preceding lemma. By (6.2.11) we have Ng(Sf) 0

for each non-zero dj e Fa. We now consider the possibility that

Nt(Tf)=0 for every d; e F.. This can occur if and only if (6.5.28)

lF(p, q) = 0 in X12.

In fact, for df = IF (P, q) E F. we have, for q EO11, T,.(q) = - Arwt(IF(p, q)). An example in which (6.5.28) holds is provided by the case in which

Dl is a circular disc concentrically imbedded in another circular disc 91.

In the case (6.5.28) we have S,, = P in 13I since Tf = 0. Thus, for example, the set of possible exceptions (6.5.24) to (6.5.19) must be empty, by Lemma 6.5.5. On the whole, the effect of (6.5.28) is only to exclude certain possibilities which may occur in the general case.

if either F. = S or F. = S we can show that (6.5.28) is possible only if X12 is simply connected.

Suppose that FM = S. Then, since the class F. is assumed to be symmetric, OR must be of genus zero. In this case the function

?,s(p, q) = 52s(P, q)dp is schlicht and maps OR onto a plane surface with maximal external

area (cf. Section 4.12). It follows that rs(p, q) -A 0 at each point p E OR. Now let C be the boundary of M. In the case (6.5.28) we have

for p, qeC -108 (P, q)dp dq

= Ls (p, q)dd dq = real.

Let q be a fixed point of C; then Im {.'- 8 (p, q) } = constant

on each component of C. Therefore Es (P, q), which has a simple pole

at P = q, maps R onto a parallel slit domain where each boundary component of C goes into a straight line parallel to the real axis. The boundary component of C which contains q is mapped into a

§ 6.5]


207

line containing the point at infinity, while the other components are

mapped into finite segments. But -Ts(p, q) must vanish at those points p e C which correspond to finite endpoints of these segments. 0 in R and 91 is properly imbedded in fit, the only Since -mss (P, q) possibility is that ))l is mapped onto a half-plane. Thus )t is simplyconnected. After a linear transformation we may suppose that 91 is the exterior of mo Jordan curves B1, -, B,,,o on the z-sphere and that ])1 is the exterior of a large circle which encloses the boundary curves of R

in its interior. Under these circumstances we have 1

and therefore a(Z

The boundary curves Bµ of 9R are characterized by the differential equations dzdg (z - C)2

=real,

x,CeB.

Integrating with respect to z and C we find that z1-C1 x2-b2=real (6.5.29)

p

z1 - b2

x2 - ,r1

provided that z1, z2 lie on the same component Bo and C1, C2 on the same component B,. From (6.5.29) we concluded that each boundary B. must be a circle through z1 and z2, i.e. that SJR is bounded by a

single circle. Thus, in the case F., = S, the hypothesis (6.5.28) implies that the imbedding of 4 into R is conformally equivalent to

the concentric imbedding of one disc in a larger one. Assume next that F. is the class S. Then l must be of genus zero. In this case the function !s(P, q) = JLS(P, q)dp is schlicht on ,Tl and maps it onto a subdomain of the sphere with

maximum external area. Hence Y8(p, q) is schlicht in a slightly

INTEGRAL OPERATORS

208

[CHAP. VI

larger domain fi't'. Under the hypothesis (6.5.28), the function 5g(P, q) = J2(p., q)dd coincides with 1 S (P, q) in )2 and therefore in fi't'. In particular, it follows that YF (', q) :7-1 0 for fi a fit' and we conclude as before that !1l must be simply-connected. 6.6. BouNDs FOR THE OPERATORS T AND We suppose that the hypotheses used in Section 6.5 are satisfied, namely: IR is properly and essentially embedded in fR and

3F (p, q) = 2' (q, P) and F. is minimal. We remark that if Fe = S, the assumption concerning essential imbedding may be dropped. We recall that a differential dl of F. is defined to be zero in 91- Ul, and therefore a differential of F,, is transformed by the operators into a differential of E,,, and not necessarily into a differential of F.. We may, however, define a new differential which coincides

with the transformed differential in 't, and vanishes in 91 - St. This differential belongs therefore to F.. In this way we define new operators tf(q), If (q) and sf(q), which have F,,, as domain and range.

Thus, foi instance, 4 (q)

f Tf (q),

q e T Z,

In the following formulas we shall be concerned with operations in the domain It only. Therefore norms and scalar products without

subscripts designating the domain of integration are assumed to be taken over V. Let the non-zero differentials of F., be normalized by the condition that N(df) = 1. (6.6.1) We shall prove the following theorem: THEOREM 6.6.1. There are real numbers R and e, each greater than

unity, such that (6.6.2)

N(tf) S

(6.6.3)

N(if)

2

§ 6.01

BOUNDS FOR THE OPERATORS T AND 2'

209

There is equality in (6.6.3) for at least one differential of F. satisfying (6.6.1). The same statement applies to (6.6.2) except in the case (6.5.28).

Thus t and I have bounds which are less than one. The proofs of (6.6.2) and (6.6.3) turn on the fact that, for q e V,

tr(q) = - (lF(q, $), f'(P))wt (6.6.5) £r(q) = (-TF(q,.), (f'(P))-)0, where lit (p, q) and 2°F (q, fi) are regular in the closure of TZ. The (6.6.4)

regularity in )l of 3F (q, fi) is a consequence of the proper embedding. Let {df,} be a sequence of differentials of FV satisfying the normalization condition (6.6.1), and suppose that df, converges pointwise

to a differential d t as v becomes infinite, the convergence being uniform in any compact subdomain D1' lying in the interior of M. Then NW(d f) = lim Nom. (d f,) S 1.

Letting Vt' tend to fit, we see that NVt (df) = N(df) S 1.

(6.6.6)

Thus the limit differential also belongs to the class Fa, with norm bounded by unity. Given e > 0, let V' be chosen such that, uniformly in q, JIZF(q, 1) 12 dA, < E_

(6.6.7)

Writing tt,(q),

we have, for q e V, I tf(q) -t,(q) I S f I lF(q,

dA.

W

(6.6.8) -I-

f

dAs,

M-W

By the uniform convergence there is a number vo = vo(e) such that (6.6.9)

f ( lF(q,1) a,

I

I /'(P) - f,(P) I dA, < e

INTEGRAL OPERATORS

210

[CHAP. VI

if v z vo. By the Schwarz inequality and the triangle inequality r + IF (q, P))

f dA,

a-W

S{f Ilg(q,')

dA,}l j f I /'(p)-f,(P) 12dAa }

(6.6.10)

f (lF(q,

12 dAx

} { [ f f /'(b)

12 dAzI1

+ [ f { f; (P) 12 dAt]J } S 2s, art-a,

by (6.6.6) and (6.6.7). Hence by (6.6.8), (6.6.9) and (6.6.10) we have,

for q e 9, (6.6.11)

t,(q) - t, (q) I < 36,

provided that v k vo. Thus t,(q) converges uniformly to tf(q) in the

closure of 1t and, in particular, (6.6.12)

N(t, -- 4) - . 0, N(4) --s N(t,),

as v becomes infinite. The same conclusions are valid for If. We remark that the above reasoning depends in an essential way on the fact that iF (q, p) and Yr (q, fi) are regular throughout the closure of DI. This is illustrated by the following instructive example.

Let {dpi,} be a complete orthonormal system for the class F., and write

I,(q) =-9i, (q) =

fa

We have

;(q)12 1 such that N (bd f) = 1

and we would have N(tf) = bed > d, a contradiction. Taking A2 = 1/d, we obtain (6.6.2). The case d = 0 corresponds to (6.5.28). In this case, the inequality (6.6.2) holds for every choice of A > 1, but there is never equality. A precisely similar argument gives (6.6.3). In this case, we have

d > 0 by (6.5.27). For differentials which are not normalized by the condition (6.6.1), the inequalities become (6.6.2)'

N(tf) S

(6.6.3)'

N(ff) S

N (df) A2

N(df).

e

2

INTEGRAL OPERATORS

212

[CH P Vt

6.7. SPECTRAL THEORY OF THE i-OPERATOR

Two types of convergence of sequences of differentials have been

considered, namely, (a) pointwise convergence of a sequence dl, which is uniform in each compact interior subdomain of X12, and (b)

convergence of the elements df, in the metric of the Hilbert space Fs. It is also useful to interpret the first type of convergence in terms of the concepts of Hilbert space. Let d(p be an arbitrary element

of this space. From the first type of convergence of the sequence df, to a limit differential dl we immediately conclude that lim

(df, 4) = (dl, 4).

This type of convergence of {df,} to an element df of the Hilbert space is called weak convergence. The significance of the formula (6.6.12) may then be expressed as follows: the operators tf and 1f transform each weakly convergent sequence in the Hilbert space into a properly, or strongly, convergent one. This property of operators was called by Hilbert "complete continuity" and a large part of his theory on forms of infinitely many variables was based on this concept. By the reasoning of the last section, one can show that a completely continuous operator is always bounded.

In the preceding section we proved that the operators tf and If, and therefore also s f, are bounded. The general theory of Hermitian

linear operators in Hilbert space guarantees the existence of a discrete spectrum of eigen-values for the operator If. By similar reasoning, we would be able to derive the same result for the other two operators. However, in order to make our representation of the theory self-contained, we shall build up the spectral theory for all three operators from more elementary concepts, making use of the particularly simple properties of these operators. In this way we are also led to certain interesting realizations of the abstract Hilbert space.

We suppose that the assumptions made at the beginning of Section 6.6 are satisfied, and we investigate the eigen-values and eigendifferentials of the following three integral equations for differentials

dl (q) of F.:

§ 6.73

SPECTRAL THEORY OF THE t-OPERATOR

(6.7.1)

df(q) = atf(q),

(6.7.2)

df(q) = el>(q),

(6.7.3)

df(q) = as, (q).

213

We begin with the eigen-value problem (6.7.1), andweobserve that (6.7.4)

tr(q) = (2'F(q, P), /'(p)) MI

(6.7.5)

(t2(q)), = (2F (q, P), tf(p))a (t2(q)),,

(6.7.6)

=-

= f f IF (q, PI)(IF(f I, ps))

an

f st

QF(fi2,

(IT (q, P), (IF (q, P), tl

f'(Ps)dA.,.dAx,

q)f'(ps)dA.,

where by (5.3.20) and (5.4.1) (6.7.7)

QF(fi, q) = f lF(r, q)(lip (r, j))-dA,, _ -lF(fi, q) -TF(fi, q). Mt

Here 27 is a local uniformizer at r. This shows that the iterated toperator is Hermitian. We have from (6.1.14) that (t,, dg) = (t, , df)

(6.7.8)

From (6.7.8) we observe that the t-operator is not self adjoint in the Hilbert space with the norm (d/, dg). For just this reason we introduced in Section 6.1 another scalar product, namely [df, dg]

= Re {(df, dg)},

with respect to which the operator t is self-adjoint. We remark further that in the new metric based on [d f, dg], the norm of each differential

is the same as before. We shall now prove the existence of eigen-differentials and eigen-

values of the integral equation (6.7.1) by considering extremum problems in the Hilbert space with the metric [df, dg]. Suppose the first v eigen-values A,, - - -,.I, and an orthonormal set of eigen-differentials d94,, (6.7.9)

-, d9p, are known. They satisfy

dVt(q) = 2xtq,(q).

INTEGRAL OPERATORS

214

[CHAP. VI

Consider now all differentials dl of F,,, measured by the scalar product [d/, dg]. Since [d f + Adh, d f + Adh] = [d f, d/] + 2Re {A (d f , dh )} + A 12 [dh, dh],

we observe that these differentials form a Hilbert space if multiplication with real numbers only is permitted. For in this case the above identity takes the form [dl + Adh, df + Adh] = [df, dl] + 2A[df, dh] + As [dh, dh].

We are thus led to a "real" Hilbert space Fe in which linear dependence is defined in the real sense. Each non-zero differential df of the Hilbert space FsR leads thus to two independent elements dl and idf of F.. In particular, setting df_,r = i d4p,; and A_,, = - AA;, k == 1, 2, , v, we have an orthonormal set of 2v eigen-differentials in F.. We remark that a differential d t which is orthogonal to both d9,,, and dp-,, in F, is orthogonal to d94,r in F,,, and conversely. Among all differentials df a F., satisfying the side conditions

N(df) = 1, [ d f , dipx] = 0, k = ± 1, ± 2, ..., ± v, there is at least one, say dl = dgp,+1, which maximizes (6.7.10)

[df, tf] = Re { (df, tf)}.

(6.7.11)

We remark that max [df, tf] = (d4',+1, t,,+1) Z 0. For if dl satisfies the conditions, so does the differential a °d f, 0

real, and we have (6.7.12)

(e'Bdf, ;i°f) = e2 B(df, tf).

Given any admissible dl we can, by choice of 0, always make (6.7.13)

[eiedf, ;,ef] = Re{e2'°(d f, t,)} Z 0.

Moreover, if (d f, t,) # 0,

we can choose 0 such that there will be inequality in (6.7.13). Further,

a maximizing differential dp,+1 must satisfy 141+11

1911+11 = (d,+1, tp,+l),

SPECTRAL THEORY OD THE t-OPERATOR

§ 6.71

for if Im {(dg,,+1, tq,,,+1)}

215

0 we can choose 0 such that, for df =

e'8dgp,+,, the value [df, t ] exceeds [dqi,,, tg,,+,]. On the other hand,

we observe from (6.6.2)' that (dl, t,) 1 ;5 {N(df)}1 {N(tf)}I s N(df)

(6.7.14)

where A > 1. Hence

0 S max [dl, tf] S

(6.7.15)

A > 1.

Writing

)_

max [df, t>] =

(6.7.16)

1

we therefore have 1 < A,+s S oo.

(6.7.17)

Assume that A,+i < oo. Let dp, be a differential of F. which is orthogonal to dpi, dr, , dpi, and let dg = dq,,+, + edq,. Then [dg, t,] =

1

+ 2Re{e(dq,, t9"+1 )} + Re{e2(dgq, t9,)},

and it follows that [dg, t.]

_ A.+i +

N (dg) -'

Re{e2(dp,, tq,)}

1 + 2Re(e (dp, d97,+,)) + 1 e 12 N (dp) 1

+ 2Re{ef(dq,, tq'.+i)

Al

(dp', dp'.+i)]} + O((e12)

Since a is an arbitrary complex number, we have (6.7.18)

(4, dgq',+r) = A.+i(dq', t9,

),

or (6.7.19)

(d9'' dk,+l --- A,+3L try+.)

From (6.7.19) we conclude that (6.7.20)

d4'.+, =

if,+i

0.

INTEGRAL OPERATORS

216

[CHAP. VI

In fact, the differential df.+l = dg7v+l - A,+1 t,,r+1

is orthogonal to all differentials dqq which are orthogonal to dgil, - , dr,.

We can also show that df,+1 is orthogonal to all d pe, e = 1, 2,

, v.

By (6.7.8) we have (df,+1, dqe) = (dT,+1, dTe),A,+1(tq e, dW,+1).

Hence, by (6.7.9), (df,.+1, dq'e) _

0.

Since the space F., can be decomposed into the linear space of the dVe, e = 1, 2, , v, and its orthogonal complement, df,+1 is orthogonal to every differential of F,., and hence is identically zero. Thus (6.7.20) is proved. It is clear that dp_,_1 = I dv,+1 minimizes (6.7.11) under the side conditions (6.7.10) and corresponds to the eigen-value 1_,-1= - A,+l Further d4T,+1] = Re {iN(dr,+i)} = 0;

that is, the differentials dqk, k = ± 1, ± 2, - . -, ± (v + 1), form an orthonormal set in F.. The differentials d4pk, k=1,2,---,v+ 1, form an orthonormal set in F.. Beginning with the lowest eigen-value Al, we can construct a sequence of eigen-values and eigen-differentials by repeating the above maximum problem. In this way, we obtain sequences dVk and A. The k-th eigen-differential satisfies the equation -Ak-4 (q) = f IF (q, p) (91k (p)) -dA.,

Therefore the k-f h Fourier coefficient of the differential IF (p, q) 1ti-iLh respect to the orthonormal system {dpk} is - dqk/Ak. By formula (2.3.23) (Bessel's inequality) (6.7.21)

Zr

mk

) $ S f ( IF (p, q) J2dA,.

1 6.7]

SPECTRAL THEORY OF 1HE t-OPERATOR

217

Integrating (6.7.21) over TZ, we find that (6.7.22)

Aa < f J

lF(P, q)

11

It dA, dAz.

SDt D2

Hence, if there are infinitely many eigen-differentials, then At tends to infinity with k. In particular, every finite eigen-value is of finite order.

Suppose now that there exists a differential df of class F. with norm unity which is orthogonal to all the eigen-differentials. Then (6.7.23) [dl, tf] = 0. In fact, if there are infinitely many eigen-differentials,

1S

Ak

[df, tr] S

-

Ak

,

k = 1, 2, ...,

and (6.7.23) follows. If there are only finitely many eigen-differentials, (6.7.23) is still true; otherwise we could define a further eigenvalue and eigen-differential by our maximum process. Since eief, 0 real, also has a norm equal to unity and is orthogonal to all eigendifferentials, we conclude from (6.7.12) and (6.7.23) that (df, tr) = 0.

(6.7.24)

Now (6.7.25)

(d4v,, t1) = (dl, tg,.) =

(dl, dip.) = 0

by hypothesis, so tt is orthogonal to all the eigen-differentials. If d f 1 is any other non-zero differential which is orthogonal to all eigen-

differentials we have

0 = (dfl + dl, tsi+>) = (d/1, t,) + (df, t,) + (dfl, ts) + (df, ts) = (dl, tf) + (dfl, it) by (6.7.24); = 2 (d/1, tr)

by (6.7.8). Hence (6.7.24)'

(dfl, t).) = 0.

The orthonormal system {dpk} of eigen-differentials can be made

218

INTEGRAL OPERATORS

[CHAP. VI

complete by adding an orthonormal set of differentials d/1, d/2,

.

which are orthogonal to all the eigen-differentials. By (6.7.24), (6.7.25) and (6.7.24)' we see that t1 is orthogonal to all differentials of

the complete orthonormal system, so t f(q) = 0.

(6.7.26)

Thus any differential d f which is orthogonal to all eigen-differentials

may be regarded as a solution of the integral equation (6.7.1) corresponding to the eigen-value A = oo. By admitting infinite eigenvalues, we may therefore suppose that the set of eigen-differentials forms a complete orthonormal system. In the case (6.5.28), the equation (6.7.26) holds for all d/ and any

complete orthonormal system in F, belongs to the eigen-value A = oo. The kernel lF(p, q) of the integral equation (6.7.1) may be expressed

in terms of the complete orthonormal system {dgg,} of the eigendifferentials, and its k-th Fourier coefficient is - d44k/Ak. Thus (6.7.27)

q) _ - ,-1 E A, d4,(P)4,(q) f1

Substituting from (6.7.27) into (6.7.7), we obtain (6.7.28)

QF(, q) = QF(q, fi) = E

Iterating the equation (6.7.9), we have (8.7.29)

d ,(q) = A'.(i (q))q, = A2,

J

QF(q, fi)P.(()dA,.

got

In the usual notation we may therefore write lF(q, fi) -- TF(q, )) QF(q,,) = l ) (q, (second iterated kernel). From (4.9.6)

(6.7.30)

(6.7.31)

LF(p, f)

It is thus natural to define

,-1

SPECTRAL THEORY OF THE t`-OPERATOR

§ 6.8]

(6.7.32)

219

`GF)(L', q) _ -LF(f', q), QF)(p, q) = QF(p, q),

and (6.7.33)

QF") (p, q) = f QF "-2) (j5, y)QF (r, q)dAn,

y > 1.

Mt

Then (6.7.34)

Q1,

(P, q) = E

a- d9'g(P)(d9,,(q))-,

,u = 0, 1, 2,....

The formula (6.7.34) gives the development of the iterated kernels

in terms of the eigen-differentials. In the case ,u = 0 we obtain the development for the kernel q). It is interesting to observe that in the general theory of integral equations the developments (6.7.34) for ,u = 0 and µ = 1 do not in general converge. The Hilbert spaces of differentials with which we are concerned are spanned by orthonormal systems whose convergence properties are of an especially

simple type. 6.8. SPECTRAL THEORY OF THE i-OPERATOR

We come now to the integral equation (6.7.2), and we have (see Section 6.6), m

(6.8.2)

(12(q))1 = f YF(q, fi)li(p)dA..

a

For differentials of class F,a, If(q) _ D,, (q) but (12(q) )y will not generally equal (!D2 (q)) f. We have

fJ. (6.8.3)

where

2F(q,.i)2'F(p,-Y)f'(r)dA,dA,,

f PF(q, P) f'(r)dA,,,

INTEGRAL OPERATORS

220

Pip (q,

) = f -TF(r,,i)(-TF(r,

[CHAP. VI

q))-dA,7

= f 2F(r,fi)(2F(r, q))-dA,r

-f

)(2F(r,

q))-dA,,.

Now

J

.11(r,:)(2F(r, q)) -dA, _ -2'(q,).

ot

and

f tF(r, j3)(2'F(r, q))-dA,, = Fi,(q,l). at-V

Thus (6.8.4)

PF(q,

Y1, (q,

) - rF(q,

)

If we wanted to define an operator If by applying the to t,, we would have the choice of setting

-operation

If= (2'F(q,$), (t'(P))-)M or

If =

-

(lF(q,f), (f'(p))

)sR

By our definitions of Section 6.6 we have committed ourselves to the

first of these two possible definitions. This choice has the decisive advantage of providing us with a completely continuous operator I,. This property does not hold for the other choice, since lF(fi, j) is not regular for q and P in the closure of V. As we have already remarked, in Section 6.1, 1' is a self-adjoint operator in F. The same clearly holds for If in F., hence we shall

be able to construct a spectral theory for the I, operator in FF, where the scalar product is (df, dg).

,-e

Assume that the eigen-values - Pi, - es, and eigendifferentials dtpi, d 2, , dip of the integral equation (6.7.2) have

SPECTRAL THEORY OF THE T-OPERATOR

§ 6.81

221

been defined, and consider all differentials d t of F. for which N(df) = 1, (df, dipk) = 0, k = 1, 2, , v. (6.8.5) Among these differentials there is one, say if = dip.+1, which maximizes the non-negative expression (compare (6.2.3)) :

- (df.1,) = - f f3r(P. q)(/'(p))- f'(q)dAdA

(6.8.6)

as Writing

max { - (dl, I,)}

(6.8.7)

(dw.+v 1v,+1)

we have 1 < e.+1 S ao,

(6.8.8)

since

(df, I,) I S {N(df)}1 {N(If)}* S

1 Q

where e > 1. Suppose that B.#-1 < oo. Let d4 be a differential of F,., which is orthogonal to dy1, die, . - , dip, and write dg = dip,+1 + edge. Then

- (dg, Ig) =

1 e'tI

- 2Re {e (d9,1,_ ,)} - e i! (d4, 1'F),

since (6.8.9)

(If, dg) = (df, 1,)

by j6.i,15). Hence -(dg, 1.) N(dg)

- 2Re {e (dye, iv,+1)} - I e 12 (dye,1v) +ML

1 + 2Re {e(dp, dip.+1)} + I e I2N(d44) e1+1

- 2 Re i e [(d p, 1r ) + +1 t

1

(d4', dV.+l)]JJf

Lov+l

Thus (6.8.10)

(d4', diV.+l)

e.+i(4, lv,+i),

+00812)

INTEGRAL OPERATORS

222

[Cxwr. VI

or (6.8.11)

(dq, d+ ,+1 + &+11'0.

+1

) = 0.

We conclude, as in Section 6.7, that d y,+1(q) = - @.+l1,0y+1(q)

(6.8.12)

Repeating the above maximum problem, we obtain a sequence of eigen-values - N,, and eigen-differentials dzpk. By arguments which

are entirely similar to those used in Section 6.7 we see that < oo. ek

In this case the set of eigen-differentials is complete. In fact, if dl

is orthogonal to all the eigen-differentials, then 1, = 0 (compare (6.7.26)) and hence, by Section 6.5, df vanishes identically. Since an eigen-differential dip,, satisfies

dVk=-Qk"Irk in ER, we see that dip,, can be extended to a differential of FS. Moreover,

f f 2F(q, p) ,a(p)(V,(q))-dAxdAC

(dV,s, dy,)%

% 29 (6.8.13 )

=PJ

µ (P) (y', (P)YdA= _

(dy'v,

We obtain in the dp,, a system of differentials on t which are orthogonal both with respect to integration over 1't and over R. We call the {dV,,} a doubly-orthogonal set with respect to the domains

Ot and V. These sets play an important role in the theory of continuation of a differential of F. over R. We have: THEOREM 6.8.1. Let dl be a differential o l the class FU and let

f = E a,, V,, µ-l

be its Fourier development in terms of 'the eigen-differentials

A

SPECTRAL THEORY OF THE s-OPERATOR

§ 6.9]

223

necessary and sufficient condition that df e F. is the convergence condition E eµ I a,,11 < Co.

(6.8.14)

µ-1

In fact, the differentials 1

dip,* =

dy,,

form an orthonormal system in F., and (6.8.14) is the Bessel inequality

for differentials of this class. The kernel 2°F(q,, ) of the integral (6.8.1) may be expressed in terms of the complete orthonormal system {dVti} of the eigenThus differentials; its k-th Fourier coefficient is 1

E

(6.8.15)

(V,(p))''t'.(q)

v-I has the expansion ev

The iterated kernel PF(q, (6.8.16)

°D

)=E

PF(q,

1

s

-1 ev

(q)

and from (4.9.6) we have also, LF(q,

(6.8.17)

) _ - E (v y-1

6.9. SPECTRAL THEORY OF THE s-OPERATOR

Finally, consider the integral equation (6.7.3) where s,, (q) = t1(q) + 1,, (q).

(6.9.1)

Obviously, s, is self-adjoint in the Hilbert space FF with the metric [df, dg]. By (6.1.12) we observe that (6.9.2)

(Adf, su) = (Adf, tAv) + (Adf,1") = AY(df, t,) +1 -11, (df, is)

where A is an arbitrary complex number. Since (df, I,) S 0 (6.9.3) and since Ad/ is always an element of F. if d t is, we can always choose

A such that [Adf, sb.] = Re{(Adf, s,,) 1 S 0.

INTEGRAL OPERATORS

224

[CHAP. VI

We note the identity (df, so) = (dl, ta) + (df, fQ) = (dg, tf) + ((dg, If))-,

that is, (6.9.4)

[dl, s,] = [dg, sf] = [sf, dg]-

If there is a differential df e F. for which [dl, sf] > 0, let (6.9.5)

max [df, sf] =

al > 0,

,

all

the maximum being taken with respect to all differentials in FiR satisfying N(df) = 1. Denote a maximizing differential by d X-1; if df

is an arbitrary differential in F. and a is an arbitrary real number, we have by the maximum property of dxl, [dxl + edf, sxl + s:f] S

N(dxl + edt). 1

Hence, by (6.9.4) and (6.9.5),

1al + 2[edf, ssl ] + [Edf, s f]

1al + al2 Re{e(dxl, df)} +

Ez

a1

N(df),

that is, [df, sxl

- 1a1 dxl]

= 0 (e)

for arbitrary choice of dl a F. and e. Thus, by the usual reasoning als, (4) = dx1(4) We define now a sequence of eigen-differentials of (6.7.3) by the following recursive procedure. If xl, - X. have been determined, belonging to the positive eigen-values al, , a,, consider the class

of all differentials in F. which satisfy (6.9.6)

N(d1) = 1, [df, dxQ] = 0,

e = 1, ..., v.

If this class of differentials has elements dl with [d f, s,,] > 0, deter-


f 6.9]

225

mine one differential dx,+1 such that (6.9.7)

[dxr+1, Sx,+1] = max [df, s,,] =

1

.

q+1

From the maximum property of dx,+1 we derive, as in the case of dx1, that for any differential dl satisfying (6.9.6): [df,

r+1

dx,+1]

= 0.

Moreover, in view of (6.9.4) and (6.9.6) : [d,-e,

1

+1

for all o = 1, 2,

dx,+1 = [sx , dx,,+1] = 1 [dxe, dxq+1] = 0, a all

, v. Thus, the differential'+1 sx -

i U,+1

dx,+1 has

been proved to be orthogonal to all differentials of FW and is, therefore, identically zero. Thus (6.9.8)

dxr+1 = o,+1Sx,+1

We can therefore build an orthonormal system of eigen-differentials dx, by repeating the above extremum problem.

Suppose that o_1, a_2, . , a-, and dx_i, dx_2, - , d X, where dx_k = o_ksx-k, cr_k < 0, k = 1, 2, -, v, have been defined, and consider differentials d f of F. satisfying IV (d1) = 1,

[df, dx_,r] = 0,

k = 1, 2,

, v.

If [d f , s1] can assume negative values under these conditions, let

dx_,_1 be a differential minimizing [df, sr], [dx1-1, sx,-1]

1

O-r_l

v_.,,_1 < 0.

As above we find that (6.9.9)

dx_.,,_1

=

We observe that the eigen-differentials dx,,, h > 0, and dx_x, k > 0, are orthogonal to each, other in the sense that [dx,,, dx_,] = 0.

INTEGRAL OPERATORS

226

(Cxer. VI

For dxh = aksxh, dx-k = a-ksx_k,

where a,, > 0, a-k < 0. By (6.9.4), [sxh, dx-k] = [sx-k, dxh];

that is, (6.9.10)

ah[dxh, dx-k] = a-k[dxh, dx-k]

Since aa, > 0, a-k < 0, we conclude that [dxh, dx-k] = 0. Now let the eigen-values be ordered according to increasing absolute values. In this order we denote the eigen-values by ti, T$, and the corresponding eigen-differentials by d0 j, dO2, . The d4sk form an orthonormal system in F,,, that is [d0,4, d4s.] = apr.

(6.9.11)

Let IF(q, $) + 2F(p, q).

Air (-P, q,

Then Re{s,(q)} = Re { f AF(P, q,

The k-th Fourier coefficient of AF is therefore equal to Re 2k(q)}. Zk

We have by Bessel's inequality k(q)}]2

e

f I AF (p, q, ) !$ dAs

sk

(6.9.12)

r (

L1f

I lF(q, ) I = dA,l1+ l

(f I F

I'

by the triangle inequality; 2L

f I le(q, p) js dAE + f I YF(P, 7) 12 dAX]. V It

§ 6 9)


227

On the other hand, let ®F(y, q, q) = i [IF (q, p) +

1)]

Then

Im {sf(q)} = Re{ f ©(f', q. 4)(f(p))-dA, V

Hence the k-th Fourier coefficient of eF is equal to Im {{'(q)} zk

and Bessel's inequality gives C Im {{k(q)}Iz

< f i e,(P, q, ) 12 dA st

(6.9.13)

S 2[f llF(q, p) (2 dA, + f -TF(p, q) la dA.]. IM

IM

By (6.9.12) and (6.9.13)

(6.9.14) '

fib

)

I

I

S4[

f

IF (q, P) 12 dA +

f I YF

12

dA,].

Integrating (6.9.14) over Dl, we see that

4[ f f

E

IlF(q,P)J2dAdAr

k

(6.9.16)

+ f f I -F

12 dA,dAt

Ma Thus, if there are infinitely many eigen-differentials, then I -ck I tends

to infinity with k. Suppose that there exists a differential dl of class F. which is orthogonal to all the eigen-differentials, that is

[df,dO,]=0, v= 1,2, Then (compare (6.7.23) ) [d f, sf] = Re {(d f, sr)} = 0.

IN'T'EGRAL OPERATORS

x23

[CHAP. VY

We have (6.9.16)

s,) - [so,, d f ] =

[d

[do,, d f ] = 0, TV

by hypothesis. If d f 1 is any other differential orthogonal to all eigendifferentials, we have 0 = [df1 + dl, s,1+,] = [df1, s,,' -i- [d}, sf] + [dfl, s>] + [dl, s,]

= [df, sfj + [d11, sib = 2[d/7, sr],

that is, [d/1, sf,] = 0.

(6.9.17)

By adding orthonorxnal differentials d/ 1, - , d f k which are orthogonal to all the eigen-differentials, the orthonormal system of eigen-differen-

tials can be made into a complete orthonormal system. By (6.9.16) and (6.9.17) we see that s, is orthogonal to all differentials of this complete system, so s f(q) = 0.

Any differential d t which is orthogonal to all eigen-differentials may therefore be regarded as an eigen-differential belonging to the eigenvalue infinity. We admit infinite eigen-values, and we again denote

the resulting complete orthonormal system of eigen-differentials by {dO,}.

We have (6.9.19)

--- lF(q, p) + 2F(P, 4) _ £ Re ,_1

T, I

(6.9.20)

lF(q,

-I- 2F(p, f) = - i E IM ,_1

Thus

= -2, 1

(6.9.21)

1F(q, p)

and (6.9.22)

-2F(p, I) = 11 2 ,-1'[,

1

I

T,

§ 6.9]


229

By the reproducing property J,LF(ji, in

4)(o.(p))-dA, = - i (O.(q))V

Hence (6.9.23)

[Lip (p,

(6.9.24)

[i LF(p, q), 0y]

Re {0,'(q)},

41;]

Im {{;(q)}.

Thus (6.9.25)

00 LF(p, j) _ - E Re {fiy(q)} 0'(P),

rvl at

(6.9.26)

- LF(p, q) = - i Z Im .-1

Adding and subtracting (6.9.25) and (6.9.26), we see that 1

(6.9.27)

q) _ - -

(6.9.28)

0 = E 0' V-1

0*

E 0.' (P) (0.(q)) _1

0,'(q)

We shall discuss later the eigen-differentials of the integral equation

(6.7.3) which belong to the eigen-value a,, _ -1. In this case, we have (6.9.29)

M,(q) = - sp,.

We showed in Section 6.2 that the S-transformation is norm-preserving

that.is (6.9.30)

NO(dP,) = Nmt(S0,) + N -m(S0,).

Since So, = so, in D, we derive from (6.9.30) (6.9.31)

So, =0 in 9t-TI.

Thus, by our result at the end of Section 6.5,

INTEGRAL OPERATORS

230

[CHAP VI

G

(6.9.32)

0,(q) = i Z b,, Z' (q), bµ real.

Conversely, every differential (6.9.32) is an eigen-differential of

(6.7.3) with the eigen-value a, = - 1. We remark that the eigen-value a, = 1 is not possible, since this case would also imply (6.9.31). 6.10. MINIMUM-MAXIMUM PROPERTIES OF THE EIGENDIFFERENTIALS

The eigen-values of the equations (6.7.1)-(6.7.3) may be defined as minima maximorum in the usual way (see [3]). In all three cases, the eigen-values have been defined by maximizing or minimizing, under suitable side conditions, an expression [d f, n,] where n, = t f, if or sr. This was done explicitly in the first and third cases, but even the second case can be easily reduce d to such a maximum problem

in F..

We now consider the problem of maximizing [f', nf] for differen-

tials /' of Fay with N(f') = 1 which satisfy k -- 1 side conditions of the form (6.10.1)

[/',g']=0,

where gi, , gk l are arbitrarily chosen but fixed differentials of F.. The maximum of [I', n,,] depends on the choice of the differentials g,', v = 1, 2, - - -, k - 1, and will be denoted by 1/K, K =

K[g,', ..., g 1]. Nye arrange the positive eigen-values of the operator n f in increasing

order and denote them by yl, ya, ; similarly, the negative eigenvalues will be arranged in decreasing order and denoted by y_i, y_2, . The eigen-differential corresponding to y, will be denoted

by x:

Consider a differential in F. of the form

(6.10.2)

f'=clxi+...+Ckxk.

The requirement N(f') = 1 leads to the condition (6.10.2)'

c ++c = 1.

We have in the k variables ck enough parameters to satisfy the k - 1

§ 610]

MINIMUM-MAXIMUM PROPERTIES

231

side conditions (8.10.1). The differential /' will, in general, be uniquely determined by (6.10.1) and (6.10.2)'. We have k

k

Ec,n = r-1 yr V-1

nr

X

and hence k

[I" nr] _

(6.10.3)

2

c E --.

V-1 Yr

In view of the condition (6.10.2), we obtain the estimate 1

(6.10.4)

S

Z

nr] ,

?

Yk_

Let us choose now, in particular, gi = x1, - - -, A-1. = xk-1 Each

differential f' with norm 1 which satisfies the side conditions (6.10.1)

in this particular case has the form: f = E-.00dx. + E acd,xy, E ccdq = 1. 51--1 r--a* r-k

We easily compute -oc

(6.10.6)

[f', nr] = E

a+E-S

w--1 Ys

r-k Y,

.

Yk

For f' = xk we have equality in (6.10.5); thus, we have shown that x_1)] = 'k Combining this result with (6.10.4), we obtain: THEOREM 6.10.1.1 fyk is the minimum maximorum for the expression

U', nr] for differentials of FM with N (d f) = 1 which satisfy any k --1 side conditions of the form (6.10.1). This result characterizes the h-th eigen-value yk of the problem considered. The same considerations apply to the negative eigen-values of the

operator nr. They may be characterized as maxima minimorum in the corresponding extremum problem.

Let us discuss these results in the case of the particular operators t, it and s. Since we are operating in the real Hilbert space F., we have to distinguish between the linearly independent eigeni ' in the second. differentials 91V and If . belongs to the eigen-value A,,, iq7; belongs to the eigen-value -A,. V,' and $'V," belong to the same negative eigen- value --e,. We

INT-EGRAL OPERATORS

232

[CHAP. VI

remark finally that if' is an eigen-differential of the third problem, ix9 will not, in general, be an eigen-differential. In order to apply the above general theory to the case of = tf, we put: xti = 97,, y, = ,l, and x_, = i9p,,, y_, In the case n f = If, we define x2,+1 = v;, X 2, = i u,, and y-2,.,., = y-a. N,. In

the case of = sf, the previous notation is preserved for the differentials, while y, = a, for all v. We observe that

[f', Sf] - [f', If] = [f'. tf].

(6.10.6)

Hence, in view of the negative-definite character of U', If] : (6.10.7)

U', Sf] S U', tf] Choose gY = (p', g2 = Z-9),', ..., 92k-3 = Tk-11 92k-2 = Wk-11 where

the are the eigen-differentials of the equation (6.7.1). Then clearly, for each normed differential which is orthogonal to all these differentials,

[f', if]

Ak

On the other hand, the maximum value of the left side of (6.10.7) is not less than the minimum maximorum 1/a2k_1. Thus we obtain the inequality 1

that is (6.10.8)

1

S Tk,

0'2k-1

AkSQ2k-1,

k=

Consider now the inequality

U', Sf] = If, If] + [f', tf] z If, 41 - U', tf] gi = q'v g2 = 'Wl ..., 921-1 = 4'f, 92, = $9 ,1, 921+% = V1, 92i+2 = iY'i, ..., 92f+2" = V'-1, g2(i+k-1) ='M_1 where the 4'. are the eigen-differentials of (6.7.1) and the V,' belong to (6.7.2). Then for each normed df which is orthogonal to all dg, Choose,

[f', If] ? -1, ek

I

[Y, tf] I s

1 "1+1

§ 6.11]

HILBERT SPACE WITH DIRICHLET METRIC

233

therefore 1

1

LO k

2,+1

If', SA

But

min U, s,] S

1

,

-9(ltk)+1

for all differentials f' a F. which are orthogonal to 2 + k --1) differentials g;. Therefore (6.10.9)

1

a-Z(!+k)+1

1

1

ek

25+1

.

Thus, various inequalities among the eigen-values of the three operators could be derived from the above characterization of the k-th eigen-value. 6.11. THE HILBERT SPACE WITH DnuCHLET METRIC

We shall now discuss the integral equation (6.7.3) from a different point of view. In the preceding sections we stressed the role of the

differentials on a Riemann surface 1t and considered eigezi-value problems connected with them. It is equally possible to focus the attention on the class of all harmonic functions in a given domain and to develop an analogous theory. We shall show in this section that both approaches are essentially equivalent and we will translate the eigen-value problem for differentials on TZ into an eigen-value problem for harmonic functions on V. The v-th eigen-differential satisfies the equation 0. (q)

r [IF (q,

(6.11.1)

by (4.11.7). Hence

(6.11.2) (1 + ±)O.(q) = -f [IF(q,

ZF(q,

P)0,(1)]dA,.

(Cser. VI

INTEGRAL OPERATORS

234

It follows that

f [P(IF(q, p), KQ)((P,(q))

1 + 1)P(0,', KQ) Z, (6.11.3)

+ P(lF (q, fi), KQ)0,'(p)]dAZ,

where P(1F(q, p), KQ) is the period of lr(q, p) around the cycle KQ, p being held fixed, and similarly for P(1F(q, fi), Ks). From (6.5.26) we have (6.11.4)

P(lF(q,.i), KQ) = (P(lF(q, P), KQ))

If t, # -- 1, that is if G

P,(q) 0 i I b,4Z (q),

(6.11.5)

b, real,

j" -1

we conclude from (6.11.3) that P(0,', KQ) is real. Thus (6.11,6) Im {&(p)} = H,(p), where H,(p) is a single-valued harmonic function on M, if -r,

Let fi and fa be differentials of class F, and write fk = uk + ivk, k = 1, 2. where uk, vk are real. We observe that

-

{If a(dv2

UvfJ - Re f-![1(4?!) dA = 4 Re

avi(av )+dA

8va) l `8v dA, i avil1r + i Re I f \ ax ay TO ax

ravl

$}

,

where z = x + iy; av2

avl av2

&' 8 x + f (dv'

ay

F') dA, = D (vi, v2) = D (vi, v2)

That is, (6.11.7)

[fz, f:] = D (vi, v$)

where D (vl, v2) is the Dirichlet integral as in Section 2.8. In analogy with (6.1.2) write (6.11.8)

g(p, q) = G(p, q) - 9(p, q).

-1.

§ 6.111


285

Here G(p, q) is the Green's function of M2, V(p, q) that of R. Now let us specialize by taking F = M. By (4.10.3) and (4.10.2) we have (6.11.9)

1M(15, q) = --

(6.11.10)

lM(p, q") = 2 a2g(p, q)

a 8pq __ ,

x

apaq

In the case F = M there are precisely G = 2h + m -1 eigen, OG, belonging to the eigen-value r = - 1. We

differentials 0.1,

have (6.11.11)

2i

a

ap

- 2i of

Sli;(p),

(0;(p))-,

and equation (6.11.2) with F = M becomes

(1 +

2

1

aq

I

v

(6.11.12)

azg(p, q) aH.(p) dA, a. apaq

('a'g(p, q) 9H.(P) dA.. n apaq ap 2

Thus

.2 a

(1 + 1 W,(q) l r,/ aq

aq J

2a

ag(0' q) all(P) dAs ap of ag(p, q) aH,(p) dA ap afi

nag 4 a Re J

(6.11.13)

aq

ff

ag (p, q) all, (P) dA ap

a

aq

%

D(g(p, q),

'

H.(p))

Integrating both sides of (6.11.13) with respect to q and choosing the constant of integration suitably, we obtain, for r,

-1,

(6.11.14)

H,(q) = -

1

X11+

D(g(p, q), H.(p))

1

1

((

1

INTEGRAL OPERATORS

286

[CHAP. VI

Thus H,(q) is an eigen-function of the integral equation (6.11.15)

H,(q) = Y, D(g(p, q), H., (p)).

We observe from (6.11.7) that (6.11.16) D(H,,, H,) = [gyp., 0;] = a,,, so the H, form an orthonormal system. The eigen-differentials of the integral equation (6.7.3) belonging to the eigen-value r = - I are just the orthonormalized differen,iZ,. Since tials iZi, [Z Z,] = Re I',,, by (4.3.10), we see from the Gram-Schmidt orthonormalization process that the eigen-differentials 01', - - , Or, belonging to the 1,

eigen-value z = -1 are given by the formula Re I'11. . Re r1,,-,L it',i

,v= 1,2,...,Gp

(6.11.17) 0' = d4i d1

Re I',1... Re T,,,_1 iZ; where do = 1 and

[Zi, Zl] ... [Zi, Zxj (6.11.18)

dk =

EMEM

Re rki ... Re1'kk [Zk' Zi] ... [Z;, Zk] In particular, we derive from (6.11.17) :

(0 iZQ) = i P(0;, KQ) (6.11.19)

r

'Re

=d4 d.

Re f,1 ... Re r,, -1 r-9 The eigen-functions of (6.11.18) belonging to the eigen-value oo are (8.11.20)

H,= Im 45, = dZi d

Re r.... Re F1,

1L

Re Z1

,v=1,2,..., G. Re r,1 ... Re

Re Z,

These eigen-functions are not single-valued. However, using (6.11.19)

§ 611]


237

we have D(H,, Re ZQ)

P(H,, KQ)

Re r11 (6.11.21)

Re.1,r1 Rel,Q

= d 1d,}

Re r,, . ReI',,rl Re r,, Thus

D (H Re ZQ) = - P(H,, KQ) = 0, B = 1, 2, , v -1. The orthonormal system {H,} of the eigen-functions of (6.11.15) is complete. For suppose that H is a harmonic function on V with (6.11.22)

D(H) = 1 such that (6.11.23) D(H, H,) = 0, v = 1, 2, Then, writing

.

P=R+iH,

we would have

[0', 45,] = D (H, H,) = 0, so the system {0,} would not be complete, a contradiction. The vector field whose elements are the gradients of the harmonic (6.11.24)

functions H on 0 with finite Dirichlet integrals forms a Hilbert space in the metric of the Dirichlet integral. We may consider the harmonic functions themselves as elements of a Hilbert space with

the metric D (H) if we normalize them by the condition that all functions vanish at a fixed point po of V. Then the only element of the Hilbert space with zero norm is the zero element H = 0. Every harmonic function H with finite "rm has a Fourier expansion (6.11.25)

I H = E a,(H,(') - H,(po)), V-1

where

a, = D(H, H,).

(6.11.26)

In particular, by (6.11.14),

(6.11.27) .g(p q}-g(

0,

q} =-n E (1V-1

; TV

INTEGRAL OPERATORS

238

[CHAP. VI

Differentiating (6.11.27) with respect to p and q or p and q", we obtain (6.9.21) and (6.9.22), in the case F = M, using (6.9.27) and (6.9.28).

In the eigen-value theory for the harmonic functions on 0, the Green's function played a distinguished role. It might be of interest to study an analogous theory where the Neumann's function takes the corresponding central role. By (4.2.22),

(8.11.28) -

2 a2G(p, q) +

2 a9N(p, 4, q0)

x

apaq

apaq

4 c,,,Z,,(p)Z(q)

where by (4.6.4) (6.11.29)

II c,,, II = II Re 1'N, II-1.

By (4.11.1), (6.11.30)

2 a2G(p, q)

L8(p, q)

apaq

+

G

v, -i

where II-1.

(6.11.31)

IIYk1 II=11 r'ol Domains B of genus zero, in particular multiply-connected domains

of the plane, are characterized by the property that (8.11.32) 1',,, = Re r,,, ,a, v = 1, 2, -, G. Therefore, in the case of domains of genus zero, (6.11.33)

Ls(p, q)

2 82N(p, q, qo)

x

apaq

In other words, for domains of genus zero the Neumann's function is related to the class S of single-valued functions in the same way that the Green's function is related to the class M. The symmetry of the class S (L5(p, q) = Ls(q, p)) is a characteristic property of the domains of genus zero, and if R is also of genus zero we may take F = Sin (6.11.2). Let the eigen-values z, in the case of the class S be denoted by t, and let the corresponding eigen-functions be W.. Equation (6.11.2) becomes

(6.11.84) (1 + ) 1'.(q)

_-f [ls(q, p)(T.(p))-+ls(q,P)V.(p)]dA,.

§ 8.111


239

In the case of the class S we may drop the assumption that any boundary component of l which is homologous to zero on R is also homologous to zero on 9R. For this assumption was only used

to show that if G

/'(q) + Sf(q) -= Z aµ Z,,(q), N-1

then a,, = 0. This conclusion, however, is obvious in the case of class S. Let.K(p, q, qo) denote the Neumann's function of 91 and write (6.11.35)

n(p, q, qo) = N(p, q, qo) -.N'(p, q, qo).

Then q, qo) 2a a2n(p, apaq

( 6 . 11 . 36 )

l s(p, q )

(6.11.37)

ls(p, q) _ --

2 a2n(p, q, qo)

apaq . We observe that these formulas differ from the analogous formulas

(6.11.9) and (6.11.10) in that (6.11.37) has a minus sign where (6.11.10) has a plus sign. This difference in sign reflects the difference

in the symmetry of the Green's and Neumann's functions as given by formulas (4.2.6), (4.2.8). This difference in symmetry has an important consequence in establishing the analogue of formula (6.11.14) for (6.11.35), namely that we must use the harmonic functions R, = Re P, in place of Im 'F',. Formula (6.11.34) then becomes ((

'.

(6.11.38)

2 a r an (p, q, qo) 8R,(p) of nagJ L ap

1 aR. (q)

l+4/

aq

st

+ 8n(p, q, qo)

_--8q Re IfJ

It

_

1a 2

a

(p, q, q0) 8R,(p) 1 ap afi dAsJ

gD(n(p,q,go),

R(p)).

ap

dA

INTEGRAL OPERATORS

240

[CHAP. VI

Integrating both sides of this equation and using the fact that n (p, q0, q0) (6.11.39)

0, we find R,(qo) =

1

n(1

1l D(n(p, q, q0), R,(p))

+ tr!

Consider now the Hilbert space of all harmonic functions on fit which have a finite Dirichlet integral and which vanish at po. The function

n(p,q,g0)-'n(po,q,go) belongs to this space and the functions R,(p) - R,(p0) form a complete orthonormal system in this space. We have therefore (8.11.40)

n(p, q, q0) -- n(po, q, q0) _ n E (1

+ 1 (R, (q)

R,(p0))

We observe finally that we are operating within the class S and hence

(Y;, ZQ)=0. If f is, moreover, simply-connected, we have G

ls(p, q) = h.(p, q) + E c,,, Z,,(p) Z,(q) ,,,r-1

Thus, (6.11.34) can be written in the form

(1+ jr:(q) (6.11.34)'

f [I. (q, p)(W.(p))- + 1.(q, P)'q',(p).] dA,. This equation agrees with (6.11.2) if we put F = M, z,, = t,, W. Thus, when R is simply connected, the eigen-differentials {YW;} form a subset of the eigen-differentials We observe that the eigenvalue t, = --1 does not occur, since the differentials Y% must belong

to the class S. Thus the {R, (q)) are the real parts of a subset of Hence the construction of the Neumann's function uses a

§ 6.12]

CLASSICAL POTENTIAL THEORY

241

subset of the analytic functions used in the construction of the Green's function. We discussed in some detail the case of domains l of genus zero imbedded in a domain N which is simply-connected, since this case

contains the important theory of conformal mapping of multiplyconnected domains in the complex plane. In this case, t is the multiply-connected domain and R the Riemann sphere. This case has been studied by the preceding methods in [1]. 6.12. COMPARISON WITH CLASSICAL POTENTIAL THEORY

In this section, we want to relate our results to the classical methods

of Poincare-Fredholm used in the boundary-value problems for harmonic functions. We shall recognize that our eigen-functions H, (p) and the eigen-values z, are closely related to corresponding eigen-functions and eigen-values of the integral equation theory of Fredholm. It will appear that the boundary values and the values of the normal derivative of H, on the boundary give rise to two sets of classical eigen-functions. For a fixed q in the interior of 1R, we have by Green's formula (6.12.1)

f G(p, q)

D(G(p, q),

C

8

ans

)ds:.

where (8H, f an,)ds, has the interpretation described at the beginning of Section 4.3. Since G (p, q) = 0 for q in DI and p on the boundary,

we see that D(G(p, q), H.(p)) = 0. The integral equation (6.11.14) therefore becomes (6.12.2)

D(1(p, q), H,,05))

1

ill + s.) X(1

f g(p,q)6

+1 I-)

by Green's formula; that is,

c

v

INTEGRAL OPERATORS

242

(6.12.3)

H, (q)

/

J,q) I+ --

[CHAP. VI

a

an,

c

dsv.

9

Removing a small uniformizer circle at q, applying Green's formula,

and then letting the radius of the circle tend to zero, we obtain (6.12.4)

f 9F (p, q) a C

q

dsv = f

an,

ds, -

c

Eliminating the integral on the left by means of formula (6.12.3), we see that q) H,(p)ds,,.

(6.12.5)

J Ic

an,

We remark that the eigen-differentials 0,'(q) of the integral equation (6.11.1) are regular on the boundary C of V. This is a consequence of the f a c t that ll(q, p) and 3 (q, fi) are regular for q on the boundary and p any point in the closure of V. The regularity of 2'M(p, q) follows from the assumption that TZ is properly embedded in 91. Hence we may let q tend to the boundary in (6.12.5). From known theorems of potential theory concerning the behavior of a double layer distribution on the boundary, we conclude that for H, (q) =

1

1

z,-

f c

aga(n, q) H,(p)ds,

,

+

1 1

H, (q),

that is (6.12.6)

agla , g) H,(p)ds,,

H, (q) = C

q e C.

9

Here the integral must be interpreted in the principal-value sense. Similarly, writing aH,(q),

q C, e

Q

we conclude from (6.12.3) and the known behavior of the normal

CLASSICAL POTENTIAL THEORY

§ 6.12;

243

derivative of a simple distribution on the boundary that for q e C, 1

1

f

aVa , q)

1

h,(P)ds9 +

1

hv(q)

TV

that is (6 12.7)

f a a($, q

h,(q)

k,(P)ds9,

q e C.

C

Equation (6.12.7) is not in invariant form, but can be made so by multiplying both sides by ds, for h,(q)dsq =

a

(q)

ds,,

rQ

is invariant. We observe that the equations (6.12.6) and (6.12.7) are transposes of one another. Equations related in this way have the same eigenvalues, and the eigen-functions of the two equations form a biortho-

gonal set, that is JHi4(q)h(q)dsq

= 0,

p 0 v.

C

In our case, we know that k. (q) aH,(q)/anq, and the biorthogonality follows at once from the fact that

f

C

fB(q) aanq) dsa `

C

-D(H"`,

H,.) = 0

a

for ,u # v. In particular, we have shown that the eigen-functions of the equations (6.12.6) and (6.12.7) are closely related, the eigenfunctions of one being the normal derivatives of the eigen-functions of the other. In other words we have shown that the first and second boundary-value problems of two-dimensional potential theory are equivalent. The equivalence of these problems, as a consequence of

the Cauchy-Riemann equations applied to conjugate harmonic functions on the boundary, was pointed out in [1].

INTEGRAL OPERATORS

244

[CHAP. VI

6.13. RELATION BETWEEN THE EIGEN-DIFFERENTIALS OF AND 91 --

1

We shall show that the eigen-differentials of (6.7.3), the equation for the domain 9)1, automatically give rise to the eigen-differentials of the corresponding integral equation for the residual domain 91. Thus we obtain simultaneously the spectral theory of the s-operator for a domain and for its complement with respect to the domain 9{

in which it is imbedded. Let 91 = R - 'I1, and assume that % consists of a single component.

Then, according to our definition, % is a finite Riemann surface. However, we shall see below that the connectedness of 91 is not essential. Let us follow our earlier convention that differentials are zero throughout any domain in which the definition is not explicitly stated. If P is an eigen-differential, we have

10,' So,(q) = r. (q), q 0,

(6.13.1)

e.(q),

where 0,'(q) (6.13.2)

q

91,

0 in 91 and B;(q) = 0 in 1. By (6.3.8) 0,'(q), q e 2R,

1

(S'(q))O, = r SS,(q) + S8,(q)

= 10,

q

W.

Hence

1- r,) 0'(q), (6.13.3)

S. (q) _

I-

qE q E fit.

1 e.(01 TV

In particular, 9,' is an eigen-differential of the integral equation (6.13.4)

0.(q) = -r,S6,(q)

By (6.2.8) and (6.5.16), since the class Fvt is assumed symmetric,

0"] = [So,,, S0,] _' [S0 , S0,Je + =z1T,[[",0;Jn+ [O ,O]n,

[Soµ, S0,J91

§ 6.13]

THE EIGEN-DIFFERENTIALS OF TZ AND 2 --IF

245

$o

(6.13.5)

'

[e,., 0"191 _ [e;,, 8;7 = (1-

Thus (6.13.5)'

[O , e 7 = (1

-

We observe from (6.13.5)' that c, < 1 is impossible. There are at most G eigen-differentials (P*, belonging to the eigen-value a = --1. Let these eigen-differentials be 0. , , OG, some of which may be

zero. From (6.13.5)' we observe that @i(q) _

= OG(q) = 0.

Writing

_

we see that 94v 90+:,

8µ

form an orthonormal system. Thus the

eigen-differentials 0. of the integral equation (6.13.7)

0µ(q) = zosm,,(q)

automatically give rise to eigen-differentials of the integral equation (6.13.4).

Suppose that there is a differential 9' in N such that

[9',9;]=0, ,'=G+1,

(6.13.8)

Then

0 = [8" e,7 = [Se. Se,] = [Se, Se,7q + [Se, Se,]

=

t

1)}

1

I

_ (1-;) [S., 0.']. since

[Se, 8,]9t = (8', S9,] _ --fir [e', 9,7 = 0.

INTEGRAL OPERATORS

246

Thus, since Jr, I > 1 for v = G -}- 1, (6.13.9)

[CHAP. VI

, we have

0,']gft = 0,

In other words, S9 is orthogonal to all the eigen-differentials 0,' which correspond to eigen-values x I z, > 1. Since the eigen-differentials c2 form a complete orthonormal system, we conclude that (6.13.10)

c' a, v:(q), S& (q) _ L'

where a, is real, v = 1,

,-1 , G. Therefore

E a, SO(q) _ ,-1

(6.13.11)

q

qE

(q),

R' (q),

2,

q

where R'(1) denotes a differential which, vanishes identically in T't.

It follows that G

(613.12 )

( S2 (q))6=

SR(q)-Ea,O,'(q), qEV, ,-1 SR (q), qE%,

since, by Section 6.5, So" (q) = 0 in % and Spv (q) R't for u = 1, 2, , G. On the other hand, (6.13.13)

gE91,

(S'(q))9 ={8'(q),

Comparing (6.13.12) and (6.13.13), we see that G

(6.13.14)

SR(q) _ .-1

a,

B'(q), From (6.2.11) and (6.13.11),

qe q e ?ft.

G

(6.13.16)

NN(S9) = E a; -}- N (R') = Nyt (O'); r-1

from (6.2.11) and (6.13.14), G

(6.13.16)

Nst(SR) = E a ; + Na (®') = N91 (R'). r-1

P,, (q) in

1 6.131

THE EIGEN-DIFFERENTIALS OF 2 AND 82 -T1

Thus a1 = as =

= aG = 0 and we have

gefit,

SO(q)_{0,

(6.13.11)' (6.13.14)'

247

(q), q E W, 0,

SR (q) _ je'(q),

qE 9R

92.,

qE

Hence (6.13.17)

and (6.13.18)

(9'(q) - R(q)

S©_R(q),

e'(q) + R'(q) = Se+R(q)

If there is a non-trivial differential e' satisfying (6.13.8), we have shown that the equation (6.13.4) must have an eigen-value z = 1

or an eigen-value r = - 1. We have thus proved the theorem: THEOREM 6.13.1. Every eigen-differential 0q o l (6.7.3) which belongs

to an eigen-value i, ; - 1 transforms by the S-operator into an eigen-differential for the corresponding integral equation in the complementary domain and the eigen-value - r,. Conversely, every eigendi f f erential o l the latter integral equation which belongs to an eigen-

± 1 is obtained in this way. We want to show now that no eigen-differential in R belongs to the eigen-value + - 1. We recall that a domain is properly imbedded in gt if each of its boundary points is an interior point of 9t. Since value

we have assumed that 9t is properly imbedded, it is clear that R = R - 92 is not properly imbedded and this means that at least one boundary component of 9 coincides with a boundary component of 91. Hence 3F (q, fi) is not regular in the closure of 9't. Let L**(q, p) be the bilinear differential of 9't which belongs to a symmetric class F = FF corresponding to the given symmetric class F,,, and let e' be an eigen-differential belonging to the eigen-value

+ 1; that is (6.13.18)'

e'(q) = S9 (q).

Then (6.13.19)

e'(q) = S9 (q) = f2'p(q,

p)(e'(p))-dA,-}-

f.re'(q

) 0' (p)dA1.

[CH". VI

INTEGRAL OPERA TORS

248

Let us write 1 (q, 2) = LF(q,1) --F (q, v). Since LF*(q, p) is orthogonal to the class F91 and since L*F(q, +) reproduces under scalar multiplication we have in view of (6.13.19) (6.13.20)

P) (0'(p)) dAz +

f YF(q, )0'

0'(q)

f 1,* (q,

29'(q)

f lF(q, P)(0'(p))-dAF- fl

and (6.13.20)`

It follows from (6.13.20) and (6.13.20)' that 9'(q) is regular in the closure of 91. By (6.2.11) and (6.13.18)' we have

NN(S®) =N.(S9) +Nm(sp) =Nit(S®)+Nit (9') =N,, (0'). Hence (6.13.21)

S® (q)

=j

0,

q

e, g691. We conclude in the usual fashion that Sq (q) has the boundary values (0' (q) )- on that portion of the boundary of 53t which coincides with

the boundary of M. By (6.13.18)', S® (q) also has the boundary values O'(q), so 0'(q) is real on that part of the boundary of J which coincides with the boundary of W. We understand, of course,

that 0' (q) on the boundary is expressed in terms of a boundary uniformizer. Now let q be a point on that part of the boundary of SJ2 which coincides with the boundary of T. By (6.13.20)', 20' (q) =

-f4 (q, p)(0'(p))-dA.- f 1F(q, P)0'(P)dAz

(6.13.22)

)dAz 3t

_- f(l(q,

Jt

))-(©'(P))-dAz- f l8(q, )0'(p)dAz.

THE EIGEN-DIFFERENTIALS OF R AND fR -SR

i 8.13]

249

Thus 0'(q), expressed in terms of boundary uniformizers, is real on the entire boundary of 92 and is therefore a differential of W. If h' E Full

(Se. h')at = (0', h')st

But by (6.2.12) (S®, h')az = - (o',1')gc' so

(6.13.23)

(0', h')ot = 0,

for every h' of Ft. Now assume that 9t is essentially embedded in W. We recall that a similar condition was imposed upon 932. We can prove, however,

that if 9 is connected, and if D1 and R each have more than one boundary component, 92 satisfies the assumption in any case. For, let b be a boundary cycle of 91 which bounds on 91. Since 91 has more

than one boundary component, b is not a boundary component of R. Hence it is a boundary component of 9)1 which bounds on 9I and also, by hypothesis, on V. But 9It is connected and has more than one boundary component, so this is impossible. Hence no boundary cycle of 92 bounds on 9I in this case. We can now apply the reasoning of Section 6.5, and we conclude that 0' (q) = 0. Thus there are no eigen-differentials corresponding to the eigen-value + 1. However, the above argument cannot be used to exclude the eigen-value - 1. In fact, since . °i((, q) is not regular in the closure of 92, there may be infinitely many eigendifferentials belonging to the eigen-value - 1. If we include these eigen-differentials, we obtain a complete orthonormal system of eigen-differentials for the class F which corresponds to F.. In order to overcome the difficulty that infinitely many eigendifferentials may belong to the eigen-value - 1, we introduce the sub-class RR of F which is composed of those differentials 0' of FF which, expressed in terms of boundary uniformizers, are regular analytic and real on that portion of the boundary of 91 which lies on the boundary of 9Z (91-boundary of 92). Suppose that 0' belongs to Fe, 45' = 0 in 92; then for q on the 91-boundary of 9 we have

INTEGRAL OPERATORS

250

So (q) _ 52p(q, p) (O'(5))-dA. 7-

[CRAP. VI

YF(q, fi)O'(p)dA, as

aR

p) ("'(p))-dA3 + f.29P(q, fi)V(p)dA, aR

aR

f YF(q, fi)V(p)dA..

= 5 (3F(q,

a

R

Thus So belongs to R.. In particular, the eigen-functions O defined

by (6.13.6) belong to R. We remark that the differentials of R% form a complete Hilbert space. For any differential e, of it* is regular analytic in 9 and on the boundary of fit. Moreover, N91 +ai (®') = 2ND (©'),

since 4' takes values in §J which are conjugate to those taken in R. Hence points of the )l-boundary of behave like interior points of the domain so far as convergence questions are concerned. A Cauchy sequence in R% will therefore converge to a differential which is real on the FR-boundary.

If the system of eigen-differentials ©,, already constructed do not form a complete orthonormal system for R%, there is a differential e' of RK which is orthogonal to all the (9 . But then we have shown

that (6.13.24)

0'(q)

So (q),

where S,9 (q) _ 0,

q

It follows in the usual way that Se(q) _ (®'(q))on the 92-boundary of %. Here ®'(q) is expressed in terms of a boun-

dary uniformizer. But, by (6.13.24),

Se(q) = - 0'(q) on the )t-boundary, sn C)' (q) is imaginary on the SR-boundary. Since e' belongs to R9,, it is real and regular analytic on the fl't-

THE EIGEN-DIFFERENTIALS OF Dl AND 91- 1

§ 6.13]

251

boundary of Yl, and it follows that {e' (q)}

is a quadratic differential of %. Let a (a finite number) be the number

of linearly independent quadratic differentials of 9. Since the number of quadratic differentials which are squares of linear differen-

tials does not exceed a, we see that by adding a finite number of eigen-differentials belonging to the eigen-value - 1, we obtain a complete orthonormal system for R% . Let this complete orthonormal system be denoted by {e,}, where Oy is the eigen-differential of the

integral equation (6.13.25)

e'(q) = TSe(q) belonging to the eigen-value T = - r ; that is T'sev (q)

9 (q)

The differential / / AF(P, q,'7) = - lF (q, p) - lF (p, q)

clearly belongs to R., and its v-th Fourier coefficient is equal to Re{ AF (P, q, J

-dAz}= Re {S9,(q) + ef(q)} T,

Re {©.(q)}.

Similarly, the differential AP (P, q, q) = i [lF(q, P) -

4)]

belongs to (Rj, and its v-th Fourier coefficient is Re{ J AF (P, q, 4)(©.(p))-dA5} = Im{Se,(q) + e ,(q)}

Im{e(q)} 'Lr

Thus (6.13.26) IF* (q,

(6.13.27) IF* (q,

/

Re {e

) -1 (p, 4) _ -i E' I 1 - z 1 Im{4y(q)}®.(p)

INTEGRAL OPERATORS

202

[CHAP. VI

Adding and subtracting these equations, we obtain (6.13.28)

iF(5, q)

ar(ih)0ti(q),

(6.13.29)

We observe that the eigen-differentials a belonging to the troublesome eigen-value r = - I do not enter into the sums (6.13.28) and (6.13.29). The remaining eigen-differentials have the form 1

©.(q)

(6.13.30)

Sp'(q),

qe

>

where the r, are the eigen-values, the 45,; the eigen-differentials, of the corresponding integral equation for 931; that is

0.(q) = r,S0,(q), q e R The formulas (6.13.28) and (6.13.29) may therefore be written (6.13.31)

(6.13.28)'

IF* (p, q) =

-- 2

,E1

(6.13.29)'x(, 4) = - ,.

a, -'

1

.

1

-{-

S0, (q),

Therefore, by solving the integral equation 4'(q) = sSo(q), for V, we simultaneously determine the bilinear differentials for 9J and for the complementary domain 9t. 6.14. EXTENSION TO DISCONNECTED SURFACES

At the beginning of Section 6.13 we assumed that the complement

91 of V with respect to R is a single domain. The question arises whether this assumption is essential, and it turns out that it is not. We shall now extend the main formulas and results of the preceding sections to the case in which the set D1 imbedded in 91 consists of several components 91214, each of which is a finite Riemann surface.

§ 8.14)

EXTENSION TO DISCONNECTED SURFACES

253

The necessary definitions of the kernel and of the difference kernel have been given in Section 5.2. For the present case, we define k

(6.14.1)

T,(q) _ (2F (q, p), t'(p))a _ E

µ-1

(6.14.2)

(ti(p))-)et= E

P,(q)=(2'

k-1

Next we observe that (6.14.3)

Ts(q) _ - (ZF(q, ), ti(p))

since (even without assuming symmetry)

(LF(q, p), /'(p))s = 0. The following formulas are seen to be true: (6.14.4)

(6.14.5)

(T,1, V,2),, = 0,

(T,", T,1). = - f J 9t

IF

(f9, p1)(/ 1))-//(P2)dA.,dA,a,

91

1f)3

fyF(P21

fi1) fi(p1) (/ (p,))'dA,i

s2 in

(S,1, S,1)st = (t!, ti)ee - 2i E Im f ft,.} (i',, ff)%(W 4)ee) (6.14.7)

M, -1

l

+ 2i Imv A) Yl'(p1)) ts(ps) dA,l dA, } ,

if

J

tc

NR(S,) = N,1(T,) + NW(',) (6.14.8)

ce

= Na(t') - 2ik, Z+-1Im If 21"F(-6, q) = -rip (q,

((g"'

(symmetry), then (6.14.8) becomes

NE(Ss) = Na(I') Furthermore, when SPF(p, q) is symmetric, we have the three for(6.14.8)'

mulas: (6.14.9)

(T,1, f!)

(6.14.10)

(T,1, t2)

254

INTEGRAL OPERATORS

(6.14.11)

CHAP. VI

[S,,, fa] = [Sf,, fx]-

In general, we also have (6,14.12)

(Tt , h')

= 0, (P', h') = - (f', h')Bt

for any h' E F. The formulas of Section 6.3 extend immediately to the more general case where 7t is disconnected; that is, (6.14.13)

T2 = I + T,

(6.14.14)

T2

(6.14.15)

= - T,

TT =DT=0,

S2 = I. Let us now assume that JJ is properly and essentially imbedded in t and that 2F (p, q) =3F(q, p5). Then all results proved in Section 6 e-Section 6.13 remain valid for the case in which Dl is disconnected. (6.14.16)

Iti particular, we may drop the assumption that R = 91- 0 is connected; we must assume, however, that R satisfies the condition that the imbedding is essential. The fact of the matter is that if 9

satisfies this condition, the hypothesis "92 is connected" is not necessary. 6.15. REPRESENTATION OF DOMAIN FUNCTIONALS OF 11)1 IN TE1tms OF THE DOMAIN FUNCTIONALS OF 9t

For a given surface 9 t there are infinitely many imbedded surfaces 92 each of which is characterized by its boundary C relative to R. We may regard the differentials of the surface 2 as functionals of C or of Dt and have then to establish a method for calculating them in terms of the differentials of the fixed surface R which carries the surfaces M. We suppose that 9 is the union of a finite number of domains X 1, and that D2 is properly and essentially imbedded in R. We shall consider only classes F of differentials on J2 for which the corresponding

class Fs, is symmetric. We established in the preceding sections the

fact that, under the above assumptions, all eigen-values of the equation (6.7.1) are greater than unity. This enables us to solve certain integral equations for the differentials on D2 by means of Neumann-

§ 6.15]

SR-FUNCTIONALS IN TERMS OF 91-FUNCTIONALS

255

Liouville series and to express the functionals of lJl in terms of the functionals of the career surface R and of integrals of the 9t-functionals

over 9)1 and the residual set 91- 9t. Let QF"`f (p, q-) be defined as in Section 6.7. By the reproducing

property of LF(b, r) we have Q(2) (p,

q) = - f QF1(r, q) LF(p, y) dA,. 9R

Using (6.7.30) we may write (6.15.2)

IF(p, q) _ .2'F(P, q) - I'F(p, q) = LF(p, q) + Qom) (p, q) and bring (6.15.1) into the form of an integral equation for LF(p, q) : (6.15.3)

IF(P, q)

= LF(P, q) - f Q( .2)(r, q)LF(p, f) dA,. V

The term IF(P, depends, by its definition, only on differentials on 91 and upon the difference domain 91-TZ, and will therefore be considered as known. We can invert (6.15.3) since the kernel QF) (r, q) has all its eigen-values A greater than unity. We define the reciprocal kernel of QF ) (r, q") by the Neumann-Liouville series 0 i> (6.15.4) QF (p, q) _ Z QF2) (p,

.-0

which converges uniformly in each closed subdomain of R. Using the expression (6.7.34) of QF" in terms of its eigen-values and eigendifferentials, we obtain (6.15.5)

Qsi)(P, q) =P-1L' (1 - $)_1d94,(P)(dp,(q))-.

4

By means of the reciprocal kernel QF l) (p, q") we may now solve

the integral equations (6.15.3) in the form:

LF(P, q) =IF(P, q) + E f

QF.)

(r, q) IF(¢, f) dA,i

gut

(6.15.6)

=

f Q( l) (r, q) IF(p, F) dA,. an

The difficulty with the above result comes from the fact that

INTEGRAL OPERATORS

256

[CHAP. VI

QF)(p, q) still contains the differential LF(p, q") of 9)2 and hence the series development (6.15.6) does not yield LF(p, q) in terms of

91-differentials alone. However, this obstacle can be overcome as follows. Because of the reproducing property of the kernel LF(p, q), we have LF(r,q) + 21 (r, q) + IF) (r,q) (6.15.7) QF'(r,q) where (6.15.8)

IF(r,

q") dAz.

V In general

(6.15.9)

LF (r, q) + E v

QF' (r, q)

,.=1

l

IF ) (r, q)

with (6.15.10)

IF (r, q)

=

f 1, (0-1 (r, 1i) j F(p, q) dA..

a

Thus, each term QFv) is composed of iterations of the known term

IF(p, q") and of the kernel LF(r, q). We do not know this term but we do know its effect under scalar multiplication over 9)t on all differentials on 91 or 92. In particular, we can now calculate all terms

in (6.15.6) and obtain: (6.15.11)

LF(p, q) = E i E ,, o

(v)I+i(1q)}

,=o

In (6.15.11) all right-side terms involve known differentials of 91; we have thus obtained a representation of the desired type for the kernel LF(p, q") on V. The basic term on the right-side has by (6.15.2),

(6.7.31) and (6.7.34) the form IF(p, q)= LF(p, q) + QF'(p, q) (8.15.12)

_ ` E 11-

Its iterates have consequently the form (6.15.12)'

IF'(p,

E (1 .-1

0-FUNCTIONALS IN TERMS OF Ot-FUNCTIONALS

§ 6.15]

257

and the identity may be checked formally by inserting (6.15.12)' into (6.15.11) and then verifying the identity by rearranging the terms on the right-hand side. It is also instructive to express the bracketed terms occurring in (6.15.11) in terms of the eigen-values and eigen-differentials. We find: (6.15.13)

E' (v) Is +1) (p, q) _ P

1(1 --

Q-1 "e

Q

This shows that the successive brackets converge to zero like 1/1Q and we obtain an estimate for the contribution of each bracket to the sum total of (6.15.11). We summarize our main result in THEOREM 6.15.1. The kernel LF(p, q") of the domain Tl can be developed into a series in terms o t iterated integrals of the R{-differential

IF(p, q) =.2F(p, q") --I'F(p, q), which is given by formula (6.15.11). It is clear that once LF(p, q) is constructed there is no difficulty

in obtaining LF(p, q). In fact, using the reproducing property of LF(p, q) under integration over 91t, we find by (4.11.6) and (4.11.7):

iF(p, q) = LF(p, q) -2i (p, q) = -(IF (r, q), LF(r,3))m = (2F(r, q), LF(r,.P))WtThus, finally: (6.15.14)

LF(p, q) = £F(p, q) + ,(.F(r, q), LF(r, .))

We have expressed LF(p, q) in terms of the -wF-kernel on Jf and an

improper integral involving LF(p, q) over R. This integral is, of course, to be interpreted in the sense of (4.9.8). From (6.15.11) we may derive immediately a series development for the differential ZQ(q) on It in terms of differentials on N. Let us define (6.15.15)

1pµ)

(q) =

r I(N+1) (q, 3)

Z,(p) dA1;

in

(for the sake of simplicity, only the symmetric class M will be considered aAd we drop the index M). We have by (6.15.10)

INTEGRAL OPERA TORS

258

JQ")(q)=(6.15.16)

f

[CHAP. VI

if J1ct4(q, P) JQ°j (r) dAn.

The integral JQ") (q) is the conjugate of the period of the differential 4) with respect to a cycle K.; we find by definition (6.15.2)

I(11+1) (p,

and by (5.1.5), for the special case ,u = 0: .Je°)(q) =W (q) - f2(q, r)W&' ) dAn. ot-a Thus 1Q°) (q), and by (6.15.16) all other differentials Jo") (q), can be (6.15.17)

expressed in terms of fit-differentials. Consider now the series (6.15.11) which converges uniformly in each closed sub-domain of V; determine on both sides of the equation the period with respect to a cycle K. Since this determination can be done by integrating along an interior path in Dl, we obtain the identity (6.15.18)

ZQ(q) =

.o{"o

("'

Je")

(q) }

where the right-hand sum converges uniformly in each closed subdomain of V. Let us determine next the period matrix of the ZQ (p)-differentials. We observe that by (6.15.10) and (6.15.15), (6.15.19)

J004 (q)

= fI(q,V)J1s_1)(r)dA, u = 1, 2, ..

Hence, we derive from (5.1.8), (6.15.16) and (6.15.19), (6.15.20)

f J(") (r) Z.(-Y) dAn = got

-' f (J, 0) (r))

t

J("-I) (r) dAn,

u=1, 2,.

and (6.15.20)'

fJo(7)z(P) dAn = 17Q,- f

&o(r) dAn = HQQ.

§ 6.15]

212-FUNCTIONALS IN TERMS OF 8R-FUNCTIONALS

259

Consequently, using (4.3.10), we find the period relation: (6.15.21)

1

0

o

{HQQ

_-

(JQ°) (y) }- E ()Jc/1(r)dA}

Thus, we have obtained series developments for the differentials ZQ and for their periods in terms of differentials on N and of their periods.

The above formulas were obtained by using the integral equation involving T. Let us now try to carry through a similar program for the integral equation involving T. By (6.8.15) and (6.8.17) we have

lF(-b, q) = - r

(6.15.22)

i

(i

-1

(q))

[fir

This formula shows that the peigen-values of the integral equation (6.15.23)

TV"(q) = -e* f SSR

are

(6.15.24)

1 --e.

Since there are no eigen-values Lo = co, there are no eigen-values ew = 1. However, as v tends to infinity, e: tends to unity. From the point of view of Hilbert space theory alone, let us try to calculate LF(p, q) from .91F(t, q). By (6.8.15), .fig)(p,q)

(6.15.25)

=

where (6.15.26)

-n° c) (p, q) _."

(2'-11

r

w,(f')(Vr(q)}

(r, q) (.F (r, $)) AAW

in

for a > 1. We have N

(6.15.27)0-1E

ao

( N

. E Ae

q) _ v-i l a-i

-

11 a J

er

;(q))-:

INTEGRAL OPERATORS

260

[CaAP. VI

Let PN(x) be the polynomial with real coefficients A.: N

PN(x) = E Aex°.

(6.15.28)

e-i x < -6, 6 > 0, it is possible to approximate

In the interval - 1 the function

1

X

uniformly by polynomials (theorem of Weierstrass).

Hence it is possible to approximate the constant 1 in the interval

- 1 S x S - 6 uniformly by polynomials of the form (6.15.28) which vanish at x = 0. Thus, given a positive integer v0, there is an No such that for N z No it is possible to make the coefficients ! x 1e

EAa(-1), 1SvSvo, (r/

0-1

\

in the series on the right of (6.15.27) as close as we please to unity.

For a fixed point q interior to Dl we have N

(6.15.29)

NIM (E A(p, q)) = E a' E A,,

(6.15.29)'

-LF(q, q) = limN,(EA(M

(). \

z

(fir J r-1 a-1 where the right side can be made to tend to -LF(q, q) as N tends to infinity. Thus, we can represent - LF(q, q") in the form

\e-1

N

Using (6.15.26), we may bring this equation into the form (6.15.29),.

LF (q, q)

= lim N--. oo

N

E

tu, r-i

+r)

(q,

We have therefore proved THEOREM 6.15.2. The kernel L)r(q, q) may be approximated arbitrarily closely by finite combinations o l iterated kernels 3F) uniformly in each closed subdomain of fit. In these considerations we have used only the properties of the Hilbert space and we have not used all the information available concerning the imbedding of 92 into 91. Let us now try to imitate the method used above in the case of the integral equation involving T. Since the eigen-values e* are not

§ 6.15]

W-FUNCTIONALS IN TERMS OF R-FUNCTIONALS

281

bounded away from 1, we cannot f 3i m the reciprocal kernel but we

can write down its finite partial sums. Let QF1(p, q) = flF(Yq)(1F(r1))-dA l (6.15.30)

got

_ -lF(p, q) - f YF(r, q)(2F(r,fi))-dA,, by (5.3.7), and let

f

IF j, q") = PF(P, q) - 2' (r, (6.16.31)

(2F(r,.'))-dA,,

/,

OM =LF(p, q) + QF'(p, q)

We have by the reproducing property of the kernel LF(p, 1), (6.15.32)

I Fif', q) = LF(ib, q)

We write QF, N (

f QF (r, q)LF(p, #)dAPI. N

//P

(6.15.33)

-v

,

q) =' QF°) (P, q), a-o

where (6.15.34)

QF °'

q) = f (20-2 ) (p, f)QF(r, q)dA,,, Q z 2,

and (6.15.35)

LF (P, q)

OF) (p,

We have, by (6.15.22) and (6.15.30), 40

(6.15.36)

QF )(p, q) _ V-1

hence by (6.15.33),

and

0F, (fi, q) _ X

v

20

f'.(p){W.(q))

LOV

1 W+*

a (6.15.37)

11 - 1)

1-C1- ()r

or= 0,1,

INTEGRAL OPERATORS

282

(6.15.38)

[CHAP. VI

IFtp,q)=-yZ j 1-tl- '2

.tq))

Hence (6.15.39)

j)fr, F, IV

q)I stp, P)dA,, = - E [

V

V-1

1 - (1 -1) ev

2N+

2]

' (P)

Thus, for fixed p, q in the interior of 1R, we have (6.15.40)

Lr (p, q) = lim

N.. V

q) I s (p, r')dA,.

6.16. THE COMBINATION THEOREM

We conclude this chapter with some remarks concerning the case in which the domain fit is the intersection of two surfaces R1 and W2. For simplicity, we suppose that 4R1, R. and fit are 2-cells. We further suppose that the boundary curves of 91, and % intersect in just two points, p1 and P. say, and that 1't is bounded by two arcs Cl and C. joining pl and p2 where C1 lies in the boundary of t1, C2 in the

boundary of t2. Later we shall take R1 and 912 to be domains of the local uniformizers zl and z2 respectively. Since 911 and R2 can be represented as domains of the zl- and z2-planes, their bilinear differentials 91(p, q') and 32 (p, q) may be assumed known and we give a

process by which the bilinear differential .2(p, q) of the union R1 U 9R2 can be determined. This process is a variant of the Schwarz alternating procedure, and yields a proof of the existence of analytic

functions on an abstract Riemann surface. Although this proof is not simpler than the standard ones, we give it for the reason that it makes our approach self-contained; even the existence theorems of Chapter 2 can then be based on the methods developed here. Further-

more, the method which we now develop is a ' natural outgrowth of our previous investigations. It should be remembered that the boundary uniformizers of DI at p1 and p2 are not admissible uniformizers for R1 and 9i2. The intersection angles of the arcs C1, C. at p1 and P. do not exceed n and we now make the assumption that each of these angles is greater

§ 6.161

THE COMBINATION THEOREM

263

than zero. Let zi be a uniformizer of 91, at the point pi and let a Z be the angle at p1 between C2 and C2 in the zi-plane. Then z = z;a will be a boundary uniformizer of 9 near p1. A differential of R1 will be multiplied by a factor a zn _il;r if expressed in terms of the boundary uniformizer of 9)1. But since 0 < cc/ar S 1, every differential of RJR, which is regular in the closure of 91, will still lead to a different-

ial on 9t which is square integrable and which possesses a bounded integral function in V. This fact will permit us to carry out all the following considerations in spite of the fact that some differentials considered become infinite near pi or p2. Since 9)2, 91, and R2 are simply-connected, there exists on each surface only the class S, so the F-subscript is superfluous since all classes are the same. In particular, M = S so all kernels are symmetric. Let L (P, q) be the bilinear differential of 9)2 and set

(6.16.1) 11(p, q) = L(p, q) -21(p, q); 12(p, q) = L(p, q) -2'2(p, q). Similarly, we define 11(p, q) and 12(p,

We begin by proving the formula (6.16.2) f l1(p, q)(12(p, r))-dA. = f (11(p, q))-12(p, ?)dA

valid for q and r in R. We have

f l1(p, q)(12(p, r))-dA1= f [L(p, q) -Y1(p, q)] (11(p, r)) dA, f .1(p, q)(12(p, r))-dA. +nz

by (4.10.8). Let f correspond to the uniformizer circle

< a at q.

Integrating by parts, we obtain

-f m

.'1(p, q)(12(p, r))-dA,

=f

q) (12(p, r))-dE+o(1)

- c1+c.-ae

aF1(p, q) aq

1

XiJ C.

(l2(p, r))-d

INTEGRAL OPERATORS

264

[CHAP. VI

since ail (p, q) /aq = 0 on Ci and the integral over 8t is o (1) as a > 0; 1

ni

since p =

f

alj(p, q) (l2 aq

Cl

(f' r)) -dz

on the boundary; ni J

1-1(p, q) lt(p, r")dx aq

agj(p, q)12(1,

ni J

agF aq' q)12(p, y)dz.

r")dz _.

C,+C,--M

ar

As a > 0, the second integral yields the value (6.16.3)

-12(q, P) = j'(L(p))l2&1P)dA,

while the first becomes (6.16.4)

`_ f 2'i(q, p)lt(p, r)dA3 au

using (4.10.2). Combining these results, we obtain (6.16.2). We introduce now the bilinear differential (6.16.5)

Q(p, q) = fli(r, p)(12(y, q))-dA,7 aR

and the two operators (6.16.6)

(6.16.7)

T,11) (q)

())roz,

T(2) (q) = (.T2 (q,

which act on differentials V' on M. We also define the operators (6.16.6)'

t,(,l)(q)

(11 (q, p),

V,

(6.16.7)'

tvt)(q)

(12(q,

Wt

For q in X12 these give the same result as the operators T, ' and Ty').

Finally we define the iterated operator (6.16.8)

412) (q) = t;i (q)

§ 6.16]


265

By (6.16.5), this may be expressed in the form (6.16.8)'

We now prove THEOREM 6.16.1. There exists a positive number a < 1 such that for all differentials yi (fi) on 13l we have the inequality NU(t121) < aNm('')(6.16.9) In order to prove the theorem we start from the inequality (6.16.10)

Ns(tt12)) 5 No(tm2>) < NU(p')

which is an immediate consequence of (6.2.9) and (8.2.11). If we can show that NN(y,')

unless ep' (q) - 0 in 9R, then the reasoning of Section 8.6 shows that

there is a number a, 0 < a < 1, such that NIR (tr,121) S a

for all differentials ip' satisfying NOW) = 1. The reasoning can be carried through because Q(q, fi) is regular in the closure of T1, except for singularities of Down character at the points p1 and p2. It remains to investigate the possibility that (6.16.10)' Nm(t( )) No (V').

In this case we must also have (6.16.11)

Nwt (t( $)) = Nu(+,').

But- by (6.2.11) N%(TT')) +N94 (1y=1)

Hence (6.16.11) implies the two identities: (6.16.12) 0 in R2 and (6.16.13) TV=) = 0 in Sts - V. From these identities we derive the fact that a ifferential tp' foi

INTEGRAL OPERATORS

268

[CHAP V]

which (6.16.10)' holds is regular in the closure of )l, except possibly

at the two critical points pl and p2 In fact, by (6.3.9), we have (T(2) )2

(6.16.14)

= I + T(2)

By (6.16.13),

=

(T(2))2

(1(2)) 2 in 9)Z

and by (6.16.12) and (6.16.14), (t(2))2

= '(q)

Since the kernel Q(p, q) of the t(2)-transformation is regular in the closure of TZ (except for the two possible singularities at p1 and p2) we have proved the asserted regularity of y,'(q). By reasoning used in Section 6.5 we see that tya) (q) has the boundary

values (,p' (q)) - on C1. To show that tea) (q) has also the boundary value (V' (q))- on Ca, a modified type of reasoning is necessary. We have t(2) (q) = -

J

12(q,

a (8

= - 2J

16.15 )

2(q, p) - G(q, p)] ap aq

82[

a 912 (q,

since

a

ii aq'

aq

p)

('(p))'

p) vanishes on C. and aGaq p) vanishes on C1 and on

C2. If q is an interior point of the arc C2, we have

f 92aq I.

dz =

- ( a9F aq, p) (,o'(p))-dz

791

(6 . 16 . 16)

C.

C.

=(] faW2(q, p)

,

aq

dz)

Let q* be a point of Vl which is near an interior point of C2. Then


§ 6.16]

zJc d

dz = nti J

(q

[a;a

267

aG(q

dz

ci+C2

1

_-

(6.16.17)

dA,

a

= ip'(q*) +

!Pva)(q*)

_ V'(q*) by (6.16.12). Letting q* tend to an interior point q of C2, we see from (6.16.15) and (6.16.16) that t,(,2) (q) has the boundary values (u'(q))on C2. We conclude that t( I) (q) ± o' (q) and (t.12) (q) - y,' (q)) /i are real

on the boundary of 9A. Both expressions have finite integrals in )J with constant imaginary part on the boundary of Tt (even at p1 and p.). Applying the maximum principle to these imaginary parts, we see that they are constant and this shows that the differentials corresponding to them vanish identically in TZ. Thus we have shown that (6.16.10)' implies y' (q) = 0 in O1 and Theorem 6.16.1 is proved. We observe that whenever a differential 92' can be expressed in the form

p'(q) = f Q(q, fi)V'(p)dA

(6.16.18)

,

then Theorem 6.16.1 implies Nv&') ;5 oc No (V').

(6.16.19)

Now let r be a fixed point in the interior of Wz - apt, and define (6.16.20)

For n (6.16.21)

Y'0(p)=f 0ra(p,

r),

pER1- I.

1, set 'iPax(p) =

-f

'

2(p,IJ1'--1.'J dA J.. Cv (p r)

02

and

6 16.22

-

f 2'1(p, q)Wv, 2(q) dAt,

1u-21L'),

p6

- Wt.

pE

1,

p E 4.

INTEGRAL OPERATORS

268

(CHA.P. VI

have at r the same singularity as £' (p, r). Let P be a point of T2. Then

Observe that all

f -'2(P, &20-2(q) dAc gr,-got

- f 32(x, 4)sn-:(4) dAt + 2' (p. r) gut

f 32(p, q)Va0-2(1) dAt + f 2 (p, PV '"(q) dA{

- f 2 (p,

dAc + -'2(1, r)

a

= -f

4) Vz»-2 (q) - 32(q, r)] dAt

dAc + 22( , r),

+ f Y2(p, 4)Vin--a(4) dAc

by (4.10.8),

22(p, r) +

f

22(p, q)V$*-2(4) dAc

f Ys(-P,4)V' -1(4)dA+3:(p,r) a by (4.10.8)';

f a

(q)]dAc.

2

Thus, for ' E fly, (8.16.23) Yz0(p)-tv

f 32(p, w

Now let 5 be a point of R1. Then

)

-w

(4)] dAt.


§ 6.163

f

289

Vzn-s(4) dAt - f .0

8t1-V

dAt

8R

= -f

dAr + f 21(x,

y,

-f

Y[1

)

Vp

.l/L

dAr

2'1(p,

In

f3i

(q)] dAr.

4)ETV -2(4) -CVs

SR

Thus, for

a 911,

(6.16.24) y2ft -i(P) -tV$x-s(p) = -

[tV9,,_2(q)-V

-s(q)]dAC.

ant

We observe that the formula (6.16.24) is valid in the case n = 1 if we

define y1 (p) = 0. Write (6.16.25)

d.' d,'

(p)

= y' (P) - y''-1(p), m =

1, 2, .

.

For p e 2 we have

-

f L ( 4') Y -1(4) - *V

-I-

so (6.16.23) and (6.16.24) become, for p e X12,

f 12(p, q)dm.-i(4) dAr, (6.16.26)6.16.26)

SR

f

11(p,

dA r.

got

Hence, by (6.16.2) and (6.16.5), dax(P) = f (Q'(q

))-4,._1(q) dAt,

SR

(6.16.27)

f Q(P. J)d9»-s(4) dAr.

-x(4)] dAr,

INTEGRAL OPERATORS

270

[CHAP. VI

But then by (6.16.18) and (6.16.19), S aNIR(d;,,

(6.16.28)

Therefore NM(d2m) S a"n-1N,(d2'),

(6.16.29)

1 NV (d$'03

-1) S am-1 NV (di)

Thus m

N,V(d,,,) S Aa, where A is a number independent of m. If m < n we have (6.16.30)

S E {Nu(d;,)}*

a4

At E a' S At 1_al,

so N (y;, - zy;,) tends to zero as m, n approach infinity. It follows from (6.16.23) that tVSm(p) converges to a limit 1P,(p) in R2 and from (6.16.24) that 8m-1(p) converges to a limit in R1. Since these

sequences have a common limit 1"(p) in R, we conclude that W1(p) = Y'$(p) in V, and Y';(p) is the analytic continuation of f'i(p) into N. - D1. We have therefore established the existence of an analytic function 11'(p) over the union of R1 and 912. Since !'(p) has a simple pole at the point r of {2 - T1, it cannot be constant over %1 U R2.

Let 9' be any function regular in the union of 1Rr and Ng with (6.16.31) N(9') < oo. Here the integration in N(p') is over the union of R1 and J}2. Then by (4.10.8) (y'o, 9") = (Vo, 9'x)8 = 0.

Further, (W,N' 9") = (So

-v

Pz-'

f

dAr

94*2

+ fs2(P. r)(9p'(p))-dA, ffis

9'')eti-m¢

")et,

§ 6.16]


271

by (4.10.8) and (4.10.8)'. Similarly Thus (6.16.32)

0, m= 1,2,....

It follows that also for the limit function If (6.16.33)

(17', V') = 0

for every qVregular in the union of R1 and R. which satisfies (6.16.31).

Let 2(p, r) be the bilinear differential of the union of 91, and 912, Then (6.16.33)'

(2, 0 = 0.

Subtracting (6.16.33)' from (6.16.33), we obtain (6.16.34)

(P-3, 92')=0.

Taking, in particular, T= ' -- 2', we see that P= .F; 3; that is, (6.16.35) W'(p) = 2(p, r). We have thus proved the theorem: THEOREM 6.16.2. The recursive procedure (6.16.20)-(6.16.22) converges uniformly in each closed subdomain of the union R1 U t9 and leads to a representation of the 2-function of this union in terms of the 2-functions of the constituent domains tl and Rs.

Since the union of R1 and t2 is simply-connected, the function belonging to the differential

¶'2(p,r)

maps the union onto the exterior of a circle, the point r going into infinity. We have therefore proved the "combination theorem" of H. A. Schwarz and C. Neumann: if two domains, each of which can be

conformally mapped onto a circle, have a connected (and therefore a simply-connected) intersection, the union of the two domains can also be mapped onto a circle. Since the proof of the uniformization theorem essentially turns on the combination theorem (see [4]), we have established the uniformization theorem by the methods developed in this chapter.

272

INTEGRAL OPERATORS

[CHAP. VI

REFERENCES. 1. S. BERGMAN and M. SCHIFFER, "Kernel functions and conformal mapping," Gompositio Math., 8 (1951), 206--249. 2. R. COURANT, Dirichie$'s Principle. conformat mapping, and mammal surfaces, Interscience, New York, 1950. S. R. COURANT and D. HILSERT, Methoden der mathematisehen Physik, Vol. I, Springer, Berlin, 1931. (Reprint, Interscience, New York, 1943). 4. L. SCHLESINGER, Automorphe Funktionen, GSschens Lehrbticherei, Vol. 5, De Grayter, Berlin, 1924.

7. Variations of Surfaces and of their Functionals 7.1. BOUNDARY VARIATIONS

The theory of Riemann surfaces is mainly concerned with the study of the functionals on a given surface or on a given class of surfaces, and the dependence of the functionals on the surface itself in the sense of the functional calculus has never been investigated systematically. In this chapter we investigate the way in which the functionals change if the surface is altered. So far as topological properties are concerned, every finite surface may be obtained from the sphere by performing, a finite number of times, one or more of the following three types of operations: (i) cutting out holes; (ii) attaching handles; (iii) attaching cross-caps. However, the conformal type of a Riemann surface may be altered by the topologically insignificant operation of "attaching a cell". To attach a cell, we first form a hole 'by removing a 2-cell from the surface, then we fill the hole with a new 2-cell which is attached by identifying its boundary with the boundary of the hole. We therefore add a fourth operation: (iv) interior deformation by cutting out a

hole and attaching a cell. We shall compute the variation of the Green's function under these four operations. By differentiation we then obtain the variation of the bilinear differential LM(, q) and, by computing its periods around the basis cycles, we obtain the variation

of the differentials of the first kind. The operations (i), (ii), and (iii) will be performed in a special way, since their main function is to change the topology, and the resulting variations therefore will involve a minimal number of free parameters. The operation (iv), on the other hand, will supply the

missing parameters and will provide the necessary generality for the variations. In particular, the operation (iv) may be performed in such a way that the conformal type of the surface is preserved. The fact that conformal type can be preserved has an important [273]

VARIATIONS OF SURFACES

274

(CRAP. VII

application in constructing variations of conformal maps of a given surface 9`c onto subdomains of a given surface R. In Chapter 5 we derived necessary and sufficient conditions for such a mapping; in

the present chapter we give the variational procedure by means of which a given mapping from % onto a subdomain 91 of R can be varied to produce a mapping from 9 onto a neighboring subdomain

9)t* of 91. This provides a powerful tool in the investigation of extremal problems relating to one-one mappings of 91 into R. Every orientable finite Riemann surface with boundary is properly imbedded in its double, and may therefore be varied by shifting the position of its boundary on the double, the double being held fixed.

This type of variation is called a "boundary variation", and is historically the oldest variation in conformal mapping. We therefore

begin by discussing boundary variations, and we show that such variations arise automatically from the, considerations of Chapters 5 and 6 by taking the domain U?. to be such a large part of the domain

91 in which it is imbedded that the residual domain 91 - 9)2 is a thin strip. In the case of plane domains bounded by analytic curves the variation of the Green's function can be expressed by Hadamard's classical formula [3]. Let 91 be a domain of the plane bounded by

analytic curves C v = 1, 2, - - , n, and let every point of the boundary be expressed by a parameter s which measures the lengths of the curves successively. A neighboring domain 9)1* may be defined

by displacing each boundary point of 0 along the normal, thus defining new curves C,*, v = 1, 2, , n, which bound 971*. Let 6n(s) = 8v(s) be a continuously differentiable function of s which determines the normal displacement of the boundary point r = r(s) of the domain 9. We suppose that 6n (s) is positive if the displacement

is in the direction of the inner normal with respect to 91. If G(p, q)

is the Green's function of Bt, G* (p, q) that of 9)t*, Hadamard's formula states that (7.1.1)

G*

(p, q)=G(p, q)

-

s i-X

J C

8G(r, p) 8G(r, q) v(s)ds+o(s). 2n 2n

Using the notation of the functional calculus, we may write (7.1.1)

in the form

BOUNDARY VARIATIONS

1 7.11

(7.1.2)

r aG( p)

6G(p, q)

276

q)

finds.

C

If 9X is an abstractly given Riemann surface with m boundary components C,, v = 1, 2, , m, we define a surface Wl in the following manner. Let G(p, q) -}- iH(p, q)

(7.1.3)

be the analytic completion of the Green's function G(p, q) of Wt. If h is the genus of Wt, we have (7.1.4)

Im Z$»+.(q)

= in

faG(P))ds.._i_fdH(pq)>o . J

C,

c,

For a fixed qo in the interior of Wt, the function (7.1.5)

-}- iH (p, qo) C= exp - G (p,Imqo)Zm+.(qo)

}

is single-valued on C, and maps it onto the circumference + s` I = 1 in the plane of C, and it may therefore be regarded as a boundary

uniformizer in the large. Each boundary component C. of W2 is mapped onto a unit circumference in this manner, and we define the

arc-length parameter. s, 0 S s S 2vm, in terms of these circumferences. A normal displacement fin(s) in the planes of these circum-

ferences may be defined as before and, as s runs over fin(s) determines a neighboring curve y,. Let ri be a small positive number, and add to W2 the set of points

corresponding to the m annular neighborhoods 1 < I C. < 1 + r/. In this way we define a finite Riemann surface t containing WZ in its interior. It is clear that R satisfies all the conditions imposed on the finite Riemann surfaces heretofore considered, and we call fR an enlargement of M. The curves y,, v = 1, 2, -, m, bound a subdomain W1* of 81 with Green's function G*(p, q), and Hadamard's formula is valid. Since formula (7.1.2) has a form which is conformally invariant,

we may express it in terms of any local boundary uniformizers z = z(r), z x + ay, and it becomes

VA.RIATIU_YS OF SURFACES

276

(7.1.6)

f

6G(p, q)

[CHAP. VII

-

aG(ay p) aGay q)

C

aydx.

The proof of formula (7.1.6) in the general case does not differ essen-

tially from that for plane domains. Since G(p, q) does not change its value along C, we have

aG(r,p)T-dG(r,p)_- i aG(r,p)

aG(r,p)_ ax

0'

az

ax

2

ay

Hence, we may put (7.1.6) into the equivalent form 2

f

aG(p> q)

C

8G (r, p) aG(r, q) 6ydx ax az

or, invariantly, (7.1.6)"

2

6G(p, q)

or

f

C

aG r, p)

ar

q) G aq

ands.

Differentiating this formula with respect to p, q and q we find, by (4.10.1) and (4.10.2), (7.1.7)

6LM(p, q) = - f LM(r, P) LM(q, P)6nds; C

(7.1.8)

SLM(p, q)

=

-f

LM(r, p) (Lm (r, q))-Bends;

C

(7.1.8)'

6LM(p, q)

f LM(7P,

LM (r, q") ands.

C

Taking in both sides of (7.1.7) the periods as q describes the cycle K,, we obtain by means of (4.10.15) (7.1.9)

J'LMfr ,p)Z.(P)8nds = f LM(9', p)Z,(r)&nds. c

G

If we, take now the periods with respect to a cycle K0 on both sides of (7.1.9), we derive by (4.3.10) (7.1.10)

or", = JZ.()Z(r)ou1s. C

§ 7.2]

VARIATION OF FUNCTIONALS

277

From these variational formulas one can then readily derive (7.1.11)

6LF(q, p) = - f LF(r,1i) LF(q, f) ands, c

(7.1.12)

8LF(q. ) _

f LF(r, P) (LF(r, q))-ands. C

These formulas express the first variations of the functionals LF, Z,' and I',,, in their dependence on the domain. In the case of plane domains these formulas have been derived from Hadamard's formula

in [6a]. 7.2. VARIATION OF FUNCTIONALS AS FIRST TERMS OF SERIES DEVELOPMENTS

Instead of deriving the formulas (7.1.7)-(7.1.12) from Hadamard's variation formula for the Green's function, we now proceed differently.

Let P be a subdomain of 91 whose boundary is obtained from that of 91 by an analytic deformation (defined below) depending on a parameter e. For a fixed point q interior to 9 we show that, under these circumstances, !F(P, q) = 0(e),

ZF(p, f) = 0(e),

uniformly for P in the closure of 9l. Formulas (7.1.7) and (7.1.8) for a positive an then follow at once from formulas (5.3.27) and (5.3.27)', and (7.1.9), (7.1.10) are obtained by computing periods from (7.1.7). By subtraction of the variation formulas for two domains

embedded in K we eliminate the condition that an is positive. Finally, by applying the results of Section 6.15 we obtain formulas which enable us to compute the variations of the domain functionals to arbitrarily high orders, that is to say, to arbitrarily high powers of the parameter e. Our method therefore yields more information than the one which is based on Hadamard's formula alone. We suppose that 8 is given and define a domain 1 embedded in SJR in the following manner. Let the v-th boundary curve of RJR be B,. The function (P, qo) + i.r°(p, q0)1 (7.2.1)

= exp

Im Zy,+I(qo)

J


278

[CHAP. VII

maps B. onto the circumference I C I = 1. Moreover, it maps a boun-

dary strip of R near B, onto a ring 1 - 36, < I ( < 1, 6, > 0. < 1 with

Let 2,(C) be regular in the closed ring I - 26, 5 Re{11'c(l)I

(7.2.2)

0, the level curve 1

V(p, po; qv qo) = loge will be a Jordan curve C1 = Cl (po, qo, q1, e) which bounds a domain

¶ of I2 containing q1 in its interior. Here po, qo are fixed points of TZ distinct from q1. Let p, q be two points lying in a neighborhood of q1, and let the coordinates of p and q, expressed in terms of the same uniformizing variable, be C and respectively. Then (7.4.2)

exp {Q,,,(p) - D,,,(po)} = a1(C - r!) + a2 (y -77)2 + ..., I a,

0,

where a1= a1(q), a2 = a2(q), etc. We suppose that and vanish at q1, and by appropriate choice of the uniformizing variable we may suppose that a,e(gl) = 0 for k z 2. Taking logarithms of both sides

of (7.4.2), we obtain (7.4.3)

Q,., (p) - Q.,Q(po) = log (C -'7) + bo + b1(C -

+b2(C-19)2+.. ,

r!)

where bo = log a1, b1= as/a1, b2 = (2a1as-a2)/ai, etc. Differentiating

the real part of (7.4.3) twice with respect to q and then taking q = q1, p e C1, we obtain, since b,(q1) = 0 for v > 0, (7 . 4. 4)

2

aV (q1, qo; p, po)

_

1

aq1 (7 . 4. 5)

2

- 2 aRe bo

1

a2V (ql, qo; p, po) ag19

+0(6),

aql

= -+0(1). 1

`s

s

Taking q = q1, p e C1, in (7.4.2) we find that (7.4.6)

_

exp {s2,,,1(P) 1

(7.4.7)

=

e2

al I2

-

,,,1(po)}, e4

1

,

1

2 = I a1(42


292

[CHAP. VII

Thus for 5 e C1 we have 282

_

(7 .4.8)

2

I a1

aV (qi, q0; P, po)

+ aRe ho +

aq1

aqi

{

C2 = 284

(7.4.9)

a2V(g1, q0; P, Po)

0(8)

+ 0(1).

aqi

I ai 14

Let lt* be the surface 9)1- Z, G* (p, q) the Green's function of 2R*, and consider the function h. (P, Po; q, qo; q1) = G* (P, q) -G'*(P, qo) V (P, po; q, q0) + V(-q1, po; q, qo) As a function of ', ha is regular harmonic in 9)1*. For p e Cl, q a fixed

(7410)

interior point of 9)'t*, the first two terms on the right side of (7.4.10)

vanish, and we have by Taylor's Theorem

h,=- V(P,Po;q,g0) ±V(g1,po;q,g0) aV (gi, Po; q, q0)

2Re j

l

+

aqi

482

a12 284 I a1 I4

1 a2V (qv po; g, qo)

q0; P,

)

+ a Rebo'll

[dV(ql, aqi

aqi

+0(83)

aqi

2 18V(gi, p0; q, qo)

Re

2+

Re a2V (ql, PO; q, qo) a2V (q1, qo; p, Po) + 0 (

I

J

aqi )

aqi

aqi

Let

g _ hs {-

482 I

+I

a1

12

Re aV (q1, Po; q, q0) àV (q1, q0; 0, Po)

2e4 a1 14 Re

aqi

L

aq1

8Re 1.]

+

aqi

I

a2V (q1, po; q, q0) a2V (q1, q0;', po) aqi I aqi

As a function of p, g8 is regular harmonic throughout V* and it is 0(e$) on the boundary Cl of ER*. From the maximum principle we

conclude that g, = 0(e8) throughout 9)1*. In any compact subdomain of 9)1* we therefore have 482 h.+ a1I2Re aV (q1, p0; q, q0) [aV(l. q0; (7

4 11 )

8q1

= 0(83).

8q1

8 Re bo

,

+

aqi

§ 7.5]

ATTACHING A HANDLE

293

Thus G* (p, q) = G* (p, q0) + V (p, po; q, q0) - V (ql, pe; q, q0) 462 8 Re bo]} Re jl aV (ql, po; q, q0) [aV(i, qo; p, po) + 12 al agl aql a4i

)-

( 7.4.12

+ O(ES)

in any compact subdomain of 0t*. Differentiating (7.4.,12) with respect to b and q, we have LM* (P, q) =LM(P, q)-I

al 12 jLM(q, gl)LM(P,gl) +LM(q, gl)LM(p, ql)}

(7.4.13)

+ 0(E2).

This formula should be compared with the corresponding formula (7.3.14) which holds for a domain x,11 with boundary. On computing

periods in (7.4.13) we find that Zµ (t') = Z" (P)

-

I al

I2{ Zµ(g1)LM(Y, ql) +

(7.4.14)

qi) }

+ 0(e2).

Finally, computing periods in (7.4.14), we obtain (7.4.15)

r = r,, +

2 I2all2

(u,v= 1,2,...,2h). The variation formula for LM(15, q") is obtained on replacing q by q in (7.4.13). 7.5. ATTACHING A HANDLE TO A CLOSED SURFACE

,

We have constructed variations which arise when holes are cut out of surfaces. Every finite orientable surface with or without boundary can be obtained from the sphere by cutting holes and attaching handles, and it is therefore desirable to study the variations

which arise by attaching handles. The handles attached will, like the holes, have a special form, but we shall later indicate a method whereby a surface can be varied without changing its topological type. By combining our variations it will then be possible to* pass

294

[Cssr. VII


from the sphere to any other finite orientable Riemann surface of given conformal type. We consider a closed surface V. Given two fixed points q1, q2 in 9 and using an unessential parameter point po, we define two Jordan curves c1 and c2 surrounding q1 and q2 by (7.5.1)

V (p, po; q1, q2) = log E, V (p, po; q1, q2) = log a,

respectively. Let v(p, po; qx, q2) be an analytic function of p in Wi whose real part is V (p, po; q3L, q2). We choose two arbitrary points pi°) and p;°° on c1 and c2 and have by (7.5.1) (7.5.2)

v(pio), fo; q1, q2) = v(p9°), P0; q1, q2) + 2 log E

-ia

where at is real. Starting from p(°), p( o) we identify points p1 on c1 and

P. on c2 if they satisfy the equation (7.5.3)

v(p1, po; q1, q2) = v(p2, po; q1, qs) + 2log

1 -ia.

We can easily see that as pl moves around cl in the clockwise sense with respect to ql, the identified point p2 moves around c2 in the counterclockwise sense with respect to q2. The surface formed by removing the hole at q1 bounded by c1 and the hole at q2 bounded

by c2 and then identifying points of c1, c2 according to the rule (7.5.3) will be denoted by W2*. It is clear that Wt* is formed from W2

by attaching a handle. We denote by Q (p) and V*(p, 150; q, q0) the Abelian integral of the third kind and the Green's function of the surface Wl*. We state now a general theorem which is useful in all variational problems to be considered. THEOREM 7.5.1. Let W2 and M* be two closed Riemann surfaces which have the domain WIo in common. Suppose that there exists a set

K of cycles K, in V. which is-canonical with respect to 9 and can be completed to a canonical set with respect to W. Then, for every quadruple p, po, q, q0 in M4, we have (7.5.4) V * (p, PO; q, qo)-V (p, po; q, qo) = Re { (t)} . 2 t i J Q 0 (t)d.Q ask

§ 7.5]

ATTACHING A HANDLE

295

Here V, V* and S2, Q* are the Green's functions and integrals of the third kind with -respect to at and MZ*. We prove this theorem by. Riemann's classical method of contour integration. We select two fixed points q and q0 in 92a and connect them by a smooth curve y e 920. Let 3 be the domain obtained from

etc by introducing cuts along K and y. Obviously, Q., (P) will be single-valued in J. Thus, we may apply the residue theorem as follows: (7.5.5) 1

J

92°°°(t)d-0406 (t)

= - tva,(P) + D..(po)

aar

On the other hand, a = Mo + K + y so some of the integrations indicated in (7.5.5) can be carried out by introducing the known periods of the integrand. We observe first that the determinations of Sho(t) on the two edges of y differ by the amount 2 ri so that the integration over both edges yields the contribution S2 9,(q) + QZ0*,0(g0)

The integration over K can be carried out by the general method of Section 3.4. We obtain sr

JJ.

'Q°'"(t)dQ9 "(t) = E i (K"' K,,)

K

dS2o.

dS2,

2n.; KA

K,

We need not calculate this expression; it is sufficient to observe that Q and D* have single-valued real parts in 3o by their definition. Hence their periods will be imaginary and the whole preceding term

has the real part zero. Thus, using (4.2.23), we may put (7.5.5) into the form (7.5.6) Re

QQ,o(t) d S 2

l

a(t)} = V* (p, P0; q, q0)-V (p, po;q, qo)

on,

This proves the theorem. Another elegant formula may be derived by the same considerations for the Abelian integral co., (p) which is characterized by the property that its periods with respect to the cycles K,1, mare zero. This integral

satisfies by (3.4.7)' the symmetry law (7.5.7)

W(p, po; q, qo) = W,QO(P) +WQQ,(P.) = w., (q) -co,,.(g0)


290

[CHAP. VII

We find the following formula for W, analogous to (7.5.4) : W* (i,, &o; q, qo) - W (p, PO; q, qo)

-- -i f 2ai

(7.5.8)

,

TV (t, to; q, go)dW (t, to; P, po)

an,

Let us apply Theorem 7.5.1 in order to calculate the Green's function

V* of the surface M* obtained by handle attachment. In this case, the domain Dlo consists of T't minus the interiors of the curves cI and c2. Thus, we have: (7.5.9) V*(p, Po; q, qo)-V(p, 2o; q, qo)=Re j

9 o(t)

2ni,!

C,+y

I c2 is to be understood where the integration along the curves ci and in the positive sense with respect to Yto. In order to calculate the integrals we introduce the local uniformizers Cx

= exp { - v(p, po; q1, q2)}, for p near

tS2 = exp { v(p, po; q1, q2)},

for P near

Then the identification formula (7.5.3) establishes relation between the local uniformizers:

(7.5.11)1

qj, q2.

the following

?7E2

i2,

where (7512)

27 = e`°`,

I

ii I = 1.

Conversely, we may prescribe the parameter a in (7.5.3) or (7.5.12)

arbitrarily and determine by this choice two particular points ki(o) and p2) on cl and c2 which correspond under the identification.

In integrating in (7.5.9) over the curve c1 we have to use the uniformizer t l in Q but can use Ca in D* since £2* is an analytic function on the surface 7t*. We may write, however, (7.5.13) aa,(t) _ avo(gl) -

QQV,(g1)Ci + .. . 2

§ 7.5]

ATTACHING A HANDLE

297

and replace Z1 by CS1 using (7.5.11). We find

(7.5.14)

2ni

f

,0 (t) = 0a0(g1) ' p

0

J

-t- terms containing e4 at least.

+ D,',0 (q1) - Zn f p1 J S2 CS

The integration over c1 is to be carried out in the positive sense with respect to STZO, i.e. clockwise. Hence, the integrations over c2 are to

be taken in the counterclockwise sense. The integrations need not be carried out over the curve c2 but may be taken over any homologous curve so long as neither P nor p0 lies in the domain bounded by these two curves. This shows that the second righthand term in (7.5.14) is 0 (e2) and the remainder term is 0 (e4) so long as P and p. stay in a closed subdomain of 9t0. Let us consider, first, the term

12vif dQ*P'0 Cl

-

1 f 2V* (t, t0; P, P0) ds . ant

t

Cl

This expression is a real-valued harmonic function of p on W. If we approach c1 from 9R0 and cross this cycle, the value of the harmonic function will jump by unity. The function V (P, PO; q, q0) is also harmonic but discontinuous in J2* and jumps by the amount 2 log e. Let us assume that the original surface 1Y has h handles. Then, we may write (7.5.15)

Im {

Z*2A1 f dS2*g0 = V (p, Po; q1' q2) 2ni

C,

2 log

e

We recognize that the integral (7.5.14) is o (1) and that the analogous integral over ca converges to zero as e > 0. Thus, by (7.5.9), we have (7.5.16)

V*(p, p0; q, q0) - V (P, p0; q, q0) = o(1) in each closed subdomain of X20. If in (7.5.14) we replace the curve c2 by a permissible homologous path of integration Y2 in some fixed

closed subdomain of0, we may replace dQ* in the integral by dQ

without affecting the result up to an order e2, since Q*' (t) _ 2 aV* (t, t0; q, q0)/at. Now, we may apply the residue theorem and


298

[CHAP. VII

we find that SQo,,(t)dD,,.(t) (7.5.17)

7"

QQao(g1) Im

c

+ g2?? D,,,. (g1)'Q9p (q2) + 0 (e4)

In a similar way, we may calculate the contribution of the integration

over c2. We observe that 27vi

f "Sn'.

27v1 J

Cl

dsr90

c,

if both integrals are taken in the clockwise sense. Hence (7.5.9), (7.5.15) and (7.5.17) lead to the result V* (.b, po; a, a.)

= V (p po: a, a.) -

V (q, q0; q1, q2)V (p, po; q1, q2)

2log-

(7.5.18)

E

+ Re {s'77 [Daa,(g1)Q ,0(q3) + D.,.(g2)Qnvo(gi)]} + o(e2). Since QQQO (ql) can also be written as 2 aV (q1, q2; q, qo) l aq1, we may

express the variation of V in terms of V itself and of its derivatives: V * (p, po; q, q0) = V (p, po; q, q0)

-

V (p, po; q1, q2)V(q, q0; q1, q2) 1

2 log8

(7.5.18)'

+ Re j 4sarj CaV (q1, q2; q, q0) aV (q1, q2; p, po) aq1

aq2

+ aV (q1, q2; q, q0) aV (q1, q2; p, po) aq1

aq2

l + 0(82).

JJ

Differentiating (7.5.18)' with respect to p and q and using definitions

(4.2.25) and (4.10.1) we obtain the formula 1

LM (p, q)'= LM (p, q)

1

log

(7.5.19) - awe {fl [LM (p, g1)LM (q, q2) + LM (p, g2)LM (q, q1)]

+ 4 [LM(p, g1)LM(q, q2) + LM(p, g2)LM(q, q1)] } + o(e2)

§ 7.6)

4F1'ACUH.JG A4 HANDLE

299

where (7.5.20)

2

Z h+i

1 a2aa () ` aiai

2logE

By formula (3.4.4) P(S?ala9, K,) = 2ri Im {Z,(q2)

(7.5.21)

so (7.5.20) gives: (7.5.22)

Im

P(Zan+1, K,)

Z=(g2)},

log E

that is (7.5.23) r2n+i,

y

=-

7z

1

Im {Z,(q1) - Z,(q2)}, v = 1, 2, ... 2h.

log E

It is interesting that the new periods _Pv*+I,,, are independent of s7, that is, of the particular way in which the curves ci and c2 are identified. We can also derive formulas for the change in the other integrals

of the first kind by computing from (7.5.18) the periods of the analytic completion of the V-functions. The new period matrix I'1* can then be obtained by considering the periods in the variational

equations for the integrals of the first kind. We do not enter into these obvious calculations. 7.6. ATTACHING A HANDLE TO A SURFACE WITH BOUNDARY

We derived in the last section a formula for the change of the Green's function of a closed surface if a handle is attached to this surface. It is evident that this result enables us to derive an analogous formula for the case of a surface V with boundary. In fact, we may imbed such a domain 931 into the closed surface = 931 + OZ obtained by doubling Wt. If 0 and q are two points of W2, the Green's

function G (p, q) has by (4.2.1) the form (7.6.1)

g) = V

300


[CHAP. VII

where V is the Green's function of the double. Instead of considering

the change of 9 obtained by identifying points along two curves cl and ca in T2, we may ask for the change of ILN if we identify not

only the points of these two curves but also the points of their images 'c 1 and c 2 in fn by the corresponding law. Such a variation will transform a into a new closed surface a* and we will be able to compute the variation V* - V by the results of the last section. We choose two points ql, q2 in B and consider the function (7.6.2)

U(1i; q1, q2) =

2

[V(p, p; q1, ql) - V (25, p; q2, q2)]

=G(p,g1)-G(P,g2) We consider the loci on la where (76.3)

U(p; g1, q2)

= log

I

,

U(b; g1, q2) = log s.

They consist of the curves cl and c2 in l and, since U(p; ql, q2) _ - U(p; ql, q2), of the image curves c2 and cl in fR. Let u(j; q1, q2) be that analytic function on which has U(p; q1, q2) as real part. We identify points on cl, c2 and a 1, c 2 as in the last section by the relation (7.6.3)'

u(P1; q1, q2) = u(P2; q1, q2) + 2 log

! -ia, 251 E c1,

t'2 E C21

where

(7.6.3) u(p1; q1, q2)=u(p2; q1, q,,)-2 log E + ia, pl f Cl, p2 E c2.

In the last section, we studied the effect of the attachment of one handle to a closed Riemann surface. If we had attached simultaneously several handles we should have been able to carry out the same considerations, integrating over several systems of identified curves instead of one. One can easily verify that, up to the order 82, the result of the attachment of several handles at the same time is obtained by adding the effects of each individual handle attachment. Thus, under the change (7.6.3)', (7.6.3)" of into t *, its Green's function will change according to the formula

§ 7.6]

ATTACHING A HANDLE

301

V* (p, po; q qa) = V (p, po; q, go) V (P, po; q1, q2)V(q, q0; q1, q2)

2 log (

V (PI po; 41, q2)V (q, qo; qj, q2)

1e

2 log

1e

[aV (q1, q2; q, q0) aV (q1, q2; ;6, p0)

+ Re j

aq1

l

aqs

+ aV (q1, qa; q, q0) aV (q1, q2; p, po)

(7.6.4)

aq,

aq2

+ aV (q1, q2; q, q0) aV (q1, q2; p. po) 42

aq,

+ aV (q1, q2, q, qo) aV (q1, q2; p, po) aqj.

aq2

+ 0(82).

This lengthy formula reduces considerably if we put P. = 3 and q0 = q. We use the following simple identities: (7.6.5)

V (q, q; q1, 42)

= - V (q, q; q1, q2)

and (7.6.6)

V (q, q; q1, q2) = G(q, q1) - G (q, qs) = U(q; q1, q2) The first identity follows from the reality and antisymmetry of V, and the second identity becomes obvious if we observe that V (q, q"; q1, q2)

is a harmonic function of q which vanishes on the boundary of Tt and has simple logarithmic poles at q1 and q2. (7.6.6) may be considered as a generalization of (7.6.1). If we differentiate (7.6.6) with respect to q1 and q2, we obtain: ^ aG(g1, q) aV(g1, q2; q, _ -aG(g2, q) (7.6.7) aV(g1, q2; q, q) aq,

aq,

422

aqa

Using all these relations, we obtain from (7.6.4)

G*(p a) = G(p a) -

U(p; q1, q2)U(q; q1, q2)

2 log-

(7.6.8)

e

- Re 4E I

raG(g1, p) aG(g2, q) + aG(g2, p) aG(g1, q) L

aq1

aq2

aq2

aq1

+o(en).


802

[CHAP. VII

This formula gives the change of the Green's function of a domain with boundary if a handle attachments is performed by means of the identification (7.6.3). Differentiating (7.6.8) witb respect to P and q, we find that 1

L* (25, q) = LM (p, q)

(7.6.9)

-

- ! log

1

Za*'+1(P) Zi+1(q)

axes{f[LM(p, g1)LM(q, q2) + LM(¢, g2)LM(q, q1)]

+ n[LM(P, 11)LM(q, ga) + LM(P, g2)Lm(q, ql)] } + o(e$).

Here we write (7.6.10)

Zh+1(() =

1

2 log

1

u(P; q1, q2)

which is indeed the new integral of the first kind added by the attachment of the new handle. Its real part is a single-valued harmonic function on D%* but has a jump of amount 1 if we cross the cycle c1, which equals ce after the identification.

We have by definition (7.6.2) and by (4.3.1)' (7.6.11)

P(Z h'+1, K.) =

1

[Im Z,(q1) -- Im Z,(qE)],

logs That is, M+V

K.)=- n1 [Im Z,(q1)-Im Z,(q2)], log

a

v=1, 2,- " ", 2h+m-1.

Therefore, computing periods in (7.6.9), we have Z*'(q) (7.6.13)

= Z.(q) - Z*u+1(q) Im {Z.(q1)

-

Z,(q2)}

-re2{1l [Z.(g1)Lm (q, qt) + Z.(g2)LM(q g1)]

+ I [Z.(gi)LM(q, q) + Z,'(g2)LM(q, q1)]} + o(as).

Finally, taking periods again in (7.6.13), we obtain

§ 7 7]

ATTACHING A CROSS-CAP

I'* (7.6.14)

303

Im {Zµ(g1)-Z. (qa)} Im{Z,(g1)---Z,(g2)}

log-

- 2,ve2 Re {r1 [Zµ(g1)Z,(gs) + Zµ(g2)Z,(g1)]} + o(es) Thus, we have obtained the variational formulas for the differentials of wt. 7.7. ATTACHING A CROSS-CAP

For the sake of completeness, we now construct a variation which

arises by attaching a cross-cap to the surface V. Means are then provided for passing, by infinitesimal steps, from the sphere to any finite Riemann surface, orientable or non-orientable. We recall that a non-orientable surface has a single-valued Green's function defined by (4.2.31). Let J2 be a surface (orientable or non-orientable) which has a single-valued Green's function G(p, q). If 932 is closed and nonorientable, it has no single-valued Green's function and we exclude this case, for the sake of simplicity. However, the treatment of the

excluded case is the same, apart from the fact that care must be taken to specify the branches of the Green's function. The method for the cross-cap attachment is closely related to the

preceding method based on Theorem 7.5.1. However, instead of working with the analytic functions

(t) we shall now work with

the single-valued Green's function on the surface and instead of Cauchy's theorem we shall use Green's formula. We shall also make

use of the Green's function in order to determine, in an invariant way, curves on 1't to which the cross-cap is to be applied. For sufficiently small e, e > 0, the level curve (7.7.1)

G (p, q j) = log

1

is a Jordan curve c which bounds a subdomain of it containing q1. For p in a neighborhood of q1, let (7.7.2) C = exp { - T (p, q1)} where T (P, q1) is the aalytic functigA whose real part is the Green's function. The curve c appears in the c-plane as the circle


304

[CHAP. VII

We remove the interior of c from the surface UI and identify the in his manner forming a new point , of c with the point surface Z* with a cross-cap. We observe that the identification is established by means of an anti-conformal (or indirect conformal) mapping. If C11 C2 are a pair of identified points on c, we have C (7.7.3)

e2

1

- - -

2

Ca

=

e2

-

-. Y bl

The pair of identified points C. C2 gives rise a single point of the surface W. In order to define a uniformizer at this point, let N1 and N. be half-neighborhoods at C1 and C. respectively which He exterior to c and set C E N1,

C, (7.7.4)

C E NE.

In the plane of t, the half-neighborhoods fit together to form a complete neighborhood, so t is the desired uniformizer. Let a/an denote differentiation with respect to the normal which points into the exterior of the circle c. If r1, r2 are a pair of identified points on c with coordinates gl, Ca respectively, we have (7.7.5)

an

(C2)ds2

and (7.7.6)

ds1

an

aG(ri, q) dsa an8

-

= - an

aG(r1, q)

an an,

ds

(C1)ds1

. 1

Suppose first that It is a non-orientable surface with boundary and let G* (p, q) be the single-valued Green's function of t* which is defined by means of the quadruple (Section 4.2). Then (7.7.7)

G*(rs, p) = G*(ri, p),

aG 8 s, p) dS2 a

_

p

AL.

Hence, writing c = cl + c=, where c1 is the upper semi-circle, c1 the lower semi-circle belonging to c, we have


§ 7.71

(7.7.8)

f

aG (r, p) ds =

C

,p

(' aG*a(

dsl +

Cl

305

f aG

(Y2, ) ds2

= 0,

C,

by (7.7.7). By Green's formula applied to the surface l1* cut along the cycle c, we have G* (p, q) - G (p, q)

(7.7.9)

aGan'I

2n J L G*(yl' c

rr

1

__

_

q) -G(rl,

(rtii, p)I

q) aG

dsi

1

aG (rv a)

ant C

- G(rl, q) aG* (raP p)j ds2, ant

by (7.7.6) and (7.7.7). We may develop G (r, q) into a series about the point q1, obtaining (7.7.10) G(r, q)=G(qi, q)+2 Re

aGagl, q)

+ ;

a2G(qj, q) C2 + ...

.

We take C = Ci = -e2/Z2 in (7.7.10) and substitute into (7.7.9). Dropping the subscript 2, we obtain (7.7.11)

G*(p, q) - G(p, q)

f [G* (rp) an(2 Re C

aG an r' p)

`

2 Re {

aG (ql, q) E2

qi

e2 aG(g1, q)

-

gi

-

+ ...

a2G(g1,

2

qi

1 a2Ga(g1, q) is' 1

qi

C2

+

})]

1

+ G(gl; q)

an fac*(r,p)

ds

'

ds,.

e

The second term on the right is zero by (7.7.8). To estimate-the first

term, let c' be the circle (C ( = a, where a is a positive number independent of e, a > s. By (7.7.11) and Green's formula

[CHAP. VII


308

G*(p,q)_G(p, q) J

[c*

aGag1

(

8n

aG**a(n, p) (2

l

2n

G (r, P)

I

an,

( 2 Re

-

q)

Re J aG(gl, q)

sel

4,

I l ]dSr + 0(84)

aG (qi, q 1 ) qi

(2Rejl

aGa,

C

I

agx.q),32

ds, + 0(e4)

Re aG(g1, q) 2s2 r [G(r, p) a an `C) (i J I a#, 2n J 0

a Re G(q1, q) a 4e' r [G(r,p) an log = 41 aqi 2n eJ

-log

-Re

1

(j -an aG(

i ds } +0(84)

UsaG(gi,P)8G(gi,q)1+O(e ),

l

aq,

aq1

j

that is,

(7.7.12) G*q)-'G(p,q) In formula (7.7.12), we have obtained an asymptotic expression for the new Green's function G* (p, q) in terms of the original Green's

function G(p, q) and its derivatives. In all variational formulas the

end result is of this form; but it is, in general obtained by first deriving an exact integro-differential equation between the new and the old Green's functions and by replacing G* by G in some places, introducing a small error but making the result more easily applicable. We want now to replace the asymptotic formula (7.7.12) by an exact integral equation for G*(p, q).

§ 7.7]


307

To obtain such a formula, we go back to (7.7.9). By (7.7.8) we have

G* (p, q) -- G(, q) ?ac

f [G*(r,p)!(G(r,q)_G(q1, aG*(r, p)

q))

G

(G(r,

q) - G(qi,

an, I Fn

fI

G*(r,

as , q) -4(r, an,

p)

aG*

q)

C

anr, p)lds,.

t

where

d(r,q)=G(r,q)-G(g1,q)

(7.7.14)

Let

T* (r, P) = G* (r, p) +iH*(r, p); (7.7.15)

Q(r, q) = T(r, q)-T(g1,q) = 4(r, q)+=J(r,q)

The.,

G*(p, q) -- G(p, q) =

J

[G*(r, p)dJ(r, q) - 4(r, q)dH*(r, p)]-

(7.7,18)

In this formula the integration around c is in the clockwise sense. By (7.7.8),

fdH*(r,p) = 4 ,

(7.7.17)

C

so, integrating by parts, G* (p, q) - G(p, q) _ (7.7.18)

j- f [G* (r, p)dJ(r, q) + H* (r, p) dd tr, 4)]

= Rei 2 f [G*(r, p)dQ(r, q) + iH*(r, p)dQ(r, q)]} l

Re{

p)dQ(r, q) J = Re{ 2 i f Q(r, q)dT*(r, p) 2ni f T*(r, t

That is, (7.7.19)

G* (p,q)--G(p,q)=Rej

2xi

f Q(r,q)dT*(r,p) C

308


(CHAP. VII

Suppose, however, that G* (p, q) is the Green's function of t* defined by means of the double. In this case, 97t may be either closed or with boundary. Instead of (7.7.7), we have (7.7.7)' G*(r2,

G* (r1, p); aG*(r2. pdss

= aG*(r1, p) ds1,

ant ant where r1, rs are identified points of c. The function G* (p, q) is not single-valued on *, but it will be single-valued on 9X* cut along c. Writing (7.7.20)

aG*(r,

1

I (p) =

f

dn*

dsf,

we do not have I (P) = 0, but, instead, {7.7.8)'

I(p) =

G(q1,

log e

In fact, if p is a point of c, the function G* (r, p) has a logarithmic pole not only at the point p but also at the point I of c. Let c be oriented in the sense in which exterior area lies to the left, and let p', p" be points on opposite edges of c, p' on the left edge, p" on the right. Then lie respectively on the right and left edges of c. We have j'aG*a(nr, fi) aGan,1)dsr + (7.7.21) I(p') =I(p") + ds,,k

k

where k is a small circle surrounding the point p and is a small circle surrounding the point fi. In the integral over k the. normal derivative is in the direction exterior to k while in the integral over A it is in the opposite direction. Hence (7.7.22) I (P') = I (p") - 2. Now let p approach the point r1 of c from the exterior of c. Then the point approaches the identified point r2 of c from the interior and we have 1(r1) + I (rs) = 0. If, however, the value I (r$) is obtained by approaching r2 from the exterior of c, then by (7.7.22) (7.7.23)

1 (r1) + I (r2) = 2.


§ 7.7]

309

By (7.7.1) and (7.7.2) (7.7.24) G(qi, rs = G(gi, ri)

aG(g1, r2)ds

an 2

=

aG(g1, rids

an an,

2

i

.

Hence, writing

=I(p)- G(q1, p)

(7.7.25)

1

log 8

we have o

u (r1) + u (r2) = 0,

(7.7.28)

an 2J ds2

2

an i

2

dsl.

Applying Green's formula to the surface l* cut along c and observing

that both u(p) and G*(p, q) vanish for P on the boundary of 92*, we have (7.7.27) a(q)

,r

[G* (r, q) 8n!)

-

* (r, q)] G* U(r)

dsr ^ 0,

an,

by (7.7.7)' and (7.7.28). This is equivalent to (7.7.8)'. c By Green's formula 1

(7.7.28)

G* (p, q) -- G (p, q) aG q) _ G(rl, q)

J [*1.

an'

0

1

f

G*(r2,

)

(ri, q)

aGan

- G(ri,4)

aGrv -

'

an

aG * (r2,

'

dsa

G

by (7.7.6) and (7.7.7)'. Substituting from (7.7.10) with _ 1= -e2f 2,

and dropping the subscript 2, we find by (7.7,8)' that G* (p, q) - G (p, q)

[G* (r,1') J

` 2 Re j

aG(gi, q) E2

a$G (q1, q) (e4\ 2 t gi

qj

r

...

Q

2G*(n, ) (2 Re 1 dG(gi, q) an,

gi

a2G a(g1, q)

2

°1

+

gi

G(qi, p)G(gi, q)

log -

.

} 11 1J ds


310

[CHAP. VII

If the sign of G* (p, q) is chosen properly, then by the same reasoning

that led to (7.7.12) we obtain the formula G* (P, q) -

(7.7.29)

G(p, q)

G(ql, p)G(q:, q) -{- Re Jl 4e $ a .(gv 1') aG(ql, q) } + 0(64). aqi aq, log

1

7.8. INTERIOR DEFORMATION BY ATTACHING A CELL. FIRST METHOD

To compensate for the special nature of the preceding variations, we now make a general interior deformation which results from "attaching a cell" to the surface P. In the remainder of this chapter we suppose, for simplicity, that R is orientable. Let y be an analytic Jordan curve in R Although the hypothesis is not essential, we suppose for simplicity of notation that y lies in the domain of a local uniformizer z. Let r(z) be a function which is regular analytic in a

complete neighborhood of y. If e is a sufficiently small positive number, the point defined by z + er(z) will trace a neighboring Jordan curve y, as z traces y. Let fit' denote the subdomain of )'l exterior to y, Wt" the subdomain of W interior to ys. In general, WW and 9X" will not be disjoint but will overlap. We attach the surface WI" to Wt' by identifying the point z on the exterior edge of y with the point z + er(z) on the interior edge of y.. The resulting surface TV* is thus obtained by removing from 9 the interior of y

and then filling up the hole so formed by attaching the piece of surface which consists of the interior of y.. This construction of the surface WIN' is expressed in terms of a particular uniformizer z. If we change to another uniformizer r, we have (7.8.1)

r(z + er(z)) = r + ee.(r),

r = r(x).

The arcs y, y. appear in the plane of r as arcs p, p., and the point s of y is identified with the point r + se.(r) of lr.. We observe that (7.8.2)

where

e.(-r) = eo(r) + 0(e)

§ 7.8]

A T'TI4CHING A CELL. FIRST METHOD

(7.8.3)

eo(z) = r(z)

311

dz

Thus, in the limit, as a tends to zero, the function r(z) transforms like a reciprocal differential. Let us assume first that O1 is a closed surface. In this case, Ot* will

also be closed and both surfaces have the domain W, exterior to y, in common. If we again denote the integrals of the third kind for Ut and TZ* by QQQO(p) and Q *o (p) and the corresponding Green's functions by V(p, po; q, qo) and V*(p, p0; q, qo) we may determine the variation of the Green's function under the deformation considered in Theorem 7.5.1. Let p, po, q, q0 be a quadruple of points lying in a fixed closed subdomain of TV; then (7.8.4) V* (p, po; q, q0) - V (p, po; q, q0)

= Re

f QQQ°(t) dQ99p(t)1. Y

Consider next the interior J'l" of ya; here both functions QQQO(t) and S?*Poo (t) are regular analytic since their poles are, for e small enough, outside 93t". Hence, by Cauchy's theorem

0=

(7.8 5)

27Li

f Q 0(tl) Y*

Because of the identification between Y. and y on 9!71*, we have

for t1ayQ and toy, ti =t+sr(t), dQv 00 (ti) = dQ99Q(t).

Replacing the variable of integration tl in (7.8.5) by t and subtracting (7.8.5) from (7.8.4), we obtain V*(p, Po; q, q0) -V (p, po; q, q0) (7.8.6)

= Re

(t) - S2QQO (t - ar (t) ) dS299p (t)

1

2ni

.

Y

Since all functions under the integral sign are analytic, we may

replace the integration over y by an integration over any fixed curve yo in R' which does not enclose any of the points p, p0, q, q0. On yo, we clearly have ar(t)a'ffo (t) + 0(e2) SaQQO(t) - S2QQO(t + ar(t))


312

[CHAP. VII

This, V* - V will be 0(s) for p, p0, q, q0 in the closed subdomain of XJ'. We have, moreover, (7.8.7)

S199o(t) = 2

-tV*(t, to; PI P01',

hence, Q*' - £2' is 0 (a) in that subregion of !l2', and we obtain from (7.8.6)

V*(p,po;q,g0)-V(p,P0;q,90) (7.8.8)

- 2-i f r(t)S?Q' (t) "S2" (t) } + O(82).

Re j

11

I

Using (7.8.7), we may bring this result into the final form: (7.8.9)

Re

(

f

t

V* (p, p0; q, q0) -- V (p, p0; q, q0)

r(t) atV (t, t0; P, p0)aV (t, t0; q, go)dt }

+ 0(82).

Y

This formula is our main result in the variation arising from cell attachment. The variational formulas for all other functions and differentials on 9)2 are easily derived from it. Differentiating (7.8.9)

with respect top and q, we find that LM(p, q) -- LM(p, q)

22

f r(t) LM(t, p) LM(t, q)dt

r

(7.8.10)

(a

+ \2i

f

r(t)LM(t,

fi)LM(t, q)dt)

+ 0(82).

Y

By computing periods *in the usual way, we obtain the formulas: Z*'(q) - Zµ (q) =

2i

f r(t)Zµ(t)LM(t, q)dt Y

(7.8.11)

, 4 )dt)-+ O(e2) + (.._fr(t)z(t)LM(t 2i

r

1'µ _ --

d'7 (7.8.12)

2z

f Y(t)Z,, (t)Z.(t) dt V

/ 21

f r(t) Z"(t) Z,.(t)dt) + 0(e2). V

§ 7.81

ATTACHING A CELL. FIRST METHOD

313

We may avoid the integrals in the variational formulas if we choose r(t) to be analytic inside y except at the single point q0 where it has a simple pole with residue a/7r. In this case, all integrals may

be evaluated by the residue theorem and yield: (7813)

(

7 Q 14)

(7.8.15)

L*(p, q) = LM(p, q) - s[aLM(p, go)LM(q q0) -}- ULM(Yi go)LM(q, go)] -- L/ (s2 ),

Z*'(q) = Z (q) - e[aZ,(go)LM(q, q0) + dZ',(go)LM(q, 40)1 + 0(62),

1':, = I',, + 2s Re {a Z' (go)Z"(go)} ± 0(s2).

In these formulas, r(z) is to be considered as a reciprocal differential. In particular, under a change of uniformizer the value a transforms in such a way that a/d4 is invariant. Let us now consider the case of a domain J7t with boundary. We

may imbed 0 in the closed surface t = 7t + t and extend the

by attaching an analogous cell at the double y of the Jordan curve y. We use there the function (r(t))- instead of r(t) so that the deformation transforms 0 and Alt into two domains 9t* and * which operation. We easily calculate are related to each other by the the variation of the Green's function G of 9Tt under the variation considered. We use (7.6.1) and (7.6.6) and observe that the integrations over y and over y yield the same contributions in the variational formula (7.8.9). We thus obtain: _). dG(t, q) 8G(t, p) } 2e r dt + O(s G*(P, q) = G(P, q) -Re 120 J r(t) at at deformation of 9)1 by cell-attachment to a deformation of

(7.8.16)

v

From this formula, we may again derive variational formulas for the Z,, and the P,.. An easy calculation shows that we obtain the formulas (7.8.10)-(7.8.15), just as in the case of a dosed surface. We considered in Section 7.5 the function W (P, p0; q, qo) which

plays a central role in the theory of those Abelian integrals on a surface which satisfy the Riemann normalization, that is, which have vanishing periods with respect to the cycles K2,-.,. In (7.5.8) we, gave a formula comparing W* and W and now apply this result


314

[Cser. VII

in the case of a closed surface 911 which is subjected to a cell-attachment

of the above form. An easy calculation leads to the result: (7

8 17)

=

_y

W* (p, po; q, q0) - W (p, P ; q, qo)

r(t)W'(t, to; P, PO) W'(t, ta; q, go)dt + O(e2).

ci V

where the prime on W indicates differentiation with respect to t. If we use the particular choice of r(t) which makes r(t) regular analytic inside y except for the pole at r with residue a, we obtain the particularly elegant formula: (7.8.18)

W* (p, po; q, q0) = W (p, p,3; q, q0)

+ ea W' (r, ro; p, po) W' (r, ro; q, q0) + 0 (e2)

where ro is an arbitrary (unessential) point. This simple variational behavior makes the function W(p, po; q, q0) a useful tool in the study of closed Riemann surfaces and of the inter-relations between their

various differentials and periods. 7.9. INTERIOR DEFORMATION BY ATTACHING A CELL. SECOND METHOD

A somewhat different variation is obtained if, instead of taking y to be a Jordan curve, we take it to be an analytic are joining the points a, # of WI. We assume that y lies in the domain of a local uniformizer z, and we impose the condition that r(z) is regular analytic in a complete neighborhood of y with r(a) = r(i4) = 0. If

e is a sufficiently small positive number, the point defined by z + er (z) will trace a neighboring arc y, joining a with fi. Let the arcs y and y, be oriented in the direction from a to #, and let x k' be the domain "lying over" I which is bounded by the boundary of $t (if there is any), the left edge of y and the right edge of y,. In general, W' will not be a subdomain of 9R since two points of 931' may lie over a point of TZ in the neighborhood of y. By identifying

the point z on the left edge of y with the point z + er(z) on the right edge of y,, we obtain a surface 91* which is topologically equivalent to W .

Let p', p" be a pair of identified points where p' is an interior point of y lying on its left edge, p" a point on the right edge of y

§ 7.9]

ATTACHING A CELL. SECOND METHOD

315

and let N', N" be half-neighborhoods at ,8', p", N' lying on the left of y, N" on the right of y,. Define

(z + er(z), z e N',

t

jl x,

z e N" .

In the plane of t the half-neighborhoods fit together to form a complete neighborhood, so the variable t acts as a uniformizer at pairs of identified points which correspond to interior points of the arcs y, y,. A complete neighborhood of the point a on the surface JZ* can be mapped conformally onto the interior of a circle; this

follows from the uniformization theorem. The fact that a maps into a point, and not into a proper continuum, follows readily (a proof is to be found in [5]). Therefore Dt* is a Riemann surface which

is topologically equivalent to M. Suppose first that J't is a closed surface. Using Theorem 7.5.1, we have the equation (7.9.1) V* (p, p0; q, q0) - V (p, pa; q, q0) = Re {

2ni

f Q, (t) dD*VV0 (t) {

where p, p0, q, q0 are a quadruple of points lying in a closed subdomain of U1 which does not contain the curves y and y, a.nd where yo is a curve surrounding y and y,. Both Q...0(t) and Q*.V*(t) are regular analytic in that part of R' which lies inside yo. Hence, we may deform the path of integration in (7.9.1) and take instead of yo the boundary of 9Jl', i.e. the curve system y - ye. By the identification in TZ*, we have

t e y, ti = t + sr(t). Hence, (7.9.1) may be brought into the form (7.9.2)

dQ 9p(tl) = dS299o(t),

V* (p, P (7.9.3)

r

= Re{

l

2ni

f r

q; q0) -V (p, po; q, q0) er(t))] d. Q*

This corresponds exactly to formula (7.8.6). Integration is along the left edge of the curve y and we may deform the path of integration y into another curve yi which also connects a with t6 but otherwise


316

[CRAP. VII

lies to the left of y. It can be shown [6] that, in the new integration d.n5C99p can be replaced by d[4, with an error o(1). Therefore, (7.9.3)

has the form. V*(p, po; q, q0) - V(p, po; q, q0) (7.9.4)

= Re { -

J' r (t) Qavo (t) D''. (t) dt } + 0 (a).

Thus, we arrive at a formula which is formally the same as that given in the previous section for a Jordan curve V. All other results of the

last section can now obtained be by reasoning entirely analogous to that of Section 7.8 and formulas (7.8.9)-(7.8.15) remain valid for the case in which y is an arc. 7.10. THE VARIATION KERNEL

In order to construct certain further variations, it will be necessary

to introduce a surface functional n(p, q) which transforms like a reciprocal differential with respect to the point p, like a quadratic differential with respect to q, and which has a simple pole for p near q (see [7a]). Because of its fundamental role in the variational theory, we shall call this functional the variation kernel. We first define the variation kernel for the four orientable surfaces whose algebraic genus G is either 0 or 1; namely the sphere, the cell, the ring domain, and the torus. The first three domains are of genus

zero, and can therefore be mapped onto plane canonical domains, namely, the whole plane, the unit disc, and the circular ring. There is a six-parameter group of mappings of the closed w-plane onto itself, given by all transformations of the form (7.10.1)

w=

awl cw+

i+

b

d'

ad-bc00.

Let w (p) give the conformal map of the abstract sphere onto the closed complex plane. We construct the variation kernel n (p, q) in terms of the complex function w(p) defined on the abstract sphere. Let q0 be an arbitrary but fixed point of the abstract sphere, and define (7.10.2)

n(p, q) =

[w(p) - w(q°)13

[u.,(q)]¢

[w (p) - w (q)] [w (q) - w (qo)] 3 w' (p)

THE VARIATION KERNEL

§ 7.101

317

We note that n(p, q) so defined is a reciprocal differential in b and

a quadratic differential in q. It also has a simple pole at

= q,

where the residue is + 1 if P and q are expressed in terms of the same local co-ordinate system.

The right side of (7.10.2) remains formally invariant under the transformations (7.10.1). Thus n (P, q) is independent of the particular choice of the mapping function w (P) and depends only on the parameter qo. If we choose qo such that w (qo) = oo, the formula (7.10.2)

takes the particularly simple form [w' (q)] 2

n(P, q)

(7.10.2)'

w'(p)[w(P) -- w(q)] Let us next consider the case of the cell. There exists a function w(P) which maps the abstract cell onto the unit circle in the complex

plane. Every function wi(g) of the form w1(p) = e'

(7.10.3)

w-wo, 1 - wow

.a real, I wo 1 < 1,

also maps the cell onto the same domain. In order to fix the map uniquely, we choose an arbitrary fixed point qo on the abstract cell and, require that w (qo) = 0. If a function w (q) is known which performs the map of the abstract cell onto the unit circle, but for which w(qo) = wo = 0, then each linear transformation (7.10.3) will lead to another map function with the required normalization w(qo) = 0. For map functions w (q) normalized in this way let us define (7

), q) =

.10.4

n

w() w(p) + w(q) [w'(q)]2

2[w(q)]

2 w(P) -w(q) w'(P) We observe that n(p, q) is real if both P and q lie on the boundary of the disc and if boundary uniformizers are used as the coordinates of P and q. The distinguished role of the arbitrarily chosen point qo in the analytic character of the variation kernel is obvious. Given an abstract ring domain, there exists a function w(p) which maps the ring onto the circular ring domain ,u < J w I < 1, ,u > 0, where, u'1 is the modulus of the ring. There are infinitely many map functions of this type, but they are all interrelated by the transformation formulas w1= eaw (A real) and wi =,u/w. We proceed to the

818


[CRAP. VII

construction of the variation kernel as follows. Let C (w) denote the Weierstrass z-function corresponding to the periods 2a , 2w2, where co, is real and Cot pure imaginary. Let the w's be related to the modulus

A-' by the formula w ni--

(7.10.5)

=e 01

In this way co, and w2 are determined up to a constant real factor.

In the usual notation we set (7.10.6)

171 = C(W1), m = C(w2)

We define the variation kernel as follows: a ' I% )_!?.iog!S2l ni { C \ni log w (q)) ni w q) [w (q] 2 [w' (p) Then n(p, q) has a simple pole for p = q and, if we use the same

(7.10.7) n (p, q) T

uniformizer for p and q in the neighborhood of the point q, then the residue is + 1. If p and q lie on the same boundary component of the ring, log [ri(p)/w(q)] is pure imaginary and hence the bracketed expression in (7.10.7) is real. If boundary uniformizers are used, then the whole expression is real as well. Suppose next that p and yq lie on different boundary components and that j w(p)/w(q) Using (7.10.5) we can therefore write niogw(q)

602+R(p,q),

where R (p, q) is a real number depending on p and q. It should be observed that w2 is a pure imaginary, and that the bracketed term in (7.10.7) is not real. We use the theorem of elliptic function theory that (7.10.8)

C(z ± w2) = C2 (Z) ± 7721

(see [8], formula XII5), where C2(z) is real for real values of its argument. Hence the bracketed terms in (7.10.7) may be transformed into (7.10.9) C(R + w2) --- yh R _ wl

7712 wi

= C2 (R) - 'R + 272% WI

111W2

wl

¢ 7 10]


319

Using the Legendre relation - ni

we find that the imaginary part of (7.10.9) is a constant, namely -n/2w1. Thus, using boundary uniformizers chosen so that w`/w = i on the boundary, we see that the imaginary part of n(p, q) is constant if p and q lie on different boundary components. In the case of the unit circle and the circular ring, n(p, q) is essentially the complex Poisson kernel. For simplicity let w(p) = z and

let u (r) be a real continuous function on the boundary C of the unit circle; then the analytic function Q(z) in I z I < 1 whose real part assumes the values u(r) on i z 1 is given by the formula (7.10.10)

S2(z) =

-.4. f u(r)n(z, r)rdr + 1K, niz

where K is an arbitrary real constant. In the case of the circular ring let u (r) be a real-valued function defined on the boundary C of the ring which satisfies the condition

d(7.10.11) fu(r)! T = 0. c

Then the single-valued analytic function Sl (z) in the ring, a < I z I < 1

whose real part assumes the given boundary values is given by the formula (Villat's formula)

(7.10.12) Q(z) =-1.. f u(r)n(z, r)rdr-2nifu(r) dz + iK, K real. z

C

1=1 -P

Finally, we take up the variation kernel for the torus. Since the torus has genus 1, it cannot be mapped onto a subdomain of the sphere. However its universal covering surface, which is simplyconnected, can be mapped confornzally onto the complex plane punctured at infinity. Let w(p) give the conformal map of the universal covering surface onto the w-plane. The single valued functions on the torus go over into elliptic functions of w whose periods we shall denote by 2co1 and 2wE. However, in the case of the general unsym-

$20

[CHAP. VII


metrical torus, we can no longer assume that the period ratio is pure imaginary. Let P. and q0 be arbitrary but fixed points of the torus, and define n(p, q) = [C(w(p) - w(q)) - C(w(p) - w(qo)) (7.10.13)

.-C(w(po) -w(q)) + C(w(po) -w(qo))]

-

We note that the bracketed expression is single-valued on the torus.

We also observe that n(p, q) has a simple pole at p = q, with a residue, + 1 if the same uniformizer is used to express p and q. In addition, this function has fixed poles at p = qQ and q = po. In the general case in which the algebraic genus G exceeds 1, we have at least two linearly independent everywhere finite differentials Zi(p), Z,(p) (linear-independence being understood in the complex sense). Let (7.10.14)

A(p. 4) = I

Z(p)

Z;(q)

Z:(p)

Z,', (q)

I

This expression is an everywhere finite bilinear differential of which vanishes when p = q. Next, let (7.10.15)

I

Z(p) = dZ1(p) dZ2(p)

Since Z (p) = Z (p), this is a function of Tt in the sense of Section 2.2.

Differentiating this function, we obtain (7.10.16)

d (p)dzs

dz(p) _ [dZ,(p)] 2

where

(7.10.17)

A (P) =

d2Z, dz!

dZ1

d2Z2

dZ9

dx=

dx

dz

and z is a uniformizer at p. Since 4Z(6), [dZ2(p)}2 are invariant, ddz$ is also invariant and is therefore an everywhere finite cubic differential of X12 (differential of dimension 3).


§ 7.10]

321

For a surface of algebraic genus greater than I we define (7.10.18)

n($, q) = VLM(b, q)

A (P, q)

If 9t has a boundary, we observe that n(p q)dr2

(7.10.19)

dz

-

n( q)dr2 -

n(P, q)dr2

dz

dz

where z and r are uniformizers at P and q, respectively. In this sense n(p, q)dr2/dz is a differential of 'c, reciprocal with respect to b, quadratic with respect to q. Let P and q be close together, and let t be a uniformizer in a neighborhood containing both P and q. Regarding 15 as variable, we have

_

+ terms, t = t (p), t0=t(q)=O. t T, The principal part is invariant if p and q are represented by the

(

7 10.20)

n (p, q)dto dt

1 dto

same uniformizer.

In addition to the singularity at p = q, there are poles at the points where d (t) vanishes. Let zero of d (P), we have (7.10.21)

n(p, q)

be such a point. If Pk is a simple

q) A(L'k, q) + ..., _ nLM(Ck, z d (Pk)

where z = z(p) and z(Pk) = 0. The important point is that the coefficient of 1/z is a quadratic differential in its dependence on q. Let (7.10.22)

Qk(P) = nLM(Pk, P) A(Pk, p).

-Since (7.10.23)

A(Pk, Pk) = 0;

aA(pk, :P) ap

LVI%=

-

d ,/,

we see that Qk is everywhere finite. More generally, we observe that (symbolically) (7.10.24)

(ap

a 11(x, q) IQy9d + aq)I.p din

Assume now that d (t) has a zero of order . at the point 1, and let z be a uniformizer at Pk, z(Pk) = 0, Then

Ck

of


322

n(p, q) =

rTLM(:P, q)

A(P)

A(p, q) =

1

zz

[CHAP. VII

OD

rZA,(q)z'.

If q does not lie near Pk, we have by the residue theorem A, (q)

= 2-ti f n(P,

q)z'-'-"'dz

where c is a small circle around the point Pk. This shows that A,(q) is a quadratic differential in q, just as n(p, q) is, in its dependence on its second variable. We have to determine the behavior of A,(q) at the point Pk. Suppose that q lies in the neighborhood of p,., and let r = z(q). If K is a contour lying in the domain of the uniformizer z which encloses in its interior both P,, and q, but no other pole of

n(p, q), we have for 0 S v S 2 - 1, (7.10.25)

A,(q) _

T-r-1

+ 27ri fn(P,

q)zA- -xdz.

K

Letting q tend to P7, we see that A,(q) remains finite and, since z(Pk) = 0, (7.10.26)

A,(Pk) = 2ni1 fn(p,

0 S V S A - 1.

Thus at a point pk of I) where zi (p) has a zero of order 2, we have (7.10.27)

n (P, q) = A-O()

+ `4 (q)

-F- regular terms

where A°(q), A1(q), , Aa_1(q) are everywhere finite quadratic differentials on the surface 97t. The sum of the multiplicities of the zeros of A (p) on the double 16 of 1J is given by formula (3.5.1)': (7.10.28)

ord (Adz8) = 6(G - R°) = 2(6h ± 3m - 6), m > 0.

Here G is the algebraic genus. In the case m = 0, the double consists

of two components, on each of which ord (A dzs) = 6h - 6. If G > 1, then by (3.7.1) and (3.7.2),

§ 7.11

(7.10.29)


323

a= 6h+3m-6

where a denotes the dimension of the space of all classes of conformally equivalent finite Riemann surfaces. Thus (7.10.30)

ord (A dz3) = 2a, G > 1,

or, by (3.8.5) (7.10.31)

ord (A dz3) = 2 dim (dZ2), G > 1.

The number of coefficients in the principal parts of n (p, q) at the zeros of A (p) is equal to ord (A dz8), and these coefficients are quadratic differentials. By formula (7.10.31) only half of these quadratic differentials can be linearly independent in the complex sense.

If we consider n (p, q) as a function of p, it has bne pole at = q and the remaining poles are at points pk, where the pk are independent of q. The latter poles will be called p-poles for simplicity. As a func-

tion of q, n(p, q) has a pole at q = p and, in the case where G = 0 or 1, it may have an additional pole at a point qo, qo independent of p. In (7.10.2), qo is the point at oo, in (7.10.4) it is the point at the origin, while in (7.10.13) it is the arbitrary point qa. This pole of n (p, q), if it exists, will be called a q-pole. It should be remarked

that the coefficient of 1/z-ro at the'p-pole in (7.10.13) is an everywherefinite quadratic differential in q. Hence in all cases the coefficients

occurring in the principal parts at the p-poles are everywhere finite quadratic differentials. 7.11. IDENTITIES SATISFIED BY THE VARIATION KERNEL

We now establish some useful identities involving integrals of n(p, q). Let y be the Jordan curve considered in Section 7.8 and let r(p) be a local reciprocal differential which is regular analytic in a complete neighborhood of y. Let Q1, , Q,, be a basis for the everywhere finite quadratic differentials of 0, where a is the number of real moduli of 9X: (7.11.1)

or = a + 6h + 3c + 3m - 6.

We impose upon r(p) the orthogonality conditions


$24

fr(P)

(7.11.2)

0,

[CRAP. VII

v = 1, 2, ..., a.

Y

Let us consider first the case m > 0. We then have a Green's function G(p, q) and we denote by T(p, q) the analytic function of

p whose real part is this Green's function. By (4.2.1), we have (7.11.3)

=BT(p,q)= 28G

T'(p,q)

Let the Jordan curve y on ft consist of the conjugate points of y. We define the conjugate f(p) of the reciprocal differential r(p).

by the formula (compare (2.2.3) and (2.2.3)'): (7.11.4)

9( )dx-1 = (r(p)dz l)

Let (7.11.5) h(p)

n(p, pi)dzi _

= ..! f

f 2ni J (Pi)n(p, r)dx1. y

Y

By (7.11.2) and (7.10.27) we see that h(p) is regular analytic at the p-poles of n(p, q). It is, therefore, a reciprocal differential which is regular analytic in V -y. If p lies on the boundary of 't and if we use boundary uniformizers h(p) is real. We state now the following theorem: THEOREM 7.11.1. If r(p) is a reciprocal differential near y which satisfies the conditions (7.11.2) and h(p) is the reciprocal differential (7.1 [.5), we have the identity Re (7.11.6)

1

2ni

f r(pr)T'(Pl, p)T'(pi q)dzi Y

= Re {h(p)T'(p, q) + h(q)T'(q, p)}. This identity has many applications in the variational calculus of Riemann surfaces and serves to transform and simplify formulas involving reciprocal differentials and derivatives of Green's functions.

In order to prove the identity (7.11.6), we denote the left-hand side of (7.11.6) by U1(p, q) and the right-hand side by U,.(p, q). Both functions are harmonic in both variables, except on the curve Y. One readily verifies that U,,(p, q) is also regular harmonic for p = q


§ 4.113

325

since the singularities of its elements cancel for p = q. Let us choose a fixed point q in the interior of 911 - y and study the dependence of these functions upon p. Since G(p1, p) is identically zero in p1

for p on the boundary of fit, we have: T'(p1, p) = 0 for p on the boundary of Mt. Hence UL (p, q) will vanish on the boundary of M. But the same will also be true for the function U,.(p, q). In fact, using boundary uniformizers we have

h(p) = real, T'(p, q) = imaginary, for p on the boundary of 9)t. Since, moreover, T'(q, p) also vanishes on the boundary, we find U,.(p, q) = 0 if p lies on the boundary of T1. It remains to investigate the discontinuity behaviour of U, and U,, across the Jordan curve y. Since r (p) is analytic in the neighborhood

of y and since the singularities of T'(p1, p) and n (p, P1.) on the curve have the same expression if (z(p) - z(p1)) in the uniformizer on y, we derive easily from Cauchy's theorem that the expressions 1

r(p1)T' (p1, p)T' (pi,

2ni

q)dz1- [r(p)T' (

,

q)] 8

Y

and

h(p) T'(p, q) - [r(p) T'(p, q)]8, where 8 = 1 for p inside y and 8 = 0 for p outside, are regular analytic in a complete neighborhood of y. This proves that the harmonic function Uf(p, q) - U, (P, q) is regulai harmonic in 9)t and, since this function vanishes on the boundary of 9t, it is identically zero. Thus, Theorem 7.11.1 is proved. Let us proceed to the case m = 0. Now the Green's function is replaced by the function V (p, po; q, q0) defined in (4.2.23). We have: (7.11.7)

2 ap V (p, po; q, q0) _ Qaao(p)

We'dhoose a fixed point q0 E J1 which lies outside a neighborhood of the Jordan curve y. We define the reciprocal differential (7.1l 8)

k(p) = _

5 r(1'1)n(p, p1)dzi Y


326

[CHAP. VII

and again subject the (local) reciprocal differential r(p) to the orthogonality conditions (7.11.2). We then have the theorem: THEOREM 7.11.2. Let M be a closed surface and r (p) a reciprocal differential which is defined near`the Jordan curve y and which satisfies

(7.11.2). If the reciprocal differential h(p) is defined by (7.11.8), we have the identity: Re j c

2ni

f

r(px)Q

Y

0(qo) + Re {q0()h(P) -F Q 0(q)h(q) qqQ + 2h (qo)

aqo

Re {Slgno (p) + -Q.0 (q)} }

Here 0(qo) is a real number which depends on h(p) and the choice

of q0 but is independent of p and q. In order to prove this identity let us observe that both sides of (7.11.9) represent harmonic functions of p and q in Y2 - y. The right-hand side is easily seen to be regular harmonic even if p approaches q or if either p or q converge to qo. We verify as before that the difference of the two sides of (7.11.9) is regular harmonic across the Jordan curve y. Hence, the two sides can differ only by a constant, i.e. a number which cannot depend

on p or q. This proves the theorem. The constant 0 (qo) is obtained in the easiest way by the following

remark. In view of (3.4.7), we have (7.11.10)

2 app

Re {D,,, (q) -".,sa(go)}

and, letting q - qo, we find (7.11.11)

lim Slaq,(p)

0.

This remark shows that the left side of (7.11.9) vanishes for q = q0

or p = qo. Thus - 0 (qo) is the limit of the real part on the right as

qo or gig0.

From the identities (7.11.6) and (7.11.9) we may derive further

identities. We take on both sides the normal derivatives of the harmonic functions and integrate them around the basis cycle K',.


§ 7.11]

327

Comparison of the results leads to the new identities. We make use of the formula

f

(7.11.12)

JKµ8n.

2ni ZK(q)

T,(q, )ds9

and correspondingly, for the case of (7.11.9),

J'_Q,0(q)ds, = - 2ni Zu(q)'

(7.11.13)

KA

and derive from (7.11.6) the identity (7.11.14)

Im j 11

2xi

f

q)dzi} = Is {h(q)Z.1.1(q)1

for the case of a surface with boundary. We can make an analogous calculation in (7.11.9). We observe that (7.11.15)

Re {f/ DO. (P) ds, } = 2n Im j aodZ, }, a

KE,

since the real part of Sla(p) is single-valued. Further by (3.4.7) Re 1

f 8n,1J'a,(q)dsn - f Q

,

(7.11.15) and (7.11.15)' we deduce easily that (7.11.16)

2 aq. _ Re

} f asnffl[Qaa, (p) + .,v, (q)]ds,

«(qo)

K#

is a complex number which is independent of q. Hence, we derive from (7.11.9) the equation (7.11.17) Im {

2aci J

=Im {Zµ(q)h(q)-{qo)}.

Y

The constant fl (go) can be determined by means of (7.11.,11). We see


328

[CHAP. VII

that the left side of (7.11.17) vanishes for q = q0. Hence ¢(q0) = - Z},(ga)h(ga), and we obtain the final identity Im { (7.11.18)

2I Y

= Im {Z,(q)h(q)

-

j Z,(g0)h(g0)}.

Formulas (7.11.14) and (7.11.18) will be used in the sequel. Their significance is obvious; the left side of each formula is a harmonic function of q and the right side indicates the simple form of its analytic completion.

From these integral identities we can derive formulas connecting

the variation kernel n(p, q) with the complex completion of the Green's function G(p, q) and with the differentials of the first kind

Z(p). These formulas are given for completeness, but may be omitted by the reader if he is so inclined. The formula (7.11.14) may be written in the form Im

(7.11.19)

2n.J r(pi)ZµPi)T'(pv q)dzi} Y

= Im{

2ni

f

r(pi)n(q, pi)dzi -

Zµ(4)

2iJ

r(pi) n(4, pz) dzi. }t

we have used the fact that (7.11.20)

Zµ(q) tare

f

1)dx1)

_ -Z'µ(4)

r(pi)n(4. pi)dzi.

2ni Y

by (7.10.19). If r(p) satisfies the conditions (7.11.2), so does ar(p) where a is any complex factor, and it follows from (7.11.14) that J

(7.11.21)

= Z'µ(4)

2ni

r(pi)Lµ(pi)T'(pi, q)dzi

fr(pi)n(q, pi)dzi-ZN() f 2ni

r(pi)n(4, pi)dzi.

THE VARIATIOV KERNEL

§ 7.11]

329

Formula (7.11.21) is valid provided that r(p) is regular analytic in a neighborhood of y and that (7.11.2) is satisfied. In the same way, we derive from (7.11.18) 1

(p1}dzl

27Li

(7.11.22)

v

= Z' (q)

27i J r(p1)n(q, p1)dzl - Zµ(go) 2n. J Y

r(pi)n(g0, pl)dz1.

Y

It is now easy to derive from the integral identities (7.11.21) and (7.11.22) new equations connecting the functionals themselves. For this purpose we choose a function r(z) of the uniformizer z which

is regular analytic inside the whole Jordan curve y except for N points PQ, where it has simple poles with residues aQ. We are not quite

free in our selection of the function r(z). We must satisfy the conditions (7.11.2) which have now, by virtue of the residue theorem, the form N v = 1, 2, - , E aQ Q,(pe) = 0, (7.11.23) e-1

On the other hand every function r(z) satisfying (7.11.23) will be a permissible choice. We introduce r (z) into (7.11.21) and by means of the residue theorem

we obtain N

(7.11.24) Eae [ZO,(pe)T'(pe q) -ZE,(q)n(q, pe) + Z,,(4)n(4, pe)] = 0. e-l

Similarly, we obtain from (7.11.22) N

(7.11.25) E ae [Z,,(pe)Dae,(pe)--Z(q)n(q, pe) + Z(go)n(go, pe)] = 0.

e-l

Since we can choose the poles pe and their residues aQ arbitrarily except for the linear conditions (7.11.23), we derive from (7.11.24) (7.11.26)

'(p)T'(p, q)=Z,(q)n(q, p)-Zk(4)n(4, p)+,-l E

To evaluate the coefficients a,,,(q). we first take q on the boundary

of 2N and we obtain


330 a

(7.11.27)

E a,.,(q)Q.(P) = 0, ,-1

[CHAP. V11

y = 1, 2, ....

Since this equation holds for arbitrary P e Dl, we conclude from the linear independence of the Q,(p) that all a,,, (q) vanish on the boundary

of fit. As q approaches a q-pole of n(p, q), the sum E a,,,(q)Q,(P) ,-1

must tend to infinity. Let the q-poles of n(q, P) which lie in the interior of J?l be qe, 2 = 1, 2, . , M, M S a - k. For simplicity, assume that all these q-poles are simple; if this assumption is not satisfied, the calculation is similar. For q near qQ, let he a uniformizer at qQ, C(q) = C, C(qe) = 0.

By (7.10.21) we have

n(q, p) = P(p) + regular terms

(7.11.28)

where PQ(p) is a quadratic differential. We readily conclude that M

E a,,, (q)Q,(P) _ -E Z0(gQ)T'(ge, q)P9(')

(7.11.29)

,-1

a-1

Thus, formula (7.11.26) becomes finally (7.11.30)

Z(p)T'(, q) = Z,(q)n(q, P) M

- E ZN(ge)T'(ge, q)PQ( ) e-1

If not all q-poles of n(p, q) are simple, higher derivatives of T(qQ, q) with respect to qQ will occur in the sum on the right side of (7.11.30).

From (7.11.25), similar considerations lead to (7.11.31) Z,

)

QO{

,-1

We observe that b,,,(q) vanishes for q = go; for q near a q-pole of n(p, q), the last sum must again counteract the increase of the first right side term. Assuming that all q-poles of n(p, q) are simple, we readily verify that

§ 7.12]

(7.11.32)

CONDITIONS FOR CONFORMAL EQUIVALENCE

Z,

Zl.(q)n(q,

331

Z"(go)n(go, p)

M

- E Z',(ge) 92e,4 (ge)Pe(P). e-1

The identities (7.11.30) and (7.11.32) between the various types of differentials on a surface TZ are due to the fact that the product of a differential and a reciprocal differential is a function on the surface and can, therefore, be expressed in terms of the fundamental functions on V. 7.12. CONDITIONS FOIL CONFOR LAL EQUIVALENCE UNDER A DEFORMATION

Let y be the Jordan curve described in Section 7.8 and let z be a uniformizer which is valid in a complete neighborhood of y. Let ro(z) be a function of z which is regular analytic in this neighborhood of y and which satisfies the orthogonality conditions (7.11.2). We

then show (under certain mild restrictions) that we can find a function er(z) such that, if (7.12.1)

r(z) = ro(z) --k- Ee (z)

then the surface R* formed from X32 by attaching a cell in the manner of Section 7.8 is conformally equivalent to V. Moreover, (7.12.2)

1 O1(z) S M,

where M is independent of e, 0 S e S ao. We show first that an everywhere finite differential Z'(p) of t may always be found which has only simple zeros. Moreover, if TZ has a boundary, all zeros of Z'(p) lie interior to 9R. If TZ has a boundary and if 2h < ,u S G = 2h + m - 1, Im Z. is single-valued on t and constant on each boundary component. It follows readily from the maximum principle that Z,,(p) is nonvanishing on the boundary of WI. Let Z,`(p) be a basis differential of Wt, where 2h 0. If z is a uniformizer at pk, z(pk) = 0, we have azzz-1 + ...) + bxzz + ..., Z' (P) = r! (ao + a1z + ....+ 0, bA 0. For sufficiently small 77, it is clear that Z'(j) has A simple zeros in a neighborhood of pk, Moreover, if Al has a boundary, then by choosing 77 still smaller if necessary, all zeros of

where ae

Z'(p) will be in the interior of V. Hence an everywhere finite differential Z'(P) of 9)1 having the desired properties exists provided

that there is no point of 9t where all basis differentials vanish simultaneously. Suppose that po is a point of 0 where all basis differentials vanish. Then every differential has a zero at P.. Let t,0(P) be the elementary integral of the second kind on the double of 9)1 which is normalized

by the condition that its periods around one of the two dual sets of cycles vanish. From (3.4.3)' we conclude that ta(p) is single-valued on , and hence 9)t is either the sphere or a simply-connected domain in which cases all differentials of the first kind vanish identically. We have therefore proved that, if there are any non-trivial differentials of 91t, there is at least one, say G

(7.12.4)

Z'(p) = T cµZ (P), µ-1

which has only simple zeros and which does not vanish on the boundary of 9)1 (if 9t has a boundary). Our proof shows that the numbers c,,, u = 1, 2, , G, occurring in (7.12, 4) may be taken real (even

in the case where 931 is closed and G = h). We therefore suppose that the numbers c,, in (7.12.4) are real and, once chosen, we shall assume that they are fixed. Let Z'(P) be the differential (7.12.4) for St. Then (7.12.5)

Z*'(p)

µ-1

where Z*'(P) is given by (7.8.11), is a corresponding differential for 931* which, if e is small enough, has only simple zeros interior to V. Let be the zeros of Z' in 911, and pi, ltia, pN the zeros of Z*' in V*. From (3.6.3) we liave (7.12.6) N = G --- R°. We formulate now the following theorem:

S 7.121

CONDITIONS FOR CONFORMAL EQUIVALENCE

333

THEOREM 7.12.1. Let M be a Riemann surface, with boundary, of algebraic genus G > 1 and let Z' (p) be an everywhere finite differential on 971 which has only simple zeros in 93 and no zero on the boundary of D1. Let !31* be obtained from fit by a variation and let Z*' (p) be the

differential on 9R* obtained from Z'(p) by this variation. If p,, v = 1, 2, , N, and pY, v = 1, 2, , N, are the zeros of Z'(p) and Z*'(p), respectively, and if Z'(p) is expressed in the form (7.12.4) we have the following necessary and sufficient conditions for the conformal equivalence of 931 and 97l*: (a)

Im Z(p,) = Im Z* (P*), v = 1, 2,

(b)

Re I

fV,

N;

dZ 1 = Re I f 9ysdZ J , v = 2, 3,

, N;

y1,

yl

a = 1, 2, , G. We remark in connection with (b) and (c) that 91* should be

(c)

Re {P(dZ,

Re {P(dZ*, K,,)},

regarded as lying over 97t. Points of 91* and 971 may then be denoted

by the same symbol, the point of 931* over the point p of 971 also being denoted by p. This convention enables us to use the same symbol Ku in the left and right sides of (c). So far as (b) is concerned, we suppose that the path of integration on the right lies over the path on the left except for small neighborhoods of p1 and p,. In order to prove the theorem we define the following relation between points p c 931 and p* a 971*: (7.12.8)

fdZ = f'dz*. 91

m1*

If the conditions (a), (b) and (c) are fulfilled this correspondence can be extended in a unique way over 931 so as to give an everywhere conformal mapping of V onto 971*. Thus, the conditions are clearly sufficient for equivalence. They are also necessary, since the harmonic functions Im Z,,(p), µ = 1, 2, . , G, are uniquely deter-

mined modulo 1 and since for small enough variation there is no room for the discrete increments which Im Zw*, (p) might possibly have with respect to Im Z,,(p). Thus, conditions (a), (b) and (c) express just the conformal invariance of the differential Z'(p) and are, therefore, necessary.

334


[CHAP. VII

If D1 is closed we have a very similar situation. However in this case Im is not uniquely determined. We define therefore Im Z,(q)

1

= 2 ,r Kµ

8V(p, po; q, q0) dsq 8n9

with V (P, po; q, qo) = Re

S2..(P0)},

that is, we require Im 4(qo) = 0 (mod 1). We may choose q0 to be one of the zeros of the differential Z' (q) on 9A and of Z*' (q) on ?R*. Thus, one equation in condition (a) can be satisfied by arbitrary normalization. The remaining set of conditions will again lead to a necessary and sufficient condition for conformal equivalence.

If P has a boundary, there are G - 1 conditions (a), G - 2 conditions (b), and G conditions (c), giving 3G -- 3 = 6h + 3m - 6 conditions in all. If f't is closed and if one of the conditions (a) is

eliminated as above, there are G - 3 conditions (a), G - 3 conditions (b) and G conditions (c), giving 3G - 6 = 6h - 6 conditions in all. In both cases the number of conditions to be satisfied therefore agrees with (3.7.1).

If G = 1, there is a single basis differential Zl and it nowhere vanishes. In this case, the condition (c) is necessary and sufficient for conformal equivalence. 7.13. CONSTRUCTION OF THE VARIATION WHICH PRESERVES CONFORMAL TYPE

In the foregoing section we have formulated a set of necessary and sufficient conditions for the conformal equivalence of the surface T't and the surface t"t* which arises from it by the interior deform-

ation. We come now to the proof that a function &(z) satisfying the conditions stated at the beginning of Section 7.12. actually exists.

In order to give this existence proof it will not be sufficient to work with the asymptotic formulas derived in Section 7.8; we shall

have to use exact integral equations connecting the differentials of the surfaces considered. We go back to the equation (7.8.6) for the variation of the Green's function of a closed Riemann surface.

VARIATION PRESERVING CONFORMAL TYPE

§ 7.13]

835

We assume that R is a surface with boundary and apply (7.8.6) to the function V of the closed surface 1 +)j D. Putting jo = and q0 = q, we obtain from (7.8 6) by use of (7.6.1), (7.13.1)

(p, q) - G(p, q)

Re

2nx

f

Q(z, q)dT* (z, p) }

where T(p, q) = D4-. (p), T* (p, q) _ DaQ(p)

are analytic functions of p whose real parts are G(p, q) and G*(p, q) respectively, and where (7.13.1)' Q(z, q) = T(z + --r (z), q) - T(z, q).

In obtaining (7.13.1), we have used (7.8.6) for cell attachments along y and ', and these cell attachments give equal contributions. This is an exact relation between G* and G and we now derive from

it corresponding relations between Z* and Z,. Keeping q (and therefore q*) fixed, we compute the integrals of the normal derivatives of both sides of (7.13. 1) around one of the basis cycles K. Using the formulas faG(p,q)ds 1 f aG*(p,g) (7.13.2)

2n

ale,

dsImZ1 9=

,. (q)

;

2;

'

Im Zµ (q).

Kµ

K11

(7.13.2)' 2n

an9

f aT

P)ds9 =

- iZ*'(q),

?n9,

T*'(q, p) =

aT

Kµ

p),

q,

we find that (7.13.3)

Im Z* (q) = Im Z,(q) - Im `

f

Q(z, q) dZ*(z)

Since the coefficients c,, iii (7.12.4) and (7.12.5) are real, we obtain

(on multiplying both sides of (7.13.3) by c,, and summing from

p=1 to =G) (7.13.4)

Im Z* (q) = Im Z(q) - Im I 2I

fQ

(z,

q)dZ* (z)

r The conditions (a), (b) and (c), are equivalent to the existence


336

(CHAP. VII

of a conformal map M: from W2* onto V. If the map (7.13.5) exists, we have (modulo the numbers c1, c2, , cc) Im Z(p) = Im Z*(p*). (7.13.6) (7.13.5)

Replacing q by p* and then Im Z*(p*) by Im Z(p) in (7.13.4), we have (7.13.7) Im Z(p) = Im Z(p*) - Im {

l 27ri

f Q(z, p*) dZ*(z) v

We assume that y does not contain any zero point p, of Z'(p). Taking

P = p, in (7.13.7), we obtain (7.13.8)

Im Z(p,,) = Im Z(p**) - Im j 2 - fQ(-,,P.*)dZ*(z)l. t

The equations (7.13.8) for v = 1, 2,

, N are seen to be equivalent

to the conditions (a). We observe that

8n Im Z(q)dsq = d(Re Z(q)).

(7.13.9)

a

Hence, integrating the normal derivatives of both sides of (7.13.4)

from pi to p,, we obtain (7.13.10)

ReI 5"czz*

Rei

l

9Z

f'iz}-Im{_L. f Y

V1

where (7.13.11)

I,(z) _ `9y a (n' p) J 91

We use again the fact that under a map (7.13.5) (7.13.12)

rdZ*(p*) _ j dZ(p) 91*

T1

Hence, we May put (7.13:10) into the form

§ 7.13]


337

(7.13.13) Rei 5"dZ*} = Rei f 9' dZ*}-Imj2I f I'(z)dZ*(z)}. t

V

V,

The equations (7.13.13) for v = 2, 3, , N express the conditions (b) of the last section and are equivalent to them. Finally, integrating the normal derivatives of both sides of (7.13.4) around a basis cycle K, and writing (7.13.14) P = P(dZ, K,,), I'µ = P(dZ*, K,,), we obtain (7.13.15)

Re I',, = Re r -- Im i

r J4 (z) dZ* (z) }

l

JY

where (7.13.16)

J, (z) =

P) Jf a Q an ,n xK

ds9

The conditions (c) are therefore equivalent to the equations (7.13.17)

Im {

1 f J, (z) dZ* (z) {

= 0,

1, 2,

, G.

Thus, the surfaces )1 and lt* will be conformally equivalent if the following conditions are fulfilled: (7.13.18)

(

Im Z(p,) = Im Z(26,) -Imi

fQ(z,p,*)dz*(z)},

.

` Y

(7.13.19) Rei f"dz*}= Re{ f

(7.13.20)

(

Im i

1 f J,, (z) dZ* (z) ] = 0,

v = 1, 2, ..., N; dZ*(z)

,u = 1, 2, ..., G.

Y

Let ro(z) be a function, assumed given, which is regular analytic in a complete neighborhood of y, and which satisfies the orthogonality

conditions (7.11.2). We set

Us

VAR.ArIONS OF SURP'4C.ES

[CHAP. VII

(7.13.11)

r(z) = ro(z) + 8 (z) and we seek the conditions imposed on Q.(z) by (7.13.18), (7.13.19) and (7.13.20).

By (7.13.1)' and (7.13.21), p*) = T(z +e r(z),

(7.13.22)

Q(z,

*) - T(z, p*)

= Fro(z)T'(z, p*) + 0(82).

Subs1 ituting from (7.13 22) into (7.13.7). we find by (7.8 14) that Tm Z(p) = Tm Z(p*) (7.13.23)

Im 1

e

f ro(z)T`(z, p*)Z'(z) dz } + o(e). r Using the identity (7.11.14) and assuming that ro(z) transforms like

a reciprocal differential, we may write (7.13.23) in the form (7.13.24) Im Z(p) = Im Z(p*) -- a Im {Z'(p)h(p*)} + o(e). Since the coefficient of a in (7.13.24) is the imaginary part of an analytic function, we have (7.13.25) Z(p) = Z(p*) -T- C + o(e), where C is a real constant. We now evaluate C. Suppose first that G = 2h + nc - 1, m > 1. Then, by (7.12.6), N 1. Diffeientiating both sides of (7.13.25) with respect to p* and then choosing p* such that the image p in the mapping (7.13.5) is a point p,, we obtain, since Z'(pY) = 0, (7.13.26)

0 = Z'(p*)Li -

o(e).

Thus

Z'(p*) = 0(e). Let the local coordinates of p, and p* , expressed in terms of the same uniformizer, be z, and z* respectively. Since Z'(p) has a simple zero at the point p,,, we conclude from (7.13.27) that (7.13.28) zy = z; 4 0(e). We develop Z(p) around the point p. into a power series of the local uniformizer z. Since Z' (p,) -- 0, we have by (1.13.28) (7.13.27)

(7.13.29)

Z(pv) == Z(p*) -}- 0(e2).


§ 7.131

330

On the other hand, for p* = P,*, formula (7.13.25) gives (7.13.30)

Z(fi,) = Z(py) + C + o(e).

Comparing (7.13.29) and (7.13.30), we conclude that C = o(e) and formula (7.13.25) may be written (7.13.31)

Z(p) = Z(p*) - eZ'(p*)h(p*) + o(e).

Therefore, if the local coordinates of p and p* are expressed in terms

of the same uniformizer, we have the formula (7.13.32)

z = z* - eh(p*) + o(e).

In the case G = 1 (doubly-connected domain) the constant C occurring in (7.13.25) is not necessarily o(e). For in this case there is a one-parameter group of conformal mappings of 92 onto itself. Each element of the group transforms a point p of 0 into a point pl, where Im Z(p) = Im Z(pl). However, we are still at liberty to normalize the mapping (7.13.25) by requiring that a given point po goes into a point po, where Im Z(po) = Im Z(po); Re

Z(25.)

= Re Z(po) + K.

For this normalization we have

C = K + e Re

o(e).

Choosing, in particular,

K=-sRe{Z'(,b)h(p*)), we see that C = o(e). In the case G = 0 (simply-connected domain) the problem of confonnal equivalence is vacuous. However, if n is represented as a domain of the z-sphere and if the mapping z* -* z is suitably normalized, it may be shown that formula (7.13.32) is valid. Since a

proof of this cage is to be found in [5], we omit details here. We summarize the results obtained so far. THEOREM 7.13.1. Let 3t be an orientable surface with boundary. If 9 can be transformed into a con f ormally equivalent surface I2* by

means of a cell attached along a Jordan curve y using a reciprocal' differential r (p) defined in a neighborhood of y, then the con f ormal


340

[CHAP. VII

mapping can be realized in terms of a local uni f ormizer of P by means of the correspondence z* = z -}- sh(z) + o(e).

where h is defined by (7.11.5). This relation holds uniformly in each closed subdomain o l 9 which does not contain the curve y. In deriving formula (7.13.32), in which z and z* are local coordinates of p and p* expressed in terms of the same uniformizer, we have assumed that r(z) is of the form (7.13.21), where o,(z) is bounded independently of s, and we have made use of formula (7.13.7) which

in turn depends on (7.13.6). Moreover, in (7.13.26), we have imunder the mapping (7.13.5). Hence, plicitly assumed that essentially, the proof of (7.13.32) is based on the validity of conditions (7.13.18)-(7.13.20). We have also made certain normalizing assumptions in the cases G = 0 and G = 1. Suppose now that the conditions (7.13.18)-(7.12.20) are fulfilled and that r(z) is of the form (7.13.21). We are then justified in assuming that there is a mapping which, expressed in local coordinates,

has the form (7.13.32). The relation (7.13.32) was derived from (7.13.18) by comparing terms of order s on both sides of the equation.

We shall now return to the same equation but work with a higher order of precision in s and will find conditions on o,(z) in order that JI and l* be conformally equivalent. By (7.13.1)' and (7.13.32), Q(z, p,(*,) = sr(z)T'(z, p,) (7.13.33)

+ 62 r(z)

[ar(zP) h(p,) + 8T'0 ,p,) h( r)' +

and, by (7.13.31), )

(7.13.34) Im Z(p; = Im Z(p,)

+

2

2

[r(z)]2T"(z,p,,)

Im {Z"(p,)[h(p,)]Z} + o(e2),

Z*(p) = Z(p) - sZ'(p) h(p) + o(e). Substituting these relations in (7.13.18), we find that (7.13.35)

} + 0(89'


1 7.183

341

Im j ys f r(z)T'(z, p,)Z'(z)dz l 2ni

-s2 [z" (py)[h(p,)]2 + (7.13.36)

17(z)T'(z, ,)d(Z'(z)h(z))

f r(z)!8T (z P,)h(p,)

l

21

,p,)h(P,))Z'(z)dz

+ 8T

V

f [(r(z)]2T"(z, p,)Z'(z)dz]J + o(e2) = 0. r

Dividing this equation by e and letting a tend to 0, we obtain Im{2

(7.13.3 7)

`ro (z) T' (z, p,) Z' (z) dz } =

0.

J

Since Z' (p,) = 0, the expression iT' (q, p,) Z' (q) is an everywhere finite quadratic differential on O1 and (7.13.37) is a consequence

of (7.11.2). Thus no new condition is imposed on ro(z) by the s-term. On the other hand, consideration of the higher order terms will lead to conditions on 99 (z) and we shall show that these conditions can be satisfied for any ro(z) which satisfies (7.11.2). However, before entering into these consideration, we want to investigate in an analogous way the conditions imposed upon ro(z) by the equations (7.13.19) and (7.13.20). We have by (7.13.1)' and (7.13.11)

"(Z) = er(z) (7. 13 . 38)

f 9v

Tanz.

)

ds, +

82

2

(r(z))2 f

11,

T

(z, P) ds,

+ .. .

91

91

-- - sr(z)2yciW''

(z) - 2

where

f

Pr aT' (q, P) anv ds,

V,

(7.13.39)

d _ 2 d f9, aGan9 (q, P) ds _Ldq Im dY 91

_ -- 2=iW;1,,(q).

q)


342

eRe

[CHAP. VII

The expression W,,,,, (q) is a differential with simple poles at p1 and y. Moreover ny

{`

dZ*-

JDVdZ*

(p1)(h(p1))2-Z"(pr)(h(pr))z}

91

v1*

+ 0(e2).

(7.13.40)

Substituting from (7.13.38) and (7.13.40) into (7.13.19), we have

j-[Z" () (h(p1) ) 2_Z" (p,) (/p,) )j

ImIe f V

'- f (y(x))2W91 qr(x)Z/(x)dx

e2J r(x)W 91 y(x)d(ZJ(x)h(z))

2

V

V

+ o(e2) = 0.

(7.13.41)

Dividing by e and letting a tend to zero, we obtain Im { f ro(x)W(z)Z'(z)dzJ)) } = 0.

(7.13.42)

7

Since Z' (p1) = 0, Z' (pr) = 0, the quadratic differential W 919r (q) Z' (q) is everywhere finite and (7.13.42) is a consequence of (7.11.2). Finally 2

er(z)2niZ".(z) - 2 (r(z))222iZ..(x) + o(s2).

(7.13.43) jµ (z)

Substituting into (7.13.20) we obtain

ImIa rr(z) Z',(z)Z'(z)dx-s25r(z)ZK(z)d(Z'(z)h(z)) (7.13.44)

V

V

+

62

r(r(x))2Z (x)Z'(z)dzI + 0 (82) = 0. V

On dividing by a and letting e tend to zero: (7.13.45)

Im fro(z) Z(z) Z'(z)dz } = V

and this again is a consequence of (7.11.2).

§ 7.13]

VA RL1 TIOiT PRESERVING CONFORMAL TYPE

343

The conditions (7.13.18)-(7.13.20) may therefore be written in the form (7.73.46)

Imi

2.(z)iT'(z,

if

P,) Z'(z)dz + F,e(ro+ ee.)

0,

Y

(7.13.47)

Im { f Qe(z) WPy9, (z)Z'(z)dz

G,.(ro + E8.) } = 0,

Y

(7.13.48)

Im { f Qe(z) Z,(z) Z'(z)dz + Ho.(ro + re.) } = 0, Y

where F,,, G/e and H,,i, are functionals of ro + 8Qe = r. For a = 0 we have by (7.13.36), (7.13.41) and (7.13.44), F,o(ro) _ ivZ"(p,)(h(P,))2-i f ro(z) T'(z, P,)d(Z'(z)h(z))

r

Y

(7.13.49) + fro(z)iIaT' PV)h(P,) +

aT

V

P, h ap.

Z'(z)dz

+ f(ro(z))2iT"(z, p,)Z'(z)dz,

(7.13.50)

-

G,o(ro) = 2 fro (z)

r

Z"(P1)(h(C1))2] 2

f (ro(z))2WZ ,(z)Z'(z)dz, r

(7.13.51) Hvo(ro)

f r,(z) Zµ(z)d(Z'(z)h(z)) + f (ro(z))2 Z (z) Z'(z)dz. r r

If G > 1, the number o = 3G - 3 = 6h + 3m - 6 is the number of real modAli of the surface 0 (which has a boundary, by hypothesis).

If G = 1, the number a of real moduli is 1. The everywhere finite quadratic differentials


344

v = 1, 2,

G

1,

v = 2, 3, ..

G

1,

iT'(q, p1)Z'(q),

,

W1V1 V (q)Z'(q),

(7.13.52)

[CHAP. VII

,u = 1, 2, ..., G, form a basis for the quadratic differentials of M. For they are a in number, and each is real on the boundary of 9`t when expressed in terms of a boundary uniformizer. The real linear independence of Z,, (q) Z'(q),

the differentials (7.13.52) is a consequence of the real independence of the linear differentials (7.13.53)

(a) iT'(q,p,) (b) W,1,(q),

v

(c) Z,,(q),

,u = 1, 2, ..

G.

The independence of these differentials follows from the fact that the residue of iT' (q, P,) at P, is - i, while the residues of W1,= (q) at p1, ', are ± 1. If a linear combination of the differentials (7.13.53) with real coefficients vanishes identically, the residues vanish and therefore no differential (a) or (b) can have a coefficient different from zero. Since the ZF, are independent, all coefficients vanish. The basis (7.13.52) will be denoted by Q1, Q2, QQ

The term 9a(q) behaves near y like a reciprocal differential. We will try to represent it, therefore, as a linear combination of reciprocal

differentials n(p, q). We assume without loss of generality that the Jordan curve y does not enclose any P-pole of n (p, q). We then select a points q, inside y for which the determinant Im Qi(q") 0 0 0. There are always such points q,, for otherwise we would have an identity (7.13.54)

Im { Z A,Qj(q)} = 0, A; real, for all q e 9, s-1

which is impossible because of the linear independence of the Q..

We then set (7.13.55)

es(p) = E

a,(e) real,

'-1

and try to choose the functions a. (a) in such a way that the tr equations

§ 7.13]


345

(7.13.46)-(7.13.48) are fulfilled. Using the residue theorem, we may

bring these equations into the form: (7.13.56) E a,(s) Im Q, (q,,) - eO,(a,, ... a0; s) = Ft(ro)

Here the functions -F,#0) are the set F,0(ro), H,a(r0) considered above. We can calculate from (7.13.56) the values of a,(0).

We observe next that the functions Oi(a1, , a0; e) are continuously differentiable functions of all their arguments. This depends ultimately on the existence of continuous derivatives of arbitrary order of the Green's function G* (p, q) with respect to its variables and with respect to the parameter e. Indeed one sees from (7.13.1) that one can develop G*(p, q) - G(p, q) into powers of e up to an arbitrary order e°C with an error term 0(a"+1). We obtain in this way more precise variational formulas; but we have confined ourselves to the first order term for simplicity. We now apply the following theorem on implicit functions [1, page 9] :

Consider the system of equations f. (X1, x2, ..., x,; t) = 0, y = 1, 2, .. (7.13.57)

a.

Let a set x°, - , xQ be given such that f,(xi, x2, ..., xe); 0) - 0, v = 1, 2, ..., a. Let all a(a + 1) functions af,/axQ, at,/at be continuous in their a + 1 variables and let the determinant I af,/8xQ ( be different from zero at the point (xi, - - , xQ, 0). There exists a uniquely determined set of continuous functions gQ(t) defined for sufficiently small values

of t such that and that

gQ(0) =xQ, e= 1,2,...a. f,(g1(t), g2(t), ..., go(t); t) - 0

is satisfied identically in t. Moreover, the first derivatives dg,(t)/dt exist and are continuous in the t-interval considered. Applying this result to our special problem we recognize that we may determine the reciprocal differential ro(p) arbitrarily except for the conditions (7.11.2). We can then always find a function e,(1i) which is continuously differentiable in a and is a reciprocal


346

[CHAP. VII

differential of p such that all conditions (7.13.18)-(7.13.20) are fulfilled identically for sufficiently small e. We have assumed in this section that Tt has a boundary; the same reasoning can be applied in the case of a closed surface. In this case the role of the boundary is taken over by a point q0 of the surface

with respect to which we normalize the ZA by the condition 0 (mod 1). It is convenient in this case to require of ro(p) in addition to (7.11.2) also the condition: Tm

(7.13.58)

h(qo) = 2xi f ro(pj)n(g0, pi)dzl = 0.

This simplifies several formulas Y and by (7.13.32) also leads to the result that q0 and qo have the same coordinate in any local uniformizer.

We summarize the results proved in this section in the form of a theorem. THEOREM 7.13.2. Let y be a Jordan curve lying in the domain of a local uni f ormizer z on the surface V. Let ro (z) be a function which is regular analytic in a complete neighborhood of y and which satisfies the orthogonality conditions fro(z) Q,(z)dz = 0,

(7.13.59)

v = 1, 2, ..., o,

r

, Q. are a basis for the everywhere finite quadratic

where Qx, Q3,

differentials of W2. In its dependence on the uni f ormizer, ro (z) transforms like a reciprocal differential. Let m denote the number of boundary

components of 9 and write (7.13.60) z

h(p) =

1

2ni

-

f ro(pi )n(p,pi )dzi

r

f ro (px )n (p, pi) dzx,

27ci

m

f-Y0 (jj. )n(P, fij )4 j,

= 0.

V

It ?R is closed, we impose on ro (p) the further condition (7.13.61) h(qo) = 0 where q0 is the point for which Im Z,, (qo) - 0 (mod 1), p = 1, 2, , 2G, G ==A.

m>0 ,

§ 7.14] VARIATIONAL FORMULAS FOR CONFORMAL MAPPING

347

Under these assumptions there exists, for all sufficiently small e, a function es(z) which is bounded independently of a such that the attach-

ment of a cell to 9t along y by means of the reciprocal differential (7.13 62) r(z) = ro(z) + 60:(z) in the manner described in Section 7.8, leads to a domain l* which is con f ormally equivalent to 1%. That is, there exists a one-one con f ormal

mapping in which the point p* of Jt* goes into the point p o t V. If the local coordinates of p* and p are expressed in terms of the same uniformizer z of 9R, z* = z(p*), z = z(p), we have by Theorem

7.13.1 the relation z* = z + ch(p) + o(s). Theorem 7.13.2 justifies the statement, made at the end of Section 3.8, that the quadratic differentials are connected with the moduli (7.13.62)

of the Riemann surface V. 7.14. VARIATIONAL FORMULAS FOR CONFORMAL MAPPING

In Chapter 5 we developed a theory of the mappings of a given surface % into another surface R. We obtained necessary and sufficient conditions for such mappings, given locally by certain power series. These conditions are expressed in terms of the coefficients of the power series considered; but they soon become so complicated that it is difficult to answer even relatively simple questions by

means of them. In the case that % is the unit circle and Dt the complex plane the above conditions should, for example, solve the coefficient problem for schlicht functions. It has been impossible, however, to deduce from them bounds even for the third coefficient of the power series for a schlicht function. It is, therefore, useful to apply variational methods and we shall characterize maps which extremalize a given functional of the map. Clearly, the main tool in such a variational approach is a sufficiently general formula for the construction of nearby mapping functions. We proceed now to establish such formulas by generalizing the methods of Section 7.3. Let 9 be a given surface which is mapped into a subdomain WI of W. The surface 91 may have a boundary or may be closed. If 91

348


[CHAP. VII

is closed, t is necessarily closed and is itself the image of fit, that is M = 91. An interesting problem arises if, for example, %is a multiply-

c6nnected domain and 91 is the sphere. This leads to the theory of schlicht functions in multiply-connected domains and will be treated

in the next chapter. We shall perform on fi an interior deformation of the type described

in Section 7.13. The given mapping of 91 onto 9Jl then defines an interior deformation, by cell attachment, of 91 and therefore of 91. If we have enough parameters at our disposal, we can find an interior deformation which preserves the conformal type of 91 and 91 simultaneously. The mapping from the domain 91 into the varied subdomain 9X* C 91 serves as an infinitesimally near comparison map for the original map of 9 onto 9)1. Let N be a mapping from 91 onto an arbitrary subdomain 91 of the given surface 91. Let Q1(p), Q=(A), , Q.,(P) and .21(q), ., . (q) be real bases for the everywhere finite quadratic 22(q), differentials of R and 91 respectively; al is the number of real moduli of 9 and a2 the number of real moduli of R. We suppose that the point P of 92 corresponds to the point q of 91 under the mapping N. We consider the linear combination (7.14.1)

a,

JdC\ 2

Z c,Q (p) +

alk

E C, .2, (q)

where z is a uniformizer at the point ' of 92, C a uniformizer at the point q of M. If there exist al + oo real numbers cl, c,l, C1, , C,1, not all zero, such that the linear combination vanishes identically,

we say that the of + oa differentials are real linearly dependent with respect to N. Suppose first that the differentials are independent with respect to N. Let y be a Jordan curve in the domain of a local uniformizer z on the surface 9, and let ro(p) be a reciprocal differential which is regular analytic in a neighborhood of y. Let q be the image point

of P in the mapping N, ' a uniformizer at q, and write (7.14.2)

ro(p) = Ro(q) dC

The Jordan curve y represented in the plane of C will be denoted

by I'. We suppose that

§ 7.14] VARIATIONAL FORMULAS FOR CONFORMAL MAPPING

f

(7.14.3)

349

v = 1, 2, ..., al,

i0,

7

(7.14.4)

ro(p) .2, (q) Y

(d

a

I) dz = f R0(q).,(q)dC=0, v=1, 2,...,a2.

r

Let ml, m$ be the numbers of boundaries, and let n(pl, P2)1 N(41, q3)

be the n-functionals of %, St respectivley. We write (7.14.5)

2ni

h(p) _

fro(p1)n(p,p1)dz1_j(p1)n(p,p1)d;i,

> 0,

Y

Y_

and (7.14.6)

Fn,i J

r

H(q) _ 2ni

r

Ro(g1)N(q, gi)dC1-

RR(q)N(q, Q1)dC1,

2ni

f1o(j)N(q,

m2>0,

1,

m2 = 0.

If either fl or 91 is closed, we suppose that the corresponding condition (7.13.58) is satisfied. In other words, we impose upon ro(p) the requirements of Theorem 7.13.2 with respect to both surfaces 9t and flit.

To the al conditions (7.13.46)--(7.13.48) for the function e, imposed by the surface YI we add the a$ corresponding conditions for e. imposed by R. giving a1 + as conditions in all for the function e,. Since the quadratic differentials involved are independent in the real sense, the reasoning of Section 7.13 shows that for all sufficiently small e there exists a function e,(z) which is bounded independently of e and which has the following properties. Writing (7.14.7)

r(z) = ro(z) + ee,(z),

we identify the point z of y with the point z + er(z) of the neighboring curve y, in the manner described in Section 7.8. Since z acts as a ziniformizer for N as well as for W. we simultaneously form in


350

ICaAP V1I

this way surfaces 9i* and 'R* from s1I and R respectively. The function

o,(z) has the property that not only 91 and %*, but also T and T4 are of the same conformal type.

Let the mapping from %* onto 1, in which the point p* goes into the point p, be denoted by N8, and let the mapping from i* onto fit, in which q* goes into q, be denoted by R,. In the mapping N from T into R, the point b with local coordinate z, say, goes into the point q with local coordinate C. By inversion of (7.13.32) we have (7.14.8)

z* = z + sh(p*) + o(s) = z + eh(p) + 0(s)

and, by the analogue of (7.13.32) forep fit, (7.14.9) C = C* -- eH(q*) ± 0(e) = S* - eH(q) ± O(s). The composite mapping N,-'NR, is a one-one conformal mapping from Q onto a subdomain VA of i which sends the point p of R.

into a point q of R and, expressed in terms of local uniformizers,

it is of the form + eh(p) dz - _-H(q) + o(s).

(7.14.10)

Here h(p) is expressed in terms of the uniformizer z while H(q) is expressed in terms of C. Thus, given a mapping N from 9 onto a subdomain 9R of R, it is possible to construct a varied mapping (7.14.10) in which R is carried onto a subdomain TtI of 91 where .l 6 is an e-variation of Mt. The possibility of varying conformal maps in this fashion enables us to apply the calculus of variations to extremal problems in conformal mapping. We have to make sure, however, that the mapping (7.14.10) does not reduce to the identity C' _ C or even to a mapping o (e), that is C + o(e). We must, therefore, investigate the possibility that for each permissible choice of ro(p) we have identically (7.14.11)

1i J

2i f

r

dc L 2ni

r

R0(g1)N(q, gi)dCi -

.

J

Ro(qi)N(q,

§ 7.141 VARIATIONAL FORMULAS FOR CONFORMAL MAPPING

351

For the sake of simplicity we restrict ourselves to the case that ml > 0 and m2 > 0; the same reasoning would also be valid in all other cases. We may assume without loss of generality ttiat 92 coincides with 9N, i.e. is already imbedded in N. Thus, we may put z = 4, p = q, y = T, ro (q) = Ro (q). Hence (7.14.11) has the simpler form: 27ci

(7.14.11)'

f ro(g1)[N(q, q1) - n(q, g1)]dC1 v

276 -.

f ro(g1)[N(q, q1) - n(q, g1)]dZ1.

This equation is assumed to hold for arbitrary choice of ra(q) so long

as (7.14.3) and (7.14.4) are fulfilled. One derives easily that this condition can be fulfilled only if Cl

aZ

N(q, q1) - n(q, q1) = E ati(q)Q,(g1) + E V-2

v=1

al

as

and

N(q, q1) - n(q, g1) = I y(q)Q+(q1) + 16, (q)-2,(q1) -1

.-1

identically in q1 for arbitrary choice of q. However, the last equation is clearly impossible in the case where q is a boundary point of 92

which is neither a q-pole of n(q, q1) nor a boundary point of R. In this case, n(q, q"1) would become infinite as q1 approaches q whereas N(q, q1) remains finite since q is not a boundary point of 92.

Thus, we have proved: THEOREM 7.14.1. 1/ a domain 92 can be mapped into a subdomain

V of 92 and if there is no real linear dependence between the basic quadratic differentials of 9)2 and of 92, then there exists an infinity o l mappings of 92 into 92 of the form (7.14.10). We have assumed, however, that the a1 + a2 quadratic differentials arising from the surfaces 92 and 92 are linearly independent in the real sense. Suppose now that this hypothesis is not fulfilled. Then there exist real numbers c1, , cal, Cl., , Ca,, not all zero, such that 472

(7.14.1)'

± 411(p) + (a 2 E C,t,(q) = 0. dz .-1 V-1

352

VARIATIONS OF SURFAC.6S

[CHAP VII

It is clear, because of the linear independence of the .1,, that not all coefficients c, are zero. Similarly, not all C, are zero. Writing (7.14.12)

Q(p) = E' C. Q. (P),

r-1

.2 (q)

X C' a,

v-1

we have the equation 2

(7.14.13)

.c(q) `dz)

= UP)

where the subscript denotes the uniformizer in terms of which the corresponding differential is expressed. This equation shows that the surface N is mapped onto a subdomain D1 of R whose boundary consists of finitely many analytic arcs on each of which we have (7.14.14)

-°Zc(q)dC2 > 0 or 2c(q)dC2 < 0.

A proof of this statement depends only on the local behavior near the

boundary and is therefore the same as the argument given in [5], Chapter VI, where .61, Q are quadratic differentials of special type. For this reason a proof is omitted here. We may also interpret equation (7.14.13) as stating that 2c(q)dC2 is a quadratic differential of l as well as of M, in which case (7.14.13), written in the form (7.14.13)'

2c(q)dC2 = Qx(j)dz2,

expresses the invariance of the quadratic differential under a change of uniformizer. In the case that the domain R C 91 possesses a quadratic differential which is finite everywhere in fl and is also a quadratic differential for fit, it might well happen that Tt cannot be varied within SJR under preservation of conformal type. In this case we say the domain DI is rigidly imbedded in M. Consider the following example of a rigid imbedding. Let M be a

triply-connected domain in the complex plane bounded by three curves B,, v = 1, 2, 3; we suppose that B1 encloses the two other curves B. and B8. If we connect B2 with B3 by a continuum in R we obtain a doubly-connected subdomain )l of R. This domain can be mapped upon a circular ring in the z9 plane, 1 < W I 0, .2(1') = finite quadratic differential on IR. if 1(z) leads to a mapping of % into RJR of this particular type, we cannot vary f (z) freely and it is, in this sense, an extremum function. In the general case, however, we may choose

h(z) =

(8.2.6)

1.l 2ni

ro(ni)n(z, ni)dzi "12ni

v

f

V

and, taking P, = f (nx), Hrf(z)1 = (826) ..

i Jf ro(vz)

()2N(f(z),pi)dzi awl

\\dW))N(/(z).i)dz

'

r

with an arbitrary Jordan curve y in 9t and a reciprocal differential :r ,(x) which has only to satisfy the conditions

§ 8.2]

THE COEFFICIENT PROBLEM FOR SCHLICHT FUNCTIONS 367

(8.2.7)

and (8.2. 8)

f r

r0(a)Q,(a)dz = 0,

f

ro(n)

v = 1, 2, ..., a1,

(')2..p)dx = 0 ,

v = 1, 2, ..., da,

where the Q,,(n) Y for a real basis for the finite quadratic differentials of % and the (v) an analogous basis for 91. By to ing H(/(z)) in the form (8.2.6)' we have assumed that 91 possesses a boundary; we do this only for the sake of definiteness but the reasoning will hold in every case. The variational formula (8.2.4) creates for each schlicht function (8.2.2) a whole neighborhood of schlicht functions. If we want them to have the same normalization as A z) we must require h(0)f'(0) = H(O) + o(1), (8.2.9) which leads to the following condition on ro(c) : 1 [n(0, x)f,(0) f

-

2ni .I ro(w)

(N(0,16)

dz

(8.2.9)'

f

[n(0, ac) f'(0) -((dac N(0, fi)] dx.

Y

Let us now compute the coefficient b

of /A (z). For this purpose,

we develop the functions N(f (z), p), N(f (x), fi) at z = 0 and the functions n(z, a), n(z, A) around the same point. We shall have (8.2.10)

n(x, ) _ 'aQ( )zQ;

n(z, x) _ Q-o

(8.2.11)

N(f(z), ) = EAQ(P)#, Q=o

Q-o Quo

The coefficients aQ(ac) and AQ(fi) will play a central role in the theory of the coefficients of functions in % which are schlicht relative to OR.

Let us consider them in greater detail. The coefficients a. (a) are to be understood as given with the domain % while the AQ(P) depend not only upon the domain 9t but also upon the coefficients of the function 1(z) in question. It can be easily seen that AQ(P) depends

THE VARIATIONAL METHOD

368

[CHAP. VIII

on the first a coefficients bl, , be of 1(z). Since n(z, n) depends on n like a quadratic differential, the coefficients aQ(n) will be quadratic differentials on %. They will not be finite, however, on the

whole surface J2, but will have poles at the point no. In fact, we have by (8.2.10), e

(8.2.10)'

ae(n)

=Q

[n(z, n) IX-0 dze

and since n(0, n) has a simple pole at no, we find that ae(n) has a pole of order a + 1 at no. The coefficients ae(i), on the other hand, are everywhere finite quadratic differentials on the double of 92. They permit us to define on % the everywhere regular quadratic differential

ae(n) = (aQ*)-

(8.2.12)

which, when expressed in terms of boundary uniformizers, satisfies (8.2.13) aa(n) = ((aQ(n)) on the boundary of 91. We can make an analogous statement with respect to the quadratic differentials on 92, namely A e (p) and (8.2.14)

AQ(') _ (AQ(()) AQ(p) has a pole of order a + 1 at P. while Ae (fi) is regular every-

where in !R, and on the boundary of 92 we have, using boundary uniformizers,

AQ(p) _ (A(p))-.

(8.2.15)

From (8.2.4)-(8.2.6) we can now calculate the coefficients

b, of f (z). We easily find: (8.2.16)

}e

, lro(n)U.(n)dz-e

'2nijro()(V.(n))-dz+0(8),

YJ

with .+s

(8.2.17)

=X eba.+i-e(n) -- A,() Q

and (8.2.18)

V -W = Z e(be)-a.+i-e(n)

(dZ'

112

l

"I

)2 .

§ 8.23


We shall call a point (bl,

-, b,,) an interior point of the coefficient

body V,, if we can choose ro(a) in such a way that each point of a sufficiently small sphere around this point in Euclidean 2n-space is attained by a variation (8.2.16) of the coefficients. We shall call the point a boundary point of V if we cannot fill a whole sphere around it by these variations. We may put (8.2.16) into the form

Re {bA - b,} = s Re{

2ni

fro@)[u() -{-

Vv(n)]dz} + 0(s),

(8.2.19)

Im{bft-b,} =eRe{_! t

y

Jro(@). i

[U,(a)-V.(a)]dz}+o(a),

and formulate the condition (8.2.9)` on ro(a) as follows: (8.2,20)

?_. f Y

ro(n)Uo(n)dz =

2ni

y

Since ro(n) is quite arbitrary except for the conditions (8.2.7), (8.2.8)

and (8.2.20), it is obvious that we can attain every point in the neighborhood of the initial point (br, b2, . , so long as there is no linear dependence, with real coefficients, between the 2n+2+ai+a2 quantities (8.2.21)

U,(n) + V,(n), %4(U,(a) -V,(n)), v = 0, 1, ..., n; (dp 2

Q,(n), v2µf0

dx) , Thus. a boundary point of V. can be characterized by the existence of n + 1 complex numbers A,o, Al -, A,, such that (8.2.22)

E yp

Q(n) + 2(P) lddo///i2,

where Q(n) and 2(p) are two finite quadratic differentials on and 91, respectively. We are now able to characterize the domains 23't C 91 which belong to the extremum functions f (z) in 91 whose first n coefficients lead

to a boundary point on V,n. Let n be a boundary point of )l; by virtue of (8.2.13) we, may then write (8.2.22) in the form:

37o


[CRAP. VIII 2(p))'dp\2=

(8.2.23)

(y o

[),A,(p) + x,A,(p)] +

real

if n lies on the boundary of R. But the expression (8.2.24)

-1r(p) = .2(p) + z [a,A,(p) +

_k(p)]

,-o is a quadratic differential on R with a pole of order n + 1 at most at the point p0. Thus, the boundary of the image domain 9't may be characterized by the simple differential equation on O't: (dp)S (8.2.26) > o.

r()

We conclude, in particular, that the extremum functions f (z) give domains PC R with boundary curves which are composed of analytic arcs with respect to uniformizers on R. We may characterize the extremum mappings in the following form. We define the quadratic differential on 9t

(8.2.26) Y(n) = ` [2 I v-0

Q-1

A, 4?-l

This differential has a pole of order n + 1 at most at no and is regular

on the rest of J't. The condition (8.2.22) may then be expressed in the form: (8 2.27)

Y(n)dx2 = rr (p)dp8.

Thus we have THEOREM 8.2.1. The mappings of a domain W into a domain N which lead to boundary points of the coefficient body V,, are characterized by the differential equation

Y(n)dc" = '(p)dp2 where Y(n) and #(P) are quadratic differentials of 9 t and 8R respectively

which are regular on their surfaces except for a pole of order n + I at most at the corresponding points no and po. This statement covers also the case that 9 is rigidly imbedded in 9t. In this case, we must consider the point (b1, , bn) of the V,, as a boundary point; on the other hand, we showed in Section 7.14

that the necessary condition for a rigid imbedding of 9 in ?R is the

§ 8.2]

THE COEFFICIENT PROBLEM FOR SCHLICHT F£1.\CTIO\'

371

differential equation (8.2.28)

Q(n)dn2 =

where Q (n) and _0(p) are two finite quadratic differentials on 91 and 91, respectively. We state next THEOREM 8.2.2. If 9X C R is the image of R by an extremum mapping then there are no exterior points of Dl on OR, that is, every extremum

function maps % into a slit domain on R. We know already from Theorem 8.2.1 that the boundary slits of X11 are all composed of analytic arcs with respect to uniformizers on 91.

In order to prove Theorem 8.2.2 let us assume conversely that there exists a point p, e R exterior to the image 1 of 91. We draw a Jordan curve T in the neighborhood of p, which also lies outside

of 92 and deform 9 by a cell attachment along r by means of a reciprocal differential R0(p) defined in a neighborhood of F. We put

on Ro(b) the restrictions

fRo().i.()dw = 0, v= 1,2,...1 Q2,

(8.2.29)

r

where the -0,(P) are a basis for all finite quadratic differentials on 91.

Then the deformation of 91 leads to a conformally equivalent surface 91A. We may map 01 A back into lR by means of a correspondence w

= w°

e f Ro(pj)N(wo, p1)dwi

r

(8.2.30)

+ 2ni f

Ji)dwl + o(e),

T

which is valid for every choice of the uniformizer w. The function

f (n) maps 91 into 91; this domain is not affected by the cell attachment at a curve outside of it, but it will change into a domain 9W under the corrective mapping (8.2.30). Thus, instead of the schlicht function /(a) mapping 91 into 9)1 we will have the nearby schlicht function


372

fA (n)

[CHAP. VIII

f

=

(8.2.31)

+

E

+ o (s) J T which maps 91 into PA. This formula corresponds to (8.2.4) but the term h(z) is omitted. We may carry out the same calculations as before and find for the v-th coefficient of /6(a) the formula:

b (8.2.32)

2ni

r

f

+ o(e).

r

, b,) will be an interior point of V except in the case that there exist n + 1 complex numbers A0, A,, , A. such that Reasoning just as before we conclude that (b1, b2,

n

(8.2.33)

E [A,A,(p) + ,-0

where /2(() is a finite quadratic differential on R. But such an equation is impossible since A,(p) has at po a pole of exactly order v + 1 and the poles in this equation cannot cancel each other. Thus, our assumption of an exterior point p, of the extremum domain 9)t leads us to a contradiction. If Tt belongs to an extremum function /(a) there can be no point fib exterior to it on 9t and the Theorem 8.2.2 is proved. Let us return now to the linear relation (8.2.22) and interpret it in geometric terms. We write briefly ab, = b f -- b, - o (s) and may then derive from (8.2.22) the equation (8.2.34)

Re { E A, db,} = 0

,-x for all variations considered. This means that by our variations we can cover only a (2n-1)-dimensional part of V,, namely a surface

element which is orthogonal to the vector (Re .1,i, - Im A,, Re A2, , - Im An). The reason for this fact lies in the special nature of our variation. Since we are at a boundary point of Vn,

§ 8 2]


the image domain T1 on i is by Theorem 8.2.2 a slit domain. All variations by cell attachment used so far will transform a slit on 91 into a slit since they are regular everywhere on 91 except in the neighborhood of the curve I'. It is clear, however, that there are deformations of W which do not change conformal type but destroy the slit nature of the domain. Since we know already that the boundary of an extremum domain consists of analytic arcs, we may use

the Julia variation of the boundary of 9J which was described in Section 7.15.

Let y be an analytic arc on the boundary of R and let r be its image on Si; T will, therefore, be an analytic arc on the boundary of X71. Let C be a boundary uniformizer on y and w a boundary unifor-

mizer on r; let Sv(C) be a real-valued analytic function of C on y and let &o = iw'(C) Sv. We shift every point of r by an amount &o; this will lead to the boundaryI'A of a new domain 9JI which is conformally equivalent to D1 if the following condition is fulfilled: (8.2.35)

=0

f 6v (C) Q(C)dC

for all finite quadratic differentials on R. By (7.15.7), the function /A (ac) which maps 9t on O1 has the form (8.2.36)

f 0 (z) = f (z) - f'(z)

J n(z,

o(Sv).

Since 9J2 is a slit domain we are forced to choose Sv z 0, that is, we can shift boundary points of 932 only in the direction of the interior normal. We again compute the coefficients bA of the varied function. Using (8.2.10), we find "+1

1

(8.2.37) b, = b,,

E ebQa.+i-e(C)l d + o(Sv). f Sv(C)(Q-1 /

r

If we wish to keep the normalization (8.2.2) for /A (z) also we must restrict 6v (C) further by the condition (8.2.38)

f Sv(C) ao(C)dC 7

= 0.

374


[CHAP. VIII

Then, using the notation (8.2.26), we have in view of (8.2.35) and (8.2.38): (8.2.39)

Re I ),8 b.

f Y(C) & (C) dC. 27c

r

We may also consider variations composed of an interior cell attachment along a Jordan curve yl in )2 and a Julia variation along a boundary arc y of fl. This type of variation is very convenient since the conditions (8.2.35) and (8.2.38) may be difficult to satisfy

with 8v > 0, as necessary under a pure boundary variation. In a variation of the mixed type, however, we may choose 6v > 0 quite

arbitrarily on the boundary of % and then adapt the interior cell attachment in such a way that the moduli are preserved and that the point no e W is still mapped into p0 a 91. The possibility of such adjustment can be shown by the same considerations that were used in Section 7.13. There are only two cases in which this adjustment is impossible, namely the case that is rigidly imbedded in R and the case that some expression 20U0(v) + ,,V0(r) depends linearly

on the finite quadratic differentials of 9 and 91. In this case, the boundary of Ot satisfies the differential equation (8.2.5)'

dt)

0 arbitrarily on y. On the other hand, we know that an interior cell attachment changes the term Re { 1 A,ab, } only in infinitesimals of higher order. Thus, the total

effect of a mixed variation with arbitrary 8v > 0 which preserves the moduli and keeps 8b0 = 0 will still be given byr (8.2.39). We see that we may indeed vary the value of Re j

.

vi

.46b, 1. In 1

general, we shall even be able to increase or decrease this number

§ 8.21

THE COEFFICIENT PROBLEM FOR SCHLFCHT FUNCTIONS 375

by appropriate Julia variations, This will be impossible only in the

case that the quadratic differential Y(') does not change its sign on the entire boundary of 9t. Let F(Re bl, Im b1, - , Re b, , Im

be a continuously different-

iable real valued function in some open set of the Euclidean 2nspace which contains V.. Then F must attain its maximum in V. at some point (b1, , b,4) and we must clearly have It aF aF (8.2.40) E{Im 8b,} Re {bb,} -f- E 0

,_IaReb,

,_1aImb,

for all permissible variations within the family of schlicht functions on 92 relative to 92. Thus, the extremum function 1(z) which leads to the maximum of F must satisfy a differential equation (8.2.27) in which the differentials do not change their sign on the boundary of 9t. Thus we have proved: THEOREM 8.2.3. Let f (x) be a mapping function of R into 91 whose first n coefficients b1, b2, , b maximize a continuously differentiable

function defined in an open set containing V. Then p = f

will

satisfy a differential equation YY (p)dp2

YXd 2 = differentials of 92 and 91 which where Y (n) and IV (p) are quadratic have poles of order n + 1 at most at the points no and po and which do not change their signs on the boundary of % and on the boundary of, respectively. An extremum mapping p = /(x) may also be considered as a realization of the given surface OR in the surface 92. In fact, if we identify those boundary points of 92 which correspond to the same point of a boundary slit of 9)2 we obtain another replica of 91. The process of identification can easily be carried out in T2 by means of the quadratic differential Y(r). In fact, let % and n;, be a pair of boundary points of % which have already been identified and suppose

that both correspond to the boundary point pl of V. If we run from p1 along an arc of the boundary till we come to a point p2, we will have one image arc on the boundary of 91 running from nl to n= and another arc running from xi to n$. But because of the differential equation we will have

376

(8.2.41)


rVj-Y-(n) d-n =

f'VI-Y(x)

[CHAP. VIII

I do 5d

Thus, we have to identify points on the boundary of 91 which give equal are length on the boundary in terms of the metric based on the quadratic differential Y (n) . Since 91 may be a surface of high genus and 92 can be chosen even simply-connected and planar, the extremum problems considered often lead to very useful realizations of a complicated surface on a simple domain by boundary identification. We shall give examples

of such realizations in the next section where we shall consider particular applications of the general theory developed here. 8.3. IMBEDDING A CIRCLE IN A GIVEN SURFACE

In order to illustrate our general result, we make the following particular application. We assume that the surface S91 is a disc and suppose that it is realized over the unit circle of the z-plane. We ask for the coefficients bx of all functions which map the unit circle into a given surface 91 such that its center corresponds to a given point PO a 91 at which a fixed uniformizer w(P) is prescribed.

First it is clear that T can be mapped into 91; for let { w { < be a neighborhood of P in the uniformizer plane. Then 91 can be mapped into 9 by the simple correspondence (8.3.1)

w = Qz.

If t (z) maps % into 9t then every function f (az) with { a I < I will lead to another imbedding of 9 in R. In fact, the correspondence z' = az maps % into a subdomain 91' C % and the mapping by f transforms 9' into a subdomain of the image 97l C 91 of 9t. If bl is the coefficient of a function f (z) which imbeds % in OR then every number abl with { a { < 1 will be an admissible coefficient,

too. This shows clearly that the coefficient region Vl is, in our special case, a circle around the origin in the complex plane and there

remains only the question of determining its radius. In order to solve this problem it is sufficient to ask for the mapping function / (z) whose first coefficient is positive and has the largest

possible value. In other words, we have to pose the extremum

§ 8.3]

IMBEDDING A CIRCLE

377

problem of maximizing Re bl. We can now apply the general theory of the preceding section, but here we have additional information about the functionals of W. In a simply-connected domain no finite quadratic differentials exist. By (7.10.4) the variation kernel of the unit circle has the form: (8.3.2)

n(z,') = z x

_.

,

n(x, )

2;2x-C

=

x

1 -}- x

2C2 1 -xC

Hence, we find from the definitions (8.2.10) and (8.2.12) : (8.3.3)

ao(C) = 0, ao(C) = 0; a1(C)

2C2

If we want to exhibit more clearly the dependence of the problem

on 1(z) it will be convenient to use instead of (8.2.11) the series developments : (8.3.4)

N(w, ) = E ae(p)wQ, N(w, ) = E ae(fi)wQ e-o Q..o

which represent the variation kernel of slt in a neighborhood of po in terms of the uniformizer w. The ocQ (p) depend only on the choice of the uniforu izer w while the coefficients AQ(p) in (8.2.11) depend

also on the unknown extremum function 1(z). We insert w = t (z) into (8.3.4) and compare the resulting series developments in z with the series (8.2.11). We obtain: (8.3.5)

ao(p) = Ao(p), blal(p) = A1(p),

ao(p) _ (ao(h) = Ao(p),

blai(p) _ b1 Ai(p) By Section 8.2, the function p = f (z) with maximum value of Re bl maps the domain )1 into Tt such that the following differential equation holds: [bi(ai(p) + ai(p)) +2oao(p) + !oa0(P) + 2(p)1dp2 = - bdz2. z2 (8.3.6)

Here, Q(p) denotes again a finite quadratic differential on R. The image domain )2 of l1 in 91 will be a slit domain according tc Theorem 8.2.2. We verify in our special case the assertion of Theorerr 8.2.3 that the quadratic differential Y(z) = 1/z2 does not change it e under the following normalization at infinity (8.3.22)

z=

P(w)

= w +po

-V'

1 +12 -}- .. .

The radius a of the image circle is a functional of Z called the mapping

radius of Z. Gr6tzsch and P61ya have considered the following problem:

Given an arbitrary bounded set of points in the w-plane, to find

§ 8.31

IMBEDDING A CIRCLE

383

a continuum C which contains the given set and whose exterior Z has the least possible mapping radius o. Let R be the w-plane from which the given point set has been removed. Let 91 be the exterior of the unit circle in the z-plane. Consider all functions w = 1(z) which are schlicllt in 92, have at infinity the series development (8.3.23)

w = 1(z) = biz -{- b2 -} za -f-

b2

-f- ...,

and map 9 into R. It is easily seen that the maximum of Re b1 for all these functions equals the reciprocal l/a of the minimal mapping radius o in the preceding problem.

It is unessential that the normalization (8.3.23) is prescribed instead of (8.2.2); we can pass from one problem to the other by linear transformations. Thus, the extremum problem of GrotzschPdlya is answered by our result. It has already been treated by similar variational methods and an analogous characterization for the extremum domain has been obtained in special cases. It appears in the present treatment as a very special case of the coefficient problem for schlicht mappings of aRiemann surface%into a Riemann surface 91.

The function z(p) defined by (8.3.10) is a polymorphic function on the surface R; that is, it undergoes a linear transformation if we continue it from some given point P e 91 along a closed path back to the same point. Moreover, it maps the boundary points of 8f into points of the unit circumference. These properties are not characteristic for z(p); consider, for example, the function C(P) which maps

the universal covering surface t of R into the unit circle. This function will also have the same properties as z(P). If 91 is a planar surface with m boundary continua, there exists according to Riemann a function ij(p) which maps S91 on a domain covering the unit circle m times, such that the image of each boundary continuum becomes

the unit circumference. We now give a property common to all analytic functions z(p) which are single-valued or polymorphic on a surface 91 with boundary and which map each boundary continuum

into a circular arc. Consider the bilinear differential 2'(p, q) _M(fi, q) of 91; if both


384

[CHAP. VIII

argument points p and q lie on the boundary of R and if we use boundary uniformizers, 2' (p, q) will be real. Let z (p) be a function on t

with the properties described above. We may use z(p) as local uniformizer and write according to (4.1.5)': (8.3.24) 2(P, q)dp dq =

dz(,+') dz(q)

;rLx(P) - z(q)7'

- 1(z (P), z(q))dz() dz(q)

where l(x, C) is a regular analytic function of both arguments in the uniformizer neighborhood considered. Now let P and q he near a boundary continuum of 91; since this corresponds to a circular arc

in the z-plane and since on a circular arc 1

(8.3.25)

dz(p)dz(q)

n [z(P) - x(q)]'

_ real,

-

we derive from the reality of 2(p, q)dpdq on the boundary of

:

l(z, C)dzdC = real

(8.3.26)

if z and C correspond to two points p and q on the same boundary continuum of )R. In particular l(z, z)dz2

will be real on the boundary of the canonical domain. It is, therefore,

a quadratic differential of R. Let z*(p) be a local uniformizer of 9R and let z = gp(z*) lead from

it to the canonical variable z. We may express 2 in terms of the local uniformizer z* in the form 00 (p, q)dpdq

(8.3.24)'

dz*(p)dz*(q)

= 2-1Z* (P)

z*(q)1

- l*(z*(p), z*(q))dz*(P)dz*(q)

Comparing (8.3.24) and (8.3.24)', we derive after easy calculation (8.3.27)

lxx=

where (8.3.28)

,F, z

1

-

3

.

s

dg, ,

1P

dz*'

is the Schwarz differential parameter. Since l* can be calculated and

§ 8.4]

CANONICAL CROSS-CUTS ON A SL RF-4CE 91

385

since ? is a quadratic differential, this result leads to a third order differential equation for each function gv(z*) which is polymorphic in 91 and maps each boundary component into a circular are. In the case of a domain of genus 0 we may also consider the mapping

of 91 onto a plane domain bounded by m circles. If T(z*) is the mapping function, we can again assert that it must satisfy the differential equation (8.3.27). This result shows the close relation between the mappings on circular domains and the bilinear different-

ial .(p, q). 8.4. CANONICAL CROSS-CUTS ON A SURFACE %

In the last section, we were led by an extremum problem for the coefficients of mapping functions to a cut system on a given surface 9R which was composed of analytic arcs, made the cut surface to a

disc and such that the disc could be mapped upon a circle with particularly simple behavior of the boundary under this mapping. In the topology of Riemann surfaces 91, a crosscut or cycle on 91 is determined only by its topological character; in particular a homotopic deformation of a cut is considered unessential. It is, however, interesting to show that we may associate with each homotopy class of cuts one particular cut which can be characterized invariantly by an extremum problem. In order to formulate the extremum problem, we choose a fixed point po a R and a fixed

uniformizer w(p) at this point. If 9Jt is a subdomain of 91 with boundary and if G (P, q) is its Green's function, we have near pa: (8.4.1)

G(p,po)=loglwl+g(po)+0(I &l).

We call g(pa) the capacity constant of I1 at p0 with respect to the uniformizer w.

We give now a curve I' on 91 which does not bound and ask for

a curve C which is homotopic to T and such that the capacity constant g(p0) of the domain U1= 9R - C be a maximum. We have to show first that there exists a curve C for which the maximum value g(p0) is actually attained. For this purpose consider all competing domains 91 -.P and map their universal covering surfaces

lr onto the unit circle so that the point po goes into its center.


388

[Cxwr. VIII

Let p = pr(C) be the inverse mapping from the unit circle onto OR - P. Consider finally the function z(p) which maps the universal covering of t itself upon the unit circle with the center corresponding

again to p0. Then the functions (8.4.2)

z = IPr(0 = z(pr(0), d5r(°) = 0,

are schlicht and bounded in the unit circle and form a normal family.

Let a(.I') be any continuous functional of r. We can then assert that there exists at least one admissible curve I' for which a (r) attains its least upper bound A. In fact, let I',, (v = 1, 2, ) be a sequence for which a (r,) converges towards A. The functions possess a subsequence 0,,(C) which converges uniformly in each closed subdomain of the unit circle towards a schlicht bounded

function. Hence, there exists a uniformly convergent sequence of functions pr, (g) for which the corresponding sequence a (r,) converges

towards the least upper bound A. Let pc (C) be the limit of this sequence of functions; it maps the unit circle into the surface 91 cut along an admissible curve C. For this particular curve C, a (C) attain its maximum value A. Having established the existence of an extremum cut C on 91, we now characterize it by a variational method. We choose a Jordan curve y on 91 outside of a neighborhood of the extremum continuum C. We define on it an analytic reciprocal differential R(p) which is orthogonal to all finite quadratic differentials on R. We perform a transformation of 91 into itself by a cell attachment which preserves conformal type. According to Theorem 7.13.1 this deformation can be realized in terms of a local uniformizer w(p) as follows: w*

= w + 2-i f R (p1.)N(w, pl)dwx - 2ni f R (6i)N(w, fi1)dwr + a(e),

where N is the variation kernel of 91 and where we suppose (for the sake of definiteness) that OR has a boundary. We wish to hold the

distinguished point po fixed under this deformation and to have at p0 (8.4.4)

dw* dw Ir 0

1,

1 8.41

CANONICAL CROSS-CUTS ON A SURFACE OR

387

this will permit us to use w* as well as w in computing capacity constants at po. We impose, therefore, on R(p) the additional restrictions (8.4.5)

- f R(pr)N(0, pi)dwf = 276

F7ri

Y

f

r)dw

V

and

(8.4.5)'

2.i f R(P1)N'(0, pl)dwx =

Zna

fR(I'N'(0, fi2)d&3, y

Y

where the prime denotes the derivative of N with respect to the first argument. By the deformation (8.4.3) the domain t = N - C has undergone a transformation, too. Its conformal type has, in general, changed but we can determine the new capacity constant g* (po) by means of the variational formula (7.8.18) for the Green's function under

cell attachment. If we use w as uniformizer at p0, we find (8.4.6) g* (po) -= g(po) - Re

8G ( 1, po) Zdwr

f R (pi) {

}

+ 0(82).

Y

If we map the deformed domain W' back onto 9t by the correspondence (8.4.3) and change over from the uniformizer w to w* we do not affect the equation (8.4.6) since w and w* as well as their deri-

vatives agree at pp. Thus, the extremum property of the curve C leads to the condition a°*(po) < g(po).

(8.4.7)

Because of the arbitrariness in the choice of the small real parameter z, we conclude (8.4.8)

Re{

2Li

po))2dwi}=O f R(pi}(?G(pv l pi

for every permissible choice of R (P). The usual application of linear algebra leads, therefore, to the following condition. There exist two complex numbers Ro and Al such that

-4- 0(N'(po,

(8G(P, x'0))2=N'(p0,*) (8 .4 .9}

(Cawr. VIII


388

t

))-

ap

+ .2(p), + AIN(po, p) + i(N(po, where 1(p) is a finite quadratic differential on M. For p e 9R, the right side of (8.4.9) is a quadratic differential which has a double pole at the distinguished point po. Thus, we have (aG(PPo)) yr(p) (8.4.10) where #'(h) is a quadratic differential on 9R with a double pole at po. On the other hand, (aG/ap)dp is a linear differential on 9J't= 9R -C which has a simple pole at po. We find, therefore, that the boundary

C of R relative to 91 satisfies the differential equation (8.4.11)

(4)

-- 1

where w is an appropriate boundary uniformizer on C. Thus, C is an analytic curve in terms of uniformizers for N. We can again give a geometric interpretation for the curve C analogous to our result in the last section. We observe that the function aG(p, PO) lap is determined at each point of 91 up to a ±-sign under arbitrary analytic continuation. If we continue aG/8p from the same initial value along two different paths to two points pi and Ps lying opposite each other on the curve C, we will arrive

at different determinations of the sign of aG/ap. In fact, by its definition, Green's function has always a positive derivative in the direction of the interior normal and since at pi and p3 the sense of the interior normals is opposite, aG/ap must have opposite sign. Thus, we see that aG/ap is two-valued on % and has the curve C as its branch line. We now define a two-sheeted covering a of SJR; we consider the universal covering surface t of 9R and identify all points of W lying over the same point p e 91 if the path connecting them cuts C an even number of times. All points over p e 91 which can be connected with p by a curve which cuts C an odd number of times are also identified and form the point P associated with p. We assumed 91 to have a boundary; we know, therefore, that 9t

§ 8.4]

CANONICAL CROSS-CUTS ON A SURFACE t

389

has a boundary, too, and possesses a Green's function G (p, q). Consider (8.4.12)

G1(p, po) = G(p, Po) -G(p, po)

This function is harmonic on 0 and has a positive logarithmic pole at the point PO and a negative logarithmic point at the associated point o. It vanishes on the boundary of 0. Clearly, the Green's function G(p, PO) of the extremum domain ? = fft - C has all the same properties on RR and must, consequently, coincide with Gl (p, po) .

Thus, we have proved: THEOREM 8.4.1. Let Jt be a surface with boundary and T a closed curve on 91 which does not bound on R. Consider the two-sheeted covering of 9't which is obtained from the universal covering surface W of OR

by identifying all points of 2t which lie over the same Point p e R and can be connected by a curve which cuts r an even number of times. Let P be the other point of R lying over the Point P. Let (P, q) be the Green's function of M. Then the curve C'

(8.4.13)

G (p, po) = G (P, 'P ',O)

is homotopic to 1' and is the curve C on OR which gives to 9 -- C the largest capacity constant at po.

It is easy to formulate the corresponding result for the case that 91 does not have a boundary. Instead of G (p, Po) we shall have to use the Green's function f (p, ; po, o) of the closed surface §t and the extremum cutting C is just the zero line of this Abelian integral. We may generalize our problem by prescribing curve systems 1', on 9 and ask for hornotopic systems which maximize the capacity constant at some point po a 91. The same reasoning can be applied and analogous results are found. We deal next with the following extremum problem. Consider a surface 9t and fix in it two points Po and pl. Let w(p) be a fixed local uniformizer at P. and v(p) a given local uniformizer at pi. The problem is to decompose 91 by a system of crosscuts into two domains 99o and 9R such that Po a 2Jto, pi e 11 and such that the sum of the capacity constant go(po) of Wo and of the capacity


390

(CHAP. VIII

constant g1t51) of Yi be as large as possible. One easily verifies that the question is significant, that is, that there exists at least one + ti such that the value of go(po) + gi(Pi) decomposition R = is the largest possible. As before, we will characterize the extremum domains D 20 and by varying Si by means of a cell attachment and comparing the sums of the capacity constants before and after the variation. We select a curve yo in 3to and a curve yi in ^)1i and prescribe on y, a reciprocal differential R,(15) such that 93

(8.4.14)

f Ri(P) .2(p)dv = 0

-Z° (p) dw +

V,

70

for every finite quadratic differential on R. We vary R. by a simultaneous cell attachment at yo and at yi by means of the reciprocal differentials R0(p) and Ri(p), respectively. The orthogonality condition (8.4.14) guarantees that the deformed surface R* will be of the same conformal type as R and by Theorem 7.13.1 we can map back from JR* onto R. We consider first the case that 9t has a boundary. We have

w* = w + E Z f R.052)N(w, t2) dw2 2i o .

7r

(8.4.15)

2ni,f.r ,o( 2N(w, fi2)dv2 + o(e). yr

This mapping affects )to and U11 and may be interpreted as a deformation by cell attachment of 9)t, along the curve y, using the reci-

procal differential R,(i). We want to keep the points P. and p, fixed under the shift (8.4.15) and also keep dw* f dw = 1, dv*fdv == 1, so that we may use w* or w as local parameter in computing go (PO)

and interchange v* with v when determining g1(p1). We require, therefore, the following conditions: (8.4.18) E f Rr(p2)N(.p , p2)dw2 _ ' rao

nr(

V-0

yr

yr

i =0,1,

§ 8 4)

CANONICAL CROSS-CUTS ON A SURFACE Ot

and

391

1('

(8.4.16)' E J R,(p2)N'(ps,p2)dw2 E 1 Pti(fi2)N'(P,, i2)dr",v2, 1=0,1. ,-0 =,,.o Y,

V,

Let G{(p, q) be the Green's function of the domain ,; using the variational formula (7.8.16) for the Green's functions we find the following new values for the capacity constants: (

(8.4.17) g(p) = g,(p) -Re j

l

.

fR(2)('' P'

2

dws I + o(e),

VS,

f or

v = 0, 1. Because of the assumed extremum property of the

domains ?'ta and (2R1 we deduce as before Re{

(8.4.18)

t

(dP2PV))2d} _Q 2ni, J R,(P2) Gy(

Y,

for every pair of reciprocal differentials R,(P) which satisfy the conditions (8.4.14), (8.4.16) and (8.4.18)'. The equation (8.4.18) leads in the usual way to equations for the quadratic differentials [aG,(p, p,)lap]2 on the surfaces 9R,. Let x0, x, and A0, A, be four complex constants and define the quadratic differential on RJR:

-0

{

x+N(p,, p) + x,(N(p,, ))- + ).N'(p,, i) + 3.(N'(p ))

(8.4.19)

+ 2(p) =gy(p),

where .9 (p) is a finite quadratic differential on M. V,(p) has a double pole at P. and at p1. From (8.4.18) and the side conditions (8.4.14), (8.4.16) and (8.4.16)' it follows that we can determine four constants

x,, A, and a finite quadratic differential .2(p) on J such that (8.4.20)

[aG(P p,)]2 =

.(p)

for p E

We deduce from (8.4.20) that the boundary curves of the domains fit, are analytic curves on 91 satisfying the differential equation


392

[CHAP. VIII

(4)2 ( 8.4.21)

= - 1.

It can easily be shown that there are no points on R whick are exterior to 1010 + 9 XI. Thus, the differential equation (8.4.21) determines a set of analytic arcs which cut the domain R into two pieces 9719 and V1 such that & = T1p + t1. Let us consider now a boundary arc y of DTto; there are two possibilities regarding y. It may

separated from I1 or it may be a division line between two subdomains of to. In other words, the edges of y may be boundaries of 9119 and 9Jt, in the one case, or may both be boundary arcs of 9)to in the second. We want to show that the extremum property of 'fR

excludes the second possibility. Let us suppose, in fact, that

both edges of y are boundaries of l ; let us remove a subarc y2 C y and identify points on both edges of yl. In this way, Uto willfioome a larger domain Oto* which still contains the point Po while 91 will not be affected by the removal of an interior slit of 97to. Let Go (p, po)

be the Green's function belonging to W. The difference function Go (P, PO) is regular harmonic in V. and non-negative Go (5, PO) on its boundary. But then it is positive inside'rd by the minimum principle and, in particular, its value at Po will be positive. Thus, we have:

-

(8.4.22)

go (po) > $o (Pd)

which shows that OJto has a bigger capacity constant at Po than 9)20.

It leads, therefore, to a value go ('0) + g1(p1) > 90 (PO) + gl(P1)

in contradiction to the assumed extremum property of

9,

0, 9711.

Thus. we have shown that all boundary arcs of to are also boundary arcs of 9711.

Sincel''(p) is single-valued on the whole surface 9R we conclude from (8.4.20) that the differentials aG,f ap coincide on the boundary

between V. and Ttl, except possibly up to a ±-sign. Since each Green s function has a positive derivative in the direction of the interior normal of its corresponding domain, we easily conclude (8.4.23)

Go(' Po) rti

ap

on boundary of T2 ,

9?21.

1 8.41

CANONICAL CROSS-CUTS ON A SURFACE tit

393

Since, moreover, both Green's functions vanish on the common boundary, we recognize that - G1(p, p1) is the analytic continuation of G0(p, po) across the boundary. Thus, G.(p, po) and - G1(p, p1) form together a harmonic function on RJR which is regular everywhere

except at the points po and p1 where it has logarithmic poles of opposite sign. Let 9(p, q) be the Green's function of RJR; then we have obviously: (8.4.24)

9(p, 16 0) -

pa) in 9(p , p ) = { - Go(p, G1(p, p1) in

1to

1

1

Thus, we have expressed the harmonic function composed of Go and - G1 in terms of the Green's function of R. The division line between Do and P1 has the simple equation (8.4.25)

9F(p, PO) = W(p, p1)

which is the integral of the differential equation (8.4.21). We are also able to calculate the maximum value of go(p0)+g1(p1)

in terms of the Green's function T(p, q) of R. Let q(po) and

(p1)

be the capacity constants of 91 with respect to po and p1 for the local uniformizers w(p) at po and v(p) at p1. We then derive from (8.4.2' ) (8.4.26)

g0(po) = S(po) - T (po, P1) g1(p1) = (P1) - 9F (po, P1)

Hence, we have proved: THEOREM 8.4.2. Let 9R be a surface with boundary and W (P, q) its

Green's function. Given two points P. and p1 on T with fixed local uni f ormizers w and v, let g (po) and g(p1) be the capacity constants of SJR at po and at p1. If SJR is subdivided into any two domains Y't1 such that po a UL and p1 a 9J21, we have the inequality: (8.4.27)

go(po) + g1(p1) S S(po) + a(p1) -

'to and

p1)

for the capacity constants go(po) of ILJ at P. and g1(p1) of P1 at p1. Equality in (8.4.27) holds for the extremum decomposition of SJR by means of the cut system

W(p, p1) = W(p, po) which divides 91 into the two extremism domains Tto and

t1.

Until now we have worked under the assumption that OR has a

394

[CHAP. VIII

THE VA.RIATIONAL METHOD

boundary. We can also carry through the same reasoning in the case of a closed surface R. We deri-ve in exactly the same way the fact that G0(p, PO) is the analytic continuation of - G1(p, p1) across the common boundary of the domains Wo and t1. But since a closed Riemanr. surface does not possess a Green's function of the type V(p, q), we have to change our reasoning from this point on. We remark that G0(p, po) and. - G1(p, p1) may be considered as the single-valued real part of an Abelian integral of the third kind on R. Thus:

in V.

- { - Gi(p, p1) GO (P, P0)

(8.4.28)

Re {S29ov1(p)}

in

't1.

The integral of the third kind Q,0,1 (p) is determined only up to an additive constant. We may take any determination of an integral Sl9op1(p) and consider the lines on OR where a = const. (8.4.29) Re {sa,,,l(p)} = a, and MI1 and their

These lines will decompose % into two domains

corresponding Green's functions will be (8.4.30)

in X20,

Go(p, po) = Re {Qgogl(p)} -a

G1(p, p1) = - Re

a

in X21.

We recognize that the value g0(po) + gl(p1) is independent of the choice of a, so every decomposition (8.4.29) leads to an extremum domain. While the extremum decomposition is uniquely determined in the case of a surface with boundary, we have an infinity of extremum domains in the case of a closed surface. In order to illustrate our result let us specialize to the case that 8t is the sphere. We introduce w as universal uniformizer on R and find the extremum domains by considering the lines: (8.4.31)

w-W Re {Q,,,l (p) } = log 1W-W81

= a.

For each fixed a the plane is decomposed into two circular domains; the one which contains wo has the Green's function (8.4.30)'

Go(w, wo)

log

w-wi -a w-wo

§ 8.4]

CANONICAL CROSS-CUTS ON A SURFACE 81

395

and the other, which contains w1, has the Green's function (8.4.30)"

w - w0 + a. w - wl

G1(w, w1) = log

Thus, we have proved: THEOREM 8.4.3. Let V. and X21 be two plane domains which have no common points. If wo a O2o and wi a SJRz, we have the following inequality between the capacity constants go (wo) of of J2,, at w1: (8.4.32)

to at wo and gi (w1)

go(wo) + gi(wi) S 2 log I wo - wl I.

The inequality (8.4.32) contains as a special case a theorem of the theory of schlicht functions in the unit circle which is due to Lavrentieff. Consider two schlicht meromorphic functions in the unit

circle with the series developments near the origin: 0

(8.4.33)

ae

w = g(z) = E b, e.

w = A z) = E a,z',

r-0 -0 Suppose, moreover, that /(z') g(z") for any points z' and z" in the unit circle, that is, the images of the unit circle by means of /(z) and g(z) considered together are still schlicht over the w-plane. There arises the question of estimates for the coefficients of these schlicht function sets. Lavrentieff showed that the product I ai b= I can be estimated if the values ao and be are given. We will derive his result from inequality (8.4.32). In fact, let go be the image domain of the unit circle by means of f (z) and let 1tl be the image by g(z). It is easily seen that Go (w, ao) = log I z = log I w (8.4.34)

ao

((

+ log i ai 1 + 0(1 w - ao I )

and

Gi(w,bi) =log (8.4.34)'

IzII

=log(w

I

b

I

o

+logI biI +0(1 w-

Applying (8.4.32) with wo = ao, wl = bo, we find (8.4.35)

log I al bl 1 S 2 log I ao - bo I

,

396

THE VARIATIQNAL METHOD

[Cs+P. VIII

that is, f a, bl 1 S 1 ao - bo' $.

(8.4.35)'

One might continue the generalization of the concept of a schlicht function in a given domain Z by considering sets of schlicht functions

f,(z) in Z such that no two image domains of Z overlap. In the theory of these "schlicht function vectors" the variational method can be easily applied. 8.5. EXTREMUM PROBLEMS IN THE CONFORMAL MAPPING OF PLANE DOMAINS

A particularly simple and interesting case of the general imbedding theory of a domain 91 into a domain R arises if 91 is a domain in the

complex z-plane with finite connectivity and SR is the complex w-plane. The theory of imbedding R into OR becomes the classical theory of the schlicht conformal mappings of %. Our general variational

method enables us to solve extremism problems connected with such schlicht mappings of the domain R. It is convenient to introduce a few general concepts which will allow us to formulate a rather general result in extremum problems of the above nature. Let 0[f] be a real valued functional defined for all functions f (z) analytic in %. We assume that {OY + ag] - 44/]} = Re {L,[g]},

(8.5.1)

e real,

41-1-14 8

exists for all analytic functions g(z) in 9 and that L,[g] is a complex

linear functional of g(z). L1[g] depends in general on 1(z); Re {L,[,-]} is called the functional derivative of o[f] for the argument function f (z). Suppose that we know that some functional 0[f] is bounded for the class df all schlicht regular functions in T and attains its maximum for at least one function of this class. We can then characterize the extremum functions by variational considerations. Suppose that the extremum function 1(z) maps the domain 91 into a subdomain 931 of R. We choose a Jordan curve y in 91 and determine a reciprocal differential r(z) which satisfies the orthogonality condition

§ 8.5)

EXTRFMUM PROBLEMS

397

fr(z)Q(z)dz = 0

(8.5.2)

Y

for all finite quadratic differentials of R. Let I' be the image in 91 of the Jordan curve 7,; we define on it the reciprocal differential (8.5.3)

R(w) = r(z) dw = r(z)f'(z)

and perform a variation of % and ;JR by a cell attachment along y and I' as described in Section 7.14. Since t does not possess any finite quadratic differentials, the conditions (8.5.2) guarantee that the conformal types of t and N are preserved under this deformation. If we map the deformed domains conformaily back into the original domains, we obtain a new one-to-one relation between z and w which

leads to a new schlichi function in %. By (7.14.10), we have (8.5.4)

/ ( z ) = f(z) + eh(z)f'(z) -eH[f(z)] + o(e)

with (8.5.5)

h(z) = ;1% f r(t)n(z, t)dt- 2 f (r{t)) .n(x, I)df Y

Y

and

(8.5.6)

H(w) =

2aci J

r(t) f'(t)2N(w, f (t))dt.

Y

The variation kernel N(q, p) for the sphere was given in (7.10.2) and an arbitrary fixed point qa occurred in the formula. Since in most

problems of conformal mapping the point at infinity plays a distinguished role in any case, we shall choose w(qo) = oo; then (8.5.7)

N(w, co) =

If we calculate the value of 0[f we obtain (8,5.8)

w-w

.

by means of (8.5.1) and (8.5.4),

0[/A] = ch[f,] + e Re {Lf [hf'-H(f)]} + o(8).

Because of the extremum property of f (z) and the arbitrariness in the

398


[CHAP. VIII

choice of the small real quantity e, we are led to (8.5.9) Re {Lf[hf' - H (f)]} = 0, for every choice of r(t), which agrees with the condition (8.5.2). Because of the linear character of L f) we may put (8.5.9) into the form (8.5.10)

Re j 2-i fr(t) [A(t) + B(t)-f'(t)2C(f (t))]dt } =

0

with (8.5.11)

A(t) = Lf[f'(z)n(z, t)], B(t) = (Lf[f'(z)n(z, l)])-

and (8.5.12)

C(w) = Lf

We observe that A (t), B(t) and C(w) depend analytically on their arguments and that we have, on the boundary of T, (8.5.13) A(t)dt2 = (B(t)dt2)Thus, A (t) + B (t) is a quadratic differential of N. The usual considerations allow us to deduce from (8.5.10) and (8.5.2) that (8.5.14) A(t) + B(t) - f'(t)2C(f(t)) = Q(t) where Q(t) is a finite quadratic differential of SJl. Thus, (8.5.14) may

be put into the simple form (8.5.15)

C(w)dw2 = Y(t)d12

where Y(t) is a quadratic differential of W. If we use boundary uniformizers on T, we can deduce that the boundary of the image domain D1 is composed of analytic arcs each of which satisfies the differential equation (8.5.16)

(dw)2 C(w)

0.

We can show by the same reasoning which was applied in Section

8.2 that the extremum domain D1 has no exterior points on W. After having shown by (8.5.16) that 9 has an analytic boundary,

EXTREMUM PROBLEMS

§ 8.61

399

we may apply to it a boundary variation of the Julia type described in Section 7.15. Using the notation of this section and formula (8.5.1). we obtain a function fe(z) such that, using a uniformizer t on the boundary y, we have:

(8.5.17) O[f '] = O[f] - Re{

fL,[/'(z)n(z, t)]v(t)dt

1

J

YY

By (8.5.11) this can be written as - 0 [ / ] = - Re

J

[A (t) + B(t)]v(t)dt t + o(e),

r and since v(t) is orthogonal to all finite quadratic differentials of l1:

(8.5.17)" P[f°]-O[f] =-Re{2nJ(' C(w) (f)2v(t)dt +o(e)Since we are quite free to vary the boundary of 92 with respect to the interior normal, that is to choose v(t) positive, so long as the orthogonality to all finite quadratic differentials is satisfied, we see as in Section 8.2 that a

(8.5.18)

C (w)

dt) : 0 on each boundary curve of Dl. l

We thus arrive at the theorem: THEOREM 8.5.1. Let 91 be a plane domain of finite connectivity; let fi U] ] be a real valued functional with the functional derivative Re {L,[9]} .

The schlicht regular function 1(z) which maximizes O [ f ] maps the domain N onto a slit domain in the w-plane whose boundary curves are composed of analytic arcs satisfying the differential equation (8.5.19)

Ly

WI(t)2

= 1,

w = w (f).

This theorem shows that in the case that R is the sphere the nature of the original domain % is rather unessential for the nature of the boundary slits of extremum domains. The differential equation (8.5.19) depends only on the nature of the functional encountered. We may also interpret the result of Theorem 8.5.1 in the following

400


(CHAP. VIII

form. Let O[f] be the functional to be maximized. Introduce into it the function E

(8.5.20)

fi(z) = f(z) + f (z) - w

and calculate

0[fl] -0[f] = Re {e C(w)} + o(e). The extremum curves will then satisfy the differential equation C(w)w'2 = 1.

This result has been obtained previously by a quite different type of boundary variation in which it was shown that there actually exist schlicht functions in 91 of the form (8.5.21) fl(z) = J (z) +

f(z)8

w

+ o(e), w = boundary point of

l)

which may be used as comparison functions [8a]. This method requires,

however, a very penetrating study into the possible singularities of the boundary curves under conformal mapping. Let us now illustrate the possibilities in applying Theorem 8.5.1.

In order to be sure that a functional has a maximum within the family of all schlicht functions in SJl, we can frequently use the fact that important subclasses of schlicht functions form normal families. Let, for example, z = 0 lie in )1; then it is well known that all functions

f (z) which are regular and schlicht in W and have at z = 0 the normalization /(0) = 0, f'(0) = 1 form a normal family S. Thus, given any bounded functional on S satisfying (8.5.1), we can assert that there exists at least one function in S for which the functional

attains its maximum value. But from this fact we can show the existence of a wide class of functionals which necessarily possess maxima in the family F of all regular schlicht functions in 1. In fact, let 0[f] be a functional which is bounded on S and attains, therefore, its maximum in this family. If 1(z) is an arbitrary function

of the class F, the transformation (8.5.22)

fo(z) =

f(z)

f(0) (0)

will transform it into an element of S. Thus, every functional fi[f]

EXTREMUM PROBLEMS

§ 8.5)

401

bounded on S gives rise to a functional (8.5.23)

W[f] = 1P[fo]

which is bounded on F and attains its maximum there. We can readily calculate the functional derivative of W[f] if the functional derivative of O[f] is known. An easy calculation shows that (8.5.23)'

![f+eg]= 1'[f]±e Re L,

°Lf (0)

(go(z)to(z))I

1+0(s)

where Re {L f.} is the functional derivative of O [ f ] at the value f0(z) of the argument function. Because of the close relation between the classes F and S most

extremum problems for regular schlicht functions in % are formulated for the class S. However, it must be observed that our variational treatment does not preserve the class S. Suppose that a function f (z) of the class S maximizes a functional 0[f], / e S. Since fo(z) = 1(z) for all elements of S, we may equally well say that &) maximizes Y'[f], f e F. Hence, by Theorem 8.5.1, the boun-

dary curves of the extremum image 9 satisfy the differential equation (8.5.24)

L,

f(z)a [1(z)

-w

w"

=

w2

1,

w = w(t),

as is easily seen by inserting the functional derivative of W[f] into (8.5.19). We summarize: THEOREM 8.5.2. Let 91 be a plane domain of finite connectivity; let S be the class of all regular schlicht functions f (z) in 9 which have at the point z = 0 e 9 the normalization f (O) = 0, f'(0) = 1. Let O[f] be a real functional satisfying (8.5.1) and bounded on S. Then O [ f ] attains its maximum in S for a function t (z) which maps % on a slit domain in the w-Mane whose boundary slits satisfy the differential equation (8.6.24)

Ly

[f(W1=

w = w(t). Let us consider, for example, the question of characterizing among all functions f (z) e S that which attains at a given fixed point zo e 91 1,

402


[CHAP. VIII

the largest possible value log i f (ze) 1. Since /(0) is prescribed to be zero, our problem is to determine for which function f(z) the distance I f (ze) -- f (0) ( is maximal; the problem is therefore called the distortion problem of conformal mapping Since here

'h[f]=log It f(zo)j, we obtain from (8-5. 3)

Lf[g] =

f(zo)

Hence, by (8.6.24) the boundary curves of the extrenilim domain satisfy the differential equation w 1

-f(zo)

which can easily be integrated: w(t) = f (xe)

4cet

( -+c e'}$

The value of the constant of integration c will depend on the boundary component of T considered and will, in general, be complex. There must be one boundary arc of 1, however, which runs up to infinity since fll cannot contain this point. On this are, c must necessarily be real and hence this particular boundary slit is a straight

line lying on the ray from the origin w = 0 to the point - f (za). The further study of the extremum function t (z) and the determination of the numerical value of log I f (ze) I belongs to the theory

of schlicht functions and will not be carried out here. We wished only to show the type of information which is available in that theory

by means of the variational method. As another example, we consider the series developments of all functions t(z) e S at the origin: (8.5.26)

t (Z) = z + E a,,z'. -2

We ask for, the maximum value of the func4ona1'.[f] - Re

Let

1 $1]

EXTREMUM PROBLEMS

(8.5.27)

f(1)

t

=z f(z)

403

+ 2' b, (--) z';

the coefficients 4-1) can easily be computed and shown to be poly-

nomials of degree v - 1 in the argument t with coefficients which depend on 1(z). We clearly have (8.5.28)

f(z)-w_/(z)-

f(1)

1 ---/(z)

a,-b,\l.1x

W

From Theorem 8.5.2 we easily derive that the function (8.5.28) which has the maximal value of Re a, maps % upon a slit domain in the w-plane whose boundaries satisfy the differential equation: 12

(8.5.29)

This differential equation may be used as a starting point for a more penetrating study of the coefficient problem for schlicht functions in multiply-connected domains. It should be observed in this connection that if i is a boundary point of fit, the auxiliary function (8.5.27) also belongs to the class S. Hence, the extremum property of f (z) leads to the inequality: (8.5.30)

Re j b, (.!-) } S Re W 1

for all boundary points of fil. This remark is very helpful in the study of the extremum domain fil; in fact, the points on the wplane for which (8.5.31)

ab

(-w )

are the critical points of arcs satisfying the differental equation (8.5.29). Let us ask, therefore, under what conditions a critical point wo of this type could lie on the boundary of the extremum domain fit. By our previous remark, the function


404

CHAP. VIII

f (z)

(8.6.32)

t* (Z) =

1- 1wo f(x)

would also belong to the class S and possess by (8.5.27) the same n-th coefficient as the extremum function f (z). But then /* (x) would itself be an extremum function and map W on an extremum domain 2R* whose boundary slits satisfy

(z.v*')2[*

(8.5.33)

1.

ct"-bn

1 )J In order to evaluate the meaning of the simultaneous equations (8.5.29) and (8.5.33), we must investigate the relations between w

and w*, b

1and b * (1) w*

.

The connection between f* (z) and

t (z) is most easiiy expressed in the form 1

1

1

f*(z)

f(z)

wo

(8.5.32),

and correspondingly

1

1

1

(8.5.32)"

w*

-W --W01

Let w1 be a boundary point of 9971; by (8.5.32)" it corresponds to a boundary point of 5712*: 1

(8.5.34)

W1*

1

1

wl

wo

To calculate bn (-1j), we consider wl

x

1- 1 * z

1

f* (z)

1

1

wi J

1

L f (x) ~ w1J

Wi

f(z) 1

W1

and compute its n-th coefficient. Hence, we have: (8.5.38)

wl

w1

Thus, the differential equation (8.5.33) can be put into the form:

EXTREMUM PROBLEMS

§ 8.51 to

405

8 w2(wow"

(8.5.37)

w (t)

(W)]>0, is the same representation of the boundary curves as that

used in (8.5.29). Dividing (8.5.37) by (8.5.29), we obtain on all boundary curves of ,T1 the relation (8.5.32)

W"'

w"

(wo - w)

2 > 0, w = w(t).

Thus, if there were a critical point wo on the boundary of V1, all boundary slits would be rectilinear and lying on the line through w = 0 and the point wo. We have thus proved that the boundary curves of the extremum domains in the coefficient problem are regular analytic arcs without any critical points. The further discussion of the equation (8.5.29) was possible because of the existence of a finite transformation of the class S into itself, given by (8.5.27) with t on the boundary of 31. While the variational method permits a comparison of the extremum function only with its immediate neighbors in the class, a finite transformation formula leads us to distant elements of the class and contains additional information. It is often possible to amplify the information gained by variation by using a simple transformation of the class of functions considered, analogously to the above treatment. Another class of schlicht functions which is normal and of great interest in the theory of conformal mapping is defined as follows. Assume that the point z = co lies in T't; consider the class of all schliicht functions in 9 which have at infinity the series development (8.5.33)

/(z) = z -}- ao -}- - -}- .. .

and are regular analytic elsewhere in W. These functions form the class F of schlicht functions in fit. Since every variation (8.5.4) transforms a function of the class F into another function of the same class, we may apply the reasoning of this section immediately to extremum problems for the class F. We obtain:


406

[Cawr. VIII

THEOREM 8.5.3. Let 92 be a plane domain of finite connectivity containing the point at infinity; let F be the class of all schlicht functions in 91 which are regular in 92 except for a pole of type (8.5.33) at infinity.

If 0 U] ] is a real functional satisfying (8.5.1) and bounded on F, then it attains its maximum in F for a function f (z) which maps 92 on a slit domain in the w-plane whose boundary slits satisfy the differential equation (8.5.34)

Lf

f(z)-wJ w" = 1, 1

w = w(t).

Let us illustrate this theorem by some applications. Let us ask for the maximum value of the functional Re {e-2 a1} where a is a fixed real constant and a1 is the coefficient of z-1 in the development (8.5.33). Applying (8.5.34), we see that the extremum function 1(z) maps 92 onto a slit domain over the w-plane with the differential equation for the boundary slits: se-2!a = 1. (8.5.35) This differential equation can be integrated and leads to (8.5.35)' w(t) = c + e{"t; that is, all boundary slits of the extremum domain 9)t are rectilinear segments with the direction e4".

We see here a new aspect of the variational method, namely the possibility of giving existence proofs for certain canonical mappings.

We may put an extremum problem for which the existence of an extremum function is ensured; we may then characterize the extremum function by variation and obtain in this way the existence of a function of the class with particular properties. For example, the preceding reasoning leads to the existence theorem: There exists a function / (z) of the class F which maps the domain 9

upon the w-plane slit along rectilinear segments in any prescribed direction.

Let zo be a given fixed point in 9't; we ask for the maximum value

of the functional Re {e-2I log f'(z0)} for all functions /(A) of the class F. Using (8.5.34), we see that the extremum function 1(z) maps

92 upon a slit domain in the w-plane and each boundary slit w (t) satisfies the differential equation:

EXTREMUM PROBLEMS

§ 8.5]

(8.5.36)

[f(zo)

H

-

w

407

w] 2

e-4ia = _ 1.

Integrating this equation we obtain the following parametric representation of the boundary slits: (8.5.37) log [w - f (zo)] = c -t- ie`"t, w = w(t), where the complex constant of integration c is, in general, different for different boundary slits. These curves are spirals with prescribed inclination around the point f (zo). For a = 0, these spirals degenerate into circular arcs with center f (zo) and for a = n/2 they become rectilinear segments pointing to the common center f (zo). We have thus proved the existence of a large class of canonical domains by raising an appropriate extremum question. REFERENCES 1. R. ConR.NT, Dirichlet's Principle, conformal mapping, and minimal surfaces, Interscience, New York, 1950. 2. G. GOLUSIN, (a) "Interior problems of the theory of schlicht functions," Uspehhi

Matem. Nauk, 6 (1936) 26-89. Translated by T. C. Doyle, A. C. Schaeffer and D. C. Spencer for Office of Naval Research, Navy Department, Washington, D. C., 1947. (b) Some problems in the theory of schlicht functions, Trudy Mat. Inst. Steklov, Moscow, Leningrad, 1949 (Russian). (c) "Method of variations in the theory of conform representation I, II, III," Mat. Sbornil,

19 (1946), 203-236; 21 (1947). 83-117; 21 (1947), 119-132 (Russian,

English summary). S. G. JuLL+, LeGons sur la reprfsentation conforms des aires multiplement connexes, Gauthiers-Villars, Paris, 1934.

4. P. MoxTEL, Lecons sur les fonctions univalentes on multivalentes, GauthierVillars, Paris, 1933. 5. A. C. SCaAzFFE}t and D. C. SPENCER, Coefficient regions for schlicht functions,

Colloquium Publications, Vol. 35, Amer. Math. Soc., New York, 1950. 6. A. C. ScHAarFER, M. ScHXFFER, and D. C. SPENCER, "The coefficient regions

of schlicht functions," Duke Math. Jour. 16 (1949), 493-527. 7. M. SCEirFER and D. C. SPENCER, "The coefficient problem for multiply-connected

domains," Annals of Math., 52 (1950), 362-402. S. M. SCa1FFRR, (a) "A method of variation within the family of simple functions,"

Proc. London Math. Soc. (2), 44 (1938), 432--449. (b) "On the coefficients of simple functions," Proc. London Math. Soc. (2), 44 (1938). 450-452. 9. D. C. SPENCER, "Some problems in conformal mapping," Bull. Amer. Math Soc., 53 (1947), 417-439.

9. Remarks on Generalization to Higher Dimensional Kahler Manifolds 9.1. KAHLER MANIFOLDS

To place the subjects discussed in the preceding chapters in a wider setting, we describe in this chapter certain special properties of Kahler manifolds of arbitrary complex dimension k, Since, as we shall presently show, a Riemann surface can always be made into a 1-dimensional Kahler manifold by the introduction of an appropriate metric, the theory of Riemann surfaces may be regarded as the special case k = I of the general theory of k-dimensional Kahler manifolds, and it is illuminating to see how some aspects of Riernann

surfaces may be generalized and others not. In this chapter proofs of some statements are omitted, references to literature being given instead. By omitting a few details we are able to give a general description of some aspects of KXhler manifolds

without making the chapter too lengthy. For the sake of completeness, we bring together in this section various known properties of complex manifolds. A complex (analytic) manifold Mk of complex dimension k is a space to each point P of which there is associated a neighborhood N(P) which is mapped topologically onto a subdomain of the Euclidean space of the complex variables xi, - , z", If q £ N(P), the coordinates

of q will be denoted by z'(q), i = 1, 2, , k. Wherever two neighborhoods intersect, the coordinates are connected by a pseudo-con-

formal mapping. We consider only those manifolds which are paracompact and Hausdorff. Following (3] we introduce a conjugate manifold Mk which is a iomeomorphic image of Mk in which the point p of Mk corresponds to the point of Vk and the neighborhood N(p) to NA(_p). Let Latin ndices run from 1 to 2k, and let

i+k (mod2k). .f q c N(T), we define [408]

§ 9.13

EAHLER MANIFOLDS

(9.1.2)

409

zT(q) = (z{(q))

where (z)- denotes the complex conjugate of the quantity z. By means of (9.1.2) the neighborhood N(T) is mapped onto a domain in the space of the variables z? = z9, i = 1, 2, , k. Now consider the product manifold Mk x M`k whose points are the ordered pairs (p, q), and let

1, 2,,

zt(p),

z'(q)= (x?(q))

i = k-(-, 1,

i

1, 2, ..., 2k.

Then (9.1.4)

x'(p, q) = (z?(q,

p))-,

,

..., 2k.

The product manifold Mk X Mk is covered by the coordinates Z' (P, q), i = 1, 2, , 2k. Introduce coordinates xt(p, q) by the formulas zt

2

x+ 1--V-1x? 2

(9.1.5)

x'- 1-- 2

z+

I +V 2

,

z?

i= 1, Then (9.1.6)

xt(p, q) = (x{ (q, p))-,

i = 1, 2,

2k.

On the diagonal manifold Dk of Mk X Mk where p = q, we have (9.1.7)

x' = z' (p, p) = (x;)-, x{ = x{(p, T) = (xi)-.

Thus D" is covered either by the self-conjugate coordinates z', x? = z', i = 1, 2, - , 2k, or by the real coordinates x'. We shall be concerned mainly with the diagonal space Dk. A tensor A whose components are real when they are expressed in the real coordinates x' will be called a real tensor. A real tensor A when expressed in self-conjugate coordinates x' satisfies (9.1.8)

Aj...!'",-11 = (A7 ..?

Let unbarred Greek indices run from I to k, and write (9.1.8)' a = oc + k, 7='%.

410

HIGHER DIMENSIONAL KAHLER MANIFOLDS

[CHAP. IX

Then (9.1.8) can also be written (9.1.9)

Aa7y7ur...A =

(Aar/t..-,u-;..

-

The tensors properly associated with the original manifold Mk are

the complex analytic ones whose indices range over values from 1 to k. On Dk there is a "quadrantal versor" which is a real tensor h,' satisfying h { h f a-

(9.1.10)

Q,

In self-conjugate coordinates z' this tensor has the components

/, 1Si=jSk,

(9.1.11)

k + 1 S i = j 5 2k,

hff (z)

0,

i* j,

or, in the real coordinates x;,

1S (9.1.11)'

hif (x) _ -- 1,

i

0, iz7

k + 1 S i S 2k,

The values (9.1.11) and (9.1.11)' are pseudo-conformal invariants. Given a vector 97{, let (9.1.12)

(I4'){ =

be the identity transformation, and let (9.1.13) (hq )i = Vri be rotation through a "quadrant". Given real numbers a and b, the operation aI + bh applied to vectors corresponds to complex multiplication in which the reality of the vector is preserved. We have

(aI + bh) (cI + dh) = (ac - bd)I + (ad + bc)h. In other words, the field obtained from the real vectors by adjoining the ope ator h is isomorphic to the complex number field. Now suppose that D" carries a Kahler metric g{, of class C°° . A Kahler metric is a Riemannian metric which satisfies the following two conditions: (9.1.14)

§ 9.1]

KAHLER MANIFOLDS

= g,. h{" hill,

(a)

gta

(b)

D,(hif 9's)

Here (9.1.15)

D99'e =

411

gyp,

8z'

denotes covariant differentiation,

=

he/D,9o5.

qJr. ji ip qjp

being the coefficients of

affine connection. Condition (a) states that the vectors y and (hp), have the same length, while (b) states that the operators h and D commute: Dh = hD. Let. (9.1.16)

hii _. gft h;i.

Multiplying both sides of (a) by h.5 and summing on j from 1 to 2k, we obtain (9.1.17)

h,. = gt5 h,f = g,v ht" k1 h i = - g,, hi' = -- he,.. Thus hit is skew-symmetric, and hence by (9.1.16) hta h,{ hvi = go hz= h,' hv! = - g,, h., = - hq, =

that is, (9.1.18)

h,v = hil h,i V.

In terms of self-conjugate coordinates z{, the formula (9.1.18) shows by (9.1.11) that any non-zero component of h,v is necessarily of the

form h,7 or h-u. In other words, h,v = 0 unless p and q are indices of opposite parity with respect to conjugation. A metric satisfying (a) is said to be Hermitian. Condition (b) gives $ qj} (9.1.19) h,4 {

}

=

h{I

Taking q = a, I _ and using self-conjugate coordinates, we obtain

-V-1{P 9},

1,2,...,2k.

412


(CHAP. IX

Hence (9.1.19)'

{jc.} _ 6 a9

J)_

j = 1, 2, ..., 2k,

= = 0,

and therefore the only possible non-zero components of the coefficients of affine connection are those with all three indices of the same parity. Since p

{

ij

_

1

g,Q

+

agt, L 8xf

2

8x{

we conclude that

NO = agy or

(9.1.20)

zy

at

agia

-

ago 8x11 T

` at at

aha3

ahra

A 1-form q, on D'' is a differential form of the first degree v = go,dz',

where q'; are the components of a covariant vector, the summation convention being used. A p-form, or exterior differential form of degree p, p > 1, is a sum of exterior products of 1-forms. Exterior

multiplication, represented by the symbol A, is associative, distributive, and satisfies (see [11]) dz' A dzf = - dz' A dx{, dz{Adz{ = 0, a A dz' = dz' A a = adz', dzt A adz' = adx' A dzi,

where a denotes a scalar. Exterior multiplication has already been defined in Chapter 1 where, however, the symbol A was omitted

for simplicity. In this chapter we adopt the notation current in tensor calculus. A p-form q' may be written in the form (

9121

9,

dz Adz- A -- AAeq . i dxil A dzk A ... A dzt,,

where ry. is a skew-symmetric covariant tensor of rank or p-vector in the language of E. Cartan, and where the parentheses

indicate that the indices are ordered according to magnitude.

KXHLER MANNF'OLDS

f 9.1)

413

Let 9"11 ... g{'i1

r{i

(9.1.22)

...{,, 11...,, =

gc{. ... g{,!,

Then ,1...1,

= I g{1" g{1!,

is just the Kronecker symbol which is usually denoted by 55.j.. We depart from the conventional notation in this instance for reasons of notational symmetry.

The differential 4 of a p-form is the (p + 1)-form

4_

(9.1.24)

1)dz

A ... A dz{,+1,

where (9.1.25)

a1

ri1...{s+1511«.9a

Di9'i1...ly

Here

a,, !Vl1...l,

!,

9

8z'

i 9 91

and we observe that

q I-

r. Hence in (9.1.25) we may replace covariant since f ? 9µ 19 differentiation D liy ordinary differentiation a/azl. We have (9.1.26) ds9 - d(dg4) - 0. 1i9.4

A form 9P satisfying dyv - 0 is said to be closed, and a form 9P - dlp

is said to be exact. rormula (9.1.26) therefore states that an exact form is closed. Let (9.1.27)

et1..{4n '°"

1 2...2 k 1/Ti r{1...{Q.

and after de Rham [11] -set

.

2 .. 2k, 1 2 .. 2k,

414


(9.1.28)

[Q" P. IX

*9 _ (*4')(i,...su-,) dx" A ... A dxs k_,

where (9.1.29)

We verify that (9.1.30)

**9'

and, for the scalar 1, (9.1.31)

*1 = el a»sxdxl A ... A dz2k.

Thus *1 is just the volume element. The co-differential 6p of a p-form q, is the (p -1)-form (9.1.32) where (9.1.33)

n ... A dx'1-1,

84' _

(4)cguys In contrast with the differential d-V, the co-differential involves the

metric structure of the manifold in an essential way. We have (9.1.34) 82p = 6(6r) = 0. A form 9' satisfying o = 0 is called co-closed; a form p = dp is said to be co-exact. Let (9.1.35)

co = h(,,)dxi A dz .

The condition (9.1.20) expresses the fact that w is closed: (9.1.36) dw = 0. The condition (9.1.19), on the other hand, asserts that DA. = 0 and hence (9.1.36)'

dw = 0.

Thus the form w is both closed and co-closed. The classical Laplace-Beltrami operator for p-forms is (9.1.37) A = 6d + d6. A p-form q: satisfying Ar = 0 will be said to be harmonic, and one satisfying dr _ hP = 0 will be said to be a harmonic field. From (9.1.36) and (9.1.36)' we see that the 2-form w is a harmonic field.

COMPLEX OPERATORS

f 9.2]

416

We recall that the Riemann curvature tensor

Rmin-al{m1}-ax=I

1}

has the symmetries (9.1.39)

J Rhill = - Rshl3 = - Rhiti t Rhil:

=

R,thi

It also satisfies the Bianchi identity (9.1.40)

Rhilc + RhatI + Rh:il = 0.

The non-commutativity of covariant differentiation is expressed by the Ricci identity (9.1.41)

(DiDI - DrDi)pi,...i,

X

Rxiµtl

h

11-1

In terms of geodesic coordinates yil, (9.1.38)'

R'"il: =

a

ay' l i

l } _'t

l

m

'

If the metric is Kahlerian, then by (9.1.19) (9.1.42)

hm" R"ìli = him R"mli

Thus, in self-conjugate coordinates R'"ili is zero unless m and i have the same parity. In other words, R,,ilt = 0 unless h, i are of different

parity and also j, 1. From (9.1.40) it follows that (9.1.43)

R.1ya=Raayj=RYXy. In other words, indices of the same parity commute. Finally, any non-zero component of the Ricci tensor (9.1.44)

Rsi = R=il:

has indices of opposite parity. 9.2. COMPLEX OPERATORS

The tensors and operators considered in Section 9.1 are all real; in other words, the operators send a real tensor into a real tensor. Now we define the complex tensors and operators introduced in [7b].

HIGHER DIMENSIONAL I AHLER MANIFOLDS

416

[PAP. IX

As in [4] let

fly=-2 1

1,

The conjugate tensor is Hii

(9.2.2)

1 h=J)

0,1

2

1,0

conjugates always being defined in terms of a real coordinate system. Let e + a = p, e 0, a 0, and set (compare [4] ) (9.2.3)

17

/7

e,a

1,o

r1 ...

ITwe re 17 s, ... -7y e

1,0

0'1-

ra J na

(r l... re )(s 1..

a)

0,1

In self-conjugate coordinates

II{ =

(9.2.4)

110

(1, 1 S i I.

=1

k,

0, otherwise.

Therefore, any non-zero component of the tensor e, a

e,a

has precisely e indices between 1 and k and a indices between k + I

and 2k. In other words (9.2.5)

17q _

A

A dte A dial A ... A dz a.

e, a

If e + or -:P > 2k or if either e < 0 or or < 0, we define II to be 0,01 zero. We plainly have (9.2.6) E II = T e+o=9 e, a

and

II, e = e', a = a'. (9.2.7)

II 17 = e, 0

e a e'a'

0, otherwise.

Thus (9.2.6) is an orthogonal decomposition of the identity operator F. Since (9.2.8)

h,! = gf vht,

grshyz = - hii,

COMPLEX OPERATORS

§ 9.23

417

we have (9.2.9)

a,¢

¢,a

¢>a

If ip =17T we say sometimes (after Hodge) that c is of type (p, a). ¢, a

We define next a complex covariant differentiator, namely (9.2.10) 2, = IIi/DJ. 1,0

The corresponding contravariant differentiator is 2i = g'121 = 1711D! = ?7/D'.

(9.2.10)'

0.1

210

The conjugate operators are

Ni = 17/D,,

(9.2.11)

0,1

17j'Dl,

(9.2.11)'

1,0

It follows from (9.1.41) and the symmetry properties of the Kahler curvature tensor that

E

v-i1,o

(9.2.12)

9

- E 17lI'RI" rs1...1 µ-1 0,1

(complex form of the Ricci identity). In the complex tensor calculus which we propose to use, the Hermitian operator 17 replaces the symmetric identity operator r, and the complex differentiator 9 replaces D. Formulas (9.1.25) and (9.1.33) may be written

r

=

!cl1...l,)D

...! )

r11...il+1 i(l1...l,)

1, 1 (11..4,)D'9201...l,).

The complex analogues of these operators are

HIGHER DIMENSIONAL KuHLER MANIFOLDS

418

17d7I=E II !14

P- ) li...V+1

(9.2.13)

Q+E a

Q+1, a

= E

Q+a-a

Q. a

lu

e44-a

s+1

Q+1, a

1

...fa

1

... f D

f(f1

Q+1, a

II 8II = - E I7{{ . (-9q)l...!E 1¢1 Q+a°a 0.0 Q+a-a P,-1 Q, a

s-1

1

(9.2.13)'

[CHAP. IX

_ - E 17!!1...!¢1,

(f1...f,)

Q+a=a ca

The conjugate operators have the forms

(9.2.14)

17 d 17 = E 17

E

sa Qfry(f1... , 1(f1...f,)

Q+a-a Q.a+,

#+e-a Q.a+1 Q.a

E 17 517= - E 17'1...!¢1(s1..a9) 1

(9.2.14)'

Q+a=a Q-',a Q,a

7

,

!q,(s...i d.

Q+a=a Q,a

The following identities are readily verified:

xH = 17

(9.2.15) (9.2.16)

*d = (- 1)

(9.2.17)

(9.2.18)

*,

k-a, k-Q

Q, a

1) ax,

a2 = 0, nq,2 = 0,

aa+ as = 0,

+ ;9'.,9' = 0.

We also verify that (dp)l1...ly = --- D!D#j.' ..1' (9.2.19)

,- _ E R! 2 1

a

.y-

!y

M ry

7t1...!/r-1A111t1 ..{y-i s{t t1...{4.

In view of the properties of the curvature tensor for a Kahler metric, we have (9.2.20)

DII=1I A. Q,a

e, 6

Now we introduce a complex Laplace-Beltrami operator (9.2.21)

Then (9.2.22)

= 5a + a5.

COMPLEX OPERATORS

§ 9.2]

419

where

0=

(9.2.23)

+ ?.,p.

The following identities are readily seen to be valid:

0=(-1)'*C7,

e=0

(9.2.24)

= %a

We note also the identity 17 0,0

= [] 17 e, o

which, in contrast with (9.2.20), is trivial. A calculation gives

9

- 1' II r.-1 1,0

(9.2.25)

- -i

'R=hgpti...z1'-ihti...£9

9

Ri4t fi R tv

/b V-

fp 1 ht/,+i

v+i

to

Taking conjugates in (9.2.25) and interchanging a and a, the find that (

47)Li...tn =

T;l .. d9 2)

- . 17 1A1 Rah 471 1...{p-ih{N+i

(9.2.25)'

p-1 0,1 I ar

4 u,V-1

Hence by (9.2.12) we have A.

(9.2.26)

If p z 2, define (9.2.27)

(L147)c...j,,

= - h(l)

97(tt)ti...{s-a,

while if p = 0 or 1, set A97 = 0. It may be verified that

420

HIGHER DIMENSIONAL XAHLER MANIFOLDS

(9.2.28)

[CHAP. IX

A8 - 8A = -- /=I &9,; Ad - dA. = h-16h

where h is the operator ry,

defined by

slit ...

3' f 1

y1...{s, 11...10

I

Also

AB=6A, AQ=QA, Ap= (-1)'*(wA*q)

(9.2.29) (9.2.30)

where 9' is a p-form. 9.3. FINITE MANIFOLDS

A relatively compact subdomain B of the Kahler manifold M will be called a finite submanifold if each boundary point p of B has a neigh-

borhood N(p) in M in which real coordinates u1, us, , us" exist satisfying the following conditions: (i) each ul is a function of the zf of class C°°, and the Jacobian a(u', , usk)/a(zl, -, zak) does not vanish in N(p); (ii) N(p) is mapped topologically onto a subdomain in the u-space in such a way that the hyperplane u211=0 corresponds u211-1. being to the points of the boundary of B, the coordinates u;, local boundary parameters; (iii) the usk-curve is orthogonal to the hyperplane usk = 0 and hence gtf, expressed in terms of the ul, satisfies on the boundary the condition that g, ak = 9' 2k = 0 for i - 1, 2, -, 2k--1. The coordinates ul will be called boundary

,

coordinates.

A finite manifold is either a finite submanifold with boundary or is a compact (closed) manifold. The topological boundary operator will be denoted by b. In N(p), e l'B, set

4 0

in N (p) r B, then tip and nqq are well-determined P-forms in a sufficiently small boundary strip 0 < u2k < s covered by the coor, u21. We verify that *t = n*, t* = *n. dinate systems The scalar product of two p-forms 97 and y, on a subdomain D of

M is defined to be (9', V) = (97, 7')D = fp A *tp, D

and the corresponding norm is (17 q ?, d

9'

, 9') =

_

We plainly have

(

ITV); e, V

that is, II is a symmetric operator. If Ca is a differentiable q-chain on Mk with real coefficients and it

97 is a (q - I)-form, we have the well-known Stokes formula

5d=f Cq

9'

bCq

where bCa denotes the boundary of C. In particular, if we take q = 2k and Ca = C2k = B, where B is a finite manifold, then for a p-form 99 and a (g5 + 1)-form V we have fd (9' A

f (d9P n

B

s

-{- (_ 1) vIP n dam) = f (dV n *iP -.P n *6f) B

-S P n *yi. b8

Thus (9.3.2)

A *y'r. bB

By specializing T and we derive at once from (9.3.2) the following well-known ,,real" Green's. formulas:


422

[CHAP. IX

(d,p, dip) -- (r, adip) = f 9' A *dy,, bB

*;3,

(daT, i) - (69, 6V) -== bB

($d4', i) - (9p, 84) =

(9.3.3)

-

(d4, =V)

f

(9 A *dy, -- tp A *d94),

bB

(gq, chip) = f (aq? A *ap` - 60 A *97), bB

(Lair, y') - (9', AV) = f (9' A *dd -rp A *dr + 61P A *17 - d A *9,). bB

Taking p analogue: (9.3.4)

II q,,

ip = U Y' in (9.3.2), we obtain its complex Q, 0+1

Q, O

(7q', Y') - (9,"...9y) = f1p A *IP. bB

From (9.3.4) we derive immediately the "complex" Green's formulas:

(a9', av) - (9', AV) = J

9 A *(D+p)-,

0r,9,9" Y') - (^94', '.'Y')

f .,9'T A *+p,

bB

bB (9.3.5)

(API iV) - (99, AV) =

f

(9' A * (aY')-

- W A *aT),

bB

(a r9'9', y')

- (9', "a 9,i) = f (.'9'99 A *f -- (-9'i)- A *97), bB

(9', A4')=2 f (91 A*(atp)-'-1pA 491 +49'91 A *f bB

A *P).

§ 9 4]

CURRENTS

423

9.4. CURRENTS

Let D be an arbitrary subdomain of M (which may coincide with M), and let C" denote the space of p-forms ip which are of class C°° and have compact carriers relative to D (that is, vanish outside a

compact subset of D). A p-current T [T] on D in the sense of de Rham [11] is a linear functional over the space CU-9 which satisfies

the following continuity restriction: for an arbitrary sequence of forms q', p, f CZk-9, whose carriers are contained in a fixed compact

subset K of D, where K is covered by a single coordinate system, T [T,,; ---, 0 (,u -+ oo) if and each partial derivative tend uniformly to zero. LEMDIA 9.4.1. (Partition of unity). Given a locally finite open covering {U,) of D, there exists a corresponding set of scalars q:, such that:

(i) Zq, = 1; (ii) q;,eCaO, 0 S 4p, everywhere, and the carrier of g', is contained in one of the open sets U,. This is a standard lemma, easily proved (see [11]). Given a (2k - p)-form W of class C°° with a non-compact carrier,

we say that T[q2] is convergent and that a

if the series is convergent for each partition of unity. The exterior product T A ip of a p-current T with a q-form +p a C°°

is defined to be Tn

T [IP A 4)],

and (after de Rham 1111) dT[p"] = (- 1)""T[4], 6T[q,] = (- 1)"T[Sq"], *T[4c]

= (- 1)'T[*9"], (T, 4,) = T[*45]

We say that T vanishes at a point of D if there is a neighborhood N of the point such that T[pi] = 0 for all q' whose carriers lie in N. The carrier of T is the set of all points of D where T does not vanish. A p-current is said to be regular at a point of D if T coincides with a p-form of class C°° in some neighborhood of the point; otherwise T is called singular at the point. The set of singular points of T is called

the singular set of T.

424


CHAP. IX

Given two p-currents S and T whose singular sets do not meet,

there exist decompositions S = S' + - S", T = T' + T", where S", T" are everywhere regular while the carriers of S' and T' do not meet. In this case we define (9.4.1)

(S, T) = (S', T") + (S", T') -4- (S", T"),

provided that the scalar products on the right converge. We observe

that the scalar products on the right side of (9.4.1) are defined since the scalar product of a current and a form has already been given and since at least one of the factors in each scalar product is regular. THEOREM 9.4.1. 1/ T is a p-current in D C M and VAT AT is regular at a point o l D, then T is also regular at that point. I l dT and 8T are

both regular at a point, then T is regular at that point. This theorem, often called Weyl's lemma, is proved in [11]. A current of degree 0 is a distribution in the sense of L. Schwartz [12], and an arbitrary p-current T may be represented as a formal differential form

where the coefficients T, 1 ... are distributions in the space of local self-conjugate coordinates z{. Hence the operator 17 may be e, applied to T:

17T[p]=T[II9)]

e, o

k-e, k- -V

We define aT[97] = (- 1)'+'T[09p], &T[m]

A p-current T is harmonic if p T[9v] = T[p T] = 0. By Theorem 9.4.1 a harmonic current is equal to a harmonic form. Similarly, if dT [T] = 6T [.p] = 0, then T is equal to a harmonic field, that is, T is equal to a form v satisfying d V = 81v = 0. If the p-current -T

is of type (e, o) and satisfies 77' = .,&T = 0 (or 8T = . T = 0), we say that T is a complex field. If T = 17 T, then nPT = 0 automatically and the relation T = 0 e, 0 implies that the coefficients of T are equivalent to hQlomorphic functions of the complex variables z1, z2,

, zk.

§ 9.5]

HERMITIAN METRICS

425

LEMMA 9.4.2. If T =17 T is holomort hic zn a domain D C M, e' ° then 8T = 0 in D. By (9.2.28) we have 0 = (A -PA) T =

;5;T, so

Since .r9,T = 0, it follows that 6T = (,+9,

T= 0.

0.

9.5. HERMITIAN METRICS

It is easy to construct a positive-definite Hermitian metric on an arbitrary complex manifold M (which is paracompact and Hausdorff - hence normal). In fact, let {U{} be a locally finite open covering of the complex manifold such that each U£ is covered by local coordinates z{l, z{2, , z{2k. Let {g,} be a partition of unity corresponding to the covering {U,}, and define k

ds2 =Z(Zq Idz,«I2)

(9.5.1)

!

a-]

The metric (9.5.1) is plainly Hermitian. Let hsi be defined by (9.1.16) using the metric (9.5.1), and consider

the 2-form w defined by (9.1.35). In the special case k = 1, this 2-form is automatically closed (dw = 0) and therefore defines a Kahler metric. Hence: LEMMA 9.5.1. A Kahler metric may be constructed on an arbitrary Riemann surface. 9.6. DIRICHLET'S PRINCIPLE FOR THE REAL OPERATORS

In this section we make no use of the complex structure, and the results are therefore valid for an arbitrary Riemannian manifold M of real dimension m. However, since m is not necessarily even, formulas (9.1.30) and (9.1.33) are to be replaced by (9.1.30)' **P = (-1)"V+VP, (9.1.33)'

8q, = (- 1)""Pi'm+1 * d

where 4' is assumed to be of degree The expression (9.6.1)

D(9,) - (d4', d4') + (8f.

B9')


42d

[CHAP. IX

is the obvious generalization of the classical Dirichlet integral. Given

a positive number s, we define (9.6.2) D,(4') = D(rp) + s(q, q) = D(9,) + s II T Moreover, we write (9.6.3)

p,-p+s.

Let AP be the Hilbert space of norm-finite differential forms of degree p and let

A=EA9. V

If V e A, then q' = tp° + V1 ± ... + tpm where tp' E An, and we define H2.

We denote by C the Subspace of A composed of forms of class C' with compact carriers. We say that a form 4' is in the (closure of) the domain of the operator d if there exists a sequence {T,,}, p,, e C°°, 970 dqp,, E A, such I hat I I cp -- 9',, 11 and I I dT, - dg), I I tend to zero (,u, v -+ oo). If such a

sequence exists with q E C, we say that 4' is in the domain of the operator do. We define the domains of 6, 6,, in a similar fashion. If V is in the domain of d, we write 9 E d; similarly for the other operators. Finally, we say 9' is in the domain of the operator A if there exists a sequence {cps,}, q)m e CO, q,, pq?,, E A, D (q,,) < oo, such that 114p -

D(q,, - T,) and I I Q4,, - p4,, II all tend to zero (es, v --+ oo). We introduce the following spaces: G = {qt T e p, D(p, t') _ (92, p p) for every tp e p}, N = {T 9 E A, D(q,, tp) _ (,Lp, y,) for every tp E p}, H = {q, T E A, Ap = 0},

F, = GnH, F=NnH.

Denote by D the closure, in the D; norm, of the space of forms 4" of class C°° with D8(q,) < oo, and let D. be the closure of C in the D,-norm. Since any two norms D,,, De, s' > 0, s" > 0, are equivalent, the spaces D, D,. are independent of the choice of the positive number s. If 99cG,wehave s11ip 1I2sD8(9)= (4', A, p) S II A,9'11 -!Im!!; that is, II p II S 11 A.p II/s, and the same inequality is true for q, e N.

§ 9.6)

DIRICHLET'S PRINCIPLE

427

Let q' c G or N; then LT = 0 implies D(T) = 0, and A,T = 0 (s > 0) implies 9' = 0. Therefore

F={g,Iq,eD, dq,=692=0}, and it will follow from (9.6.19) that

Fo={pjpeD dq,=Sq,=O}. Now let [dC] denote the closure, in the sense of the norm I I . . ((, of

the space of forms dp, cp a C, and let [b C] be defined in a similar way. Also, let [A C] be the closure of the space of forms AT, q, e C. We have the following two well known formulas of orthogonal decomposition ([9a], [11]). (9.6.4)

A = [A C] + H,

(9.6.5)

A=[dC]+[SC]+F.

These two formulas are easily proved using Theorem 9.4.1. Given a form ' e A, we have by (9.6.4)

9'= 9'i+922 where 972 e H and qoi is orthogonal to T2, that is, (q,l, q,2) = 0. We call

9'2 the harmonic component of 9' and write 9'2 = H9. We say that a current T of degree p is of class A if it is convergent (see Section 9.4) for every q, a A'"-', 99 e C°°. For currents of class A, we define HT [p] = T [Hq,] . Now let y be a point of M and consider the operator 1 = 1. satisfying (1, 92) = SP(Y)

for every 99. Strictly speaking, the operator 1 applied to forms of degree p > 0 is not a current (since 1[92] = (- 1)'"9+p(1, *92) = (-1)'"9+.v*97(y) is not a mapping into the reals), but it is a trivial extension of the notion of current and may be treated as though it were a current of class A. We define Hl' q,] = 1 [H9] = HT (y), and then we have (H1, -p) = (1, H92) = Hgo(y).

By Theorem 9.4.1, H1 is a symmetric harmonic double form and, applying the inequality of Schwarz, we obtain

IH9' (y) I S 1/(Hl, H1) IIH9'II =K, IIHTII This inequality shows at once that the space H is closed; and hence

428

HIGHER DIMENSIONAL HAHLER MANIFOLDS

[CHAP. IX

is a Hilbert space (the Hilbert space of norm-finite harmonic forms).

The operator H denotes orthogonal projection onto H. It follows that F is also closed, and hence F is the Hilbert space of norm-finite harmonic p-fields. We denote orthogonal projection

onto F by F. Let B denote the space of forms which belong to both the domain of d and the domain of 6; that is, B = {g, J 9 a d, 8}. We note the following lemma: LEMMA 9.6.1. The space B coincides with D.

To prove this lemma, we have to show that, given any form 9' in the domains of d and 6, there exists a sequence {'p}, T" a C°°, D,(Tp) < co, such that D,(97

-

TO) -> 0 (,a --> oo).

We base the proof on formula (9.6.5). Let +p e A; then (9.6 6)

'P = ZV1 + '+V2 + 'P3'

'Pi E [dC],

'P2 E [b C1

Vs=F'paF.

If 'p e C', then 'P1, tp2 e C. In fact, we regard (9.6.6) as a current

formula and apply the operator d6 to it; we obtain

d&Vi= A%dO'eC°°, and hence, by Theorem 9.4.1, Vi t C°°. Similarly 'p2 e C. Assume 99 is in the domains of d and 6. Then there exist two sequences {aµ}, {#J, such that I I T - aµ I I -> 0, 11 dpi -do t# I I -> 0, and 1 192 - Ps, I I -> 0, J J ap - ao,,v J J -- 0. Let', ap and 3, be decomposed according to

formula (9.6.6): 'p = 9'1 + T2 + 9's, aµ = aµ1 + aµ2 + q-0, We define

+ flv2 + flvs'

Then I I - 9'µ I I2 = 1191- #,I I IS +I ITS -- aµ2 I l2 - 0,

and

D(9,-ypµ) = IIdcp-damII'-1- II69,-%iII2->0 as µ oo. Thus D, (97 -'µ) -> 0 (,u -+ oo), and Lerhma 9.6.1 follows. If they exist, we define the Green's operator G, of the manifold M to be the one-one linear mapping of A onto G whose inverse mapping


§ 9.6]

429

is Q and we define the Neumann's operator N, to be the one-one linear mapping of A onto N whose inverse is Q,.

If these operators exist, they are unique. For if 97 e G, and A,cp = 0, then p, = 0. Similarly, if q e N and L,p, = 0. THEOREM 9.6.1 [13c]. For each positives the Green's and Neumann's

operators exist. They are symmetric operators and they satisfy (9.6.7)

Ds(G.9,) S IIplI$/s,

D.(N.-p) S 11plls/s.

Theorem 9.6.1 follows in a straight-forward fashion from Dirichlet's Principle, and we now give a proof which, in its main lines, does not differ from the classical Dirichlet's Principle.

Given fi e A, write y = a/s, and define D(rp, w) + s(9

y,

-y), E.(m) =

The Dirichlet's Principle which we establish asserts the existence of a

unique forma which minimizes E,(p,) and satisfies, perhaps in a generalized sense, the equation Apt = A. If we restrict the forms in the minimum problem to those with compact carriers, then a = G,f where G, is the Green's operator while, if we place no restriction on the forms p, in the minimum problem, then a = N,# where N, is the Neumann's operator. Let (9.6.8)

e = e(s) = inf E. (p), e, = e,(s) = inf E, (p) for 92 e C.

It is obvious that (9.6.9)

0 S e(s) S e,(s) S 11,112/s.

We have to show that the above minimum problems have solutions.

Since the treatment is essentially the same in both problems, we consider the problem of minimizing E,(97), T a D. The minimizing

properties will then show that the minimizing element is in the space N. We base the proof on the following variant of B. Levi's inequality: (9.6.10)

VD,(p,-+p) S E

+ VE,(y) - e.


430

[Cxar. IX

Let a, s be real numbers, a + r = 1. Since app + Tip - y = a (q' - y) + r(v, - y), we have E.(acp + ,rip) = a' E. (9)) + 2 ar E.(4', v') + rQ E.(w)

e;

that is, a'[E.(-P) -" e] + 2 ar [E. (q,, u) _. e] + ra[E.(y) - e]

0.

Since the left side is homogeneous in a and r, this inequality is valid

for arbitrary real numbers a, r and it follows that E.(T, V) - e 1

E

e.

Hence

D.(v - v') =

E. (p, '') + E.(v )

e] - 2[E,(-p,v') -- e] + [E.(v') - e]

e] + 2E

.

E

+ [E.(() -- e], and this is (9.6.10).

Let {{Pµ}, Tµ a C°°, be a sequence such that E,(97.) -* e (fc - ao). Then D, (To -- 9P,) -} 0 (,u, v -a oo) by (9.6.10), and hence by Lemma 9.6.1 there exists an a e D such that s = E,(oc). Since E,(oc + eqp) Z e for every real e, we conclude by the usual reasoning that the coefficient

of a in E, (a + Egg) must vanish, that is (9.6.11)

D, (a, 4') = (fl, 4,), c e D.

If ry e C, then D. (a, q) = (a,

Therefore a, regarded as a current, satisfies the equation (9.6.12) O.a[4'] = XT I , e C. Theorem 9.4.1 remains valid if L is replaced by the operator Q,. In fact, the proof, which is based on the existence of a local elementa-

ry solution for the operator A. does not differ essentially from the proof given in [11] for the case s = 0. Hence, if # e C°°, then a is

equal to a form of class C°° and, in this case, p,a = fi in the ordinary sense. Thus (9.6.11) becomes (9.6.13)

D.(%, p) = (Q,a, 97), 0 a D.


§ 9.63

431

The above reasoning applies equally well to the restricted minimum problem in which the competing forms belong to C and, in particular,

the formulas (9.6.11), (9.6 13) are valid with D replaced by Do. We denote the solution of the restricted minimum problem by that of the unrestricted problem by NA, and we show now that these operators are linear, bounded and symmetric, therefore self-adj oint. It is sufficient to consider N,, since the same reasoning applies to

G,. Let NI, 02 be two elements in the space A, and write ai =

N,#x, a$ = N,552, a = N,(#1 + 02). By (9.6.11) we have

D.(a--al -aa, q') = 0,

, eD.

If we choose -p = a - al - all we conclude that a = ai + a$; that is, N,($1 + N2) = N,(A1) + N.(1B2).

Since it is obvious that N,(c5) = cN,p for any real number c, the operator N, is linear. Next, taking or. = N,j, we have by (9.6.11) s Ilallz s D,(a) = (fl, a) S Ijai l and therefore s l jal j S j i flj j. Thus

II#II,

D,(N,9) = D,(a) s Il8112/s.

In particular, N, is continuous, and we may verify that the equation Q,a = is valid. In fact, let e A, be a sequence such ,9, e C°°, that 0 (,u -} oo), and let a, = N,§,,. Then and

II a-a, II < it fl-s jj/s, D,(a, -a,,) s II##-f,, Ill/s. Hence II a - au II, D(ocµ - a.),

I

I ©ac - A%, I tend to I

zerc

(Jul v -- oo), and it follows that or. T N,(3 is in the domam of Q anc

that

= f8. Finally, if p, y, e A, we have by (9.6.11 D. (N.m, N.V) = (4', N.cV)

Hence (9.6 14)

(N.(P, t,) = (q', N, ,).

This completes the proof of Theorem 9 6.1 so far as N, is concerned In the restricted minimum problem we have an additional identit3 which is obtained as follows. Let 97 e D,, and let {970}, To e C, be

HIGHER DIMENSIONAL K.9HLER MANIFOLDS

432

[CHAP. IX

sequence converging to q' in the sense of the' D,-norm. If W E p, we have Da(9p,y) = lim D.,(4'µ, v) = lim (gyp Sao

,

A.w) = (w, oar)

and therefore D.(4', V) = (9,, L.TV),

9' E D., ' e /,

Since Ga,R E D we obtain (9.6.16)

D5(G8P, v) = (GAF',

)4 E A, V E L .

This formula shows that G,5 e G, and the proof of Theorem 9.6.1 is completed. If M is a finite manifold with boundary, Green's formula shows that the difference D.(G.fl, V) - (G,fl, A.V) is the limit of integrals extended over the boundaries of a sequence o subdomains which converge to M, and these integrals involve the values of G, P, * G, fl on these boundaries. Formula (9.6.16) states that

the limit of these boundary integrals vanishes and hence, in this sense,

nG,j = 0.

(9.6.17)

Similarly, (9.6.13) shows that

1&V.#=ndN,#=0.

(9.6.18)

In the case of scalars these formulas state that G,g vanishes at the boundary and that the normal derivative of N,fi vanishes at the boundary, and therefore G,, N, have the boundary behavior of the classical Green's and Neumann's functions. We have (9.6.19) (9.6.20)

{,j9)EA, 1G.9,=97-sG,p=01f, F={g,i9,eA, AN,q,=9,-sN#=0}.

F

In fact, if 9' E A and AG,T = q, - s G,T = 0, then 97 E G, Q,qq = 0. Conversely, if q7 E G, A99 = 0, then , = G, (Q,9,) = s G,97. Similarly

for (9.6.20). From the minimum principle by which Ga, N, were

§ 9.7-1

BOUNDED MANIFOLDS

438

defined it follows that (9.6.21)

FOG, = G. F. = F,/s,

(9.6.22)

F N, = N, F = F/s.

Finally, we have the orthogonal decompositions: (9.6.23)

A = [ A G] + Fe,.

(9.6.24)

A = [AN] + F,

where (A G] denotes the closure under the norm (( (( of the space of forms 97 = App, V E G, and where [A N] has a similar meaning with respect to the space N. In fact, let 97 E A, (9,, y) = 0 for y e [AG]. Then 0 = (q', AG,w) = (9,, 'y, - sG,,p) --= (97 - sG,97, y,) for every y, E A, and this implies that 97 a F,. Similarly, for (9.6.24), 9.7. BOUNDED MANIFOLDS

A relatively compact subdomain M of a Riemannian manifold R will be called a b-manifold (bounded manifold). We have: THEOREM 9 7.1. [13b]. Let M be a b-manifold imbedded in a Riemannian manifold R. Then F,, is the subspace o l harmonic forms on R which vanish identically outside M. We define the Green's operator Go of a manifold M to be the oneone linear mapping of A - F. onto G - F, whose inverse mapping is A. If this operator exists, it is unique. For let (p e G - F0, L99 = 0 Then (p e H n G = F so p = 0. The domain of the operator G. may be extended to the whole space A by defining it to be identically zero on the space F0. Then G,, satisfies (9.7.1)

(1 - F,,)(p = A G,, 99, F0G,92 = G,F092 = 0,

9'

A.

We recall that an operatoi is completely continuous if it carries bounded sets into sets whose closure is compact. THEOREM 9.7.2. A b-manifold possesses a bounded, symmetric com-

pletely continuous Green's operator G,. The biharmonic Green's operator is the one-one linear mapping of

A - H onto (G n N) - F,, whose inverse mapping is A. We extend this operator to the whole space A. by defining it to be identically zero on H, and denote it by BO.


434

[CffieP. IX

Theorem 9.7.3. A b- nani f old possesses a bounded biharmonic Green's

operator B,. We begin by deriving Theorem 9.7.3 from Theorem 9.7.2, and we observe first that Theorem 9.7.2 contains a solution (in the generalized sense) of the first boundary-value problem of potential theory. In

fact, let 9' be an arbitrary form in the domain of the operator and set v = q, --- G. ©9p.

(9.7.2)

Then

Q+Y = Q9'- (Q9'-F,Q9') = F,Qp = 0. On the other hand, V - 9' _ --- G. Q9' E G, so yn is a (generalized) solution of the first boundary-value problem. Since we may add to w any form in F. without changing its boundary behavior, the solution

is unique if and only if F. contains only the form 0. Now define the adjoint operators (9.7.3)

B, = G, - G, H, B; = G. - HG,.

Given any form 9' in the domain of the operator Q, set

,p = (1---H)p-B,' Q'p. Then

Av = L9P so

F,Q9') = 0, Hy,= 0,

= 0. That is

(1- H)'p = B' A9% On the other hand, we obviously have (9.7.4)

(9.7.5)

(1 -- H),p = Q B,V,

,p

e A.

Let 9' be an arbitrary form in the domain of L, y' an arbitrary form in A. Smce B,+p E G, we have

('V, (1- H) 9') = (V, B' Q4p) = (B,s, Q9') By (9.7.5)

= D(B,,V, 4')-

('V,(1-H)H)(QB.V,9') Therefore (9.7.6)

D(B,y1, 4') = (QB,5, 9')

§ 9.7)

BOUNDED MANIFOLDS

435

for every 9) a A, V e A, and it follows that B, V e N. Since F, B, Io = 0, R. is the biharmonic Green's operator whose existence is asserted in Theorem 9.7.3.

From Theorem 9.7.3 we obtain a solution of the first boundaryvalue problem for biharmonic forms. In fact, given any p in the domain of L, let Then

AV= Lr-(L'---H/.p) =HA97, A2v= A ASV= 0, so p is biharmonic. Since W-97 e G n N, +p is a (generalized) solution

of the first boundary-value problem for biharmonic forms. It remains to prove Theorems 9.7.1 and 9.7.2, and we base the proof of both on the following results: Let R be an arbitrary Riemannian manifold of class CO. Given any point q e R, there is a neighborhood U = U(q) depending only on q such that, given an arbitrary integer v, a number s Z 0, and a positive number ,, there exists a double form y (x, y) with the following properties: (i) the double fotm y(x, y) is defined for x e R, y e U, and y(x, y) is of class C°° in x, y if x 0 y. If r(x, y) is the distance of the points x

and y, [r(x, y)]-1 y(x, y) is of class C°° for x = y. (ii) for fixed y e U, the carrier of y(x, y) has diameter less than i. (iii) for x e R, y e U, we have po(x) y(x, y) = g(x, y) where g(x, y)

is a double form of class C" in x and y. (iv) the operator (p(x ), d (x) y ((x, y)) is a completely continuous transformation which maps forms q7 a A (R) into A(U). The same statement applies to (T(x), 6(x)y(x, y)). The form y(x, y) with the properties (i) - (iv) can be constructed by the method of Kodaira [9a]. Kodaira assumes that the metric is real analytic, but the infinite series used by him can be replaced by finite partial sums. To prove Theorem 9.7.1, let 9; e F, and extend q, over R by defining

it to be identically zero outside M. There exists a sequence {Tµ}, T;, c C(M), converging to 'p in the D; norm. Given a point q e R, let y (x, y) be the double form associated with the neighborhood

HIGHER DIMENSIONAL KARLER MANIFOLDS

436

[CHAP IX

U = U(q). For y e U we have by Green's formula D.(q',(x), y(i., y)) - (q',,(x), g(x, y)), oo, we obtain as limit (in the sense of the norm I I As ,u .

II)

(9.7.7)

9'(y) = D,(9, (x), y(x, y)) - (T (z), g(x, y)), y e U. Formula (9.7.7.) (with y depending on s) is valid for each s z 0.

Since d9 = 6p = 0 in M and in R - M, the case s = 0 gives 9,(y)

=-(q,(x), g(x,y)), y - U.

Thus ga(y) e C'' in U, and hence everywhere in R. This proves Theorem 9.7.1.

We base the proof of Theorem 9.7.2 on the following lemma: LEMMA 9.7.1. For every b-manifold the operator G, (s > 0) is completely continuous. V, e C (M), In fact, since G, 92 e G, there exists a sequence converging to G,p in the D, norm, and we obtain formula (9.7.7)

with 9i replaced by G.97-

(9.7.8) G,q,(y)=D,(G,p(x), y(x, y))-(G,q%(x), g(x, y)), yeUnM. Since M can be covered by finitely many neighborhoods U, it follows

that G, is a completely continuous transformation from A onto G. We remark that the operator N. will not in general be completely continuous on b-manifolds. For if N, were completely continuous, the

space F would be finite dimensional. To prove Theorem 9.7.2, we begin with the following observations

concerning a completely continuous operator P, II P9 II s c I; 9' II,

c>0. Let 0

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