Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich)

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

Birkhuser

Lectures in Mathematics ETH Zurich Department of Mathematics Research Institute of Mathematics Managing Editor: Michael Struwe

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems Notes taken by Roger Moser

Author's address:

CMLA Ecolc Normale Superiture do Cachan 61. avenue du President Wilson 94235 Cachan Cedex France

2000 Mathematical Subject Classification 53C42. 53C43; 53C28, 53035, 70006

A CIP catalogue record for this hook is available from the Library of Congress. Washington D.C.. USA

Deutsche Bibliothek Cataloging-in-Publication Data HElein, FridE ic: Constant mean curvature surfaces, harmonic maps integrable systems I Freddric Hdlein. Notes taken by Roger Moser. - Basel ; Boston ; Berlin : BirkhBuser, 2001

(Lectures in mathematics : ETH Zurich) ISBN 3-7643-6576-5

ISBN 3-7643-6576-5 Birkhauser Verlag, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation. broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must he obtained. 0 2001 Birkhttuser Verlag. P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF Printed in Germany ISBN 3-7643-6576-5 987654321

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Contents Preface

7

.

Introduction: Surfaces with prescribed mean curvature .

9

From minimal surfaces and CMC surfaces to harmonic maps Minimal surfaces 2.1 Constant mean irvature.surfaces 2.2 .

15 16 18

3

Variational point of view and Noether's theorem

22

4

Working with the Hopf differential 4.1 Appendix

34 39

5

The Gauss-Codazzi condition Appendix 5.1

41

1

2

6 7

8

9

50

Elementary twistor theory for harmonic maps . Appendix 6.1 Harmonic maps as an integrable system Maps into spheres 7.1 7.2 Generalizations A new setting: loop groups 7.3 7.4 Examples . . . . . . . . . .. . .

52

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60 63 63 68 71

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.

Construction of finite type solutions Preliminary: the Iwasawa decomposition (for Rc) 8.1 Application to loop Lie algebras 8.2 The algorithm 8.3 Some further properties of finite type solutions . 8.4

Constant mean curvature tori are of finite type . . . . . . . . . . . . . The result . . 9.1 Appendix . . . . . . . . . . . . . . . . . 9.2

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83 84 87 89 89 94

Contents

6 10

Wente tori CMC surfaces with planar curvature lines 10.1 A system of commuting ordinary equations 10.2 10.3 Recovering a finite type solution 10.4

11

Spectral curves

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107 107 109 110 .

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Weierstrass type representations Loop groups decompositions . 11.1 Solutions in terms of holomorphic data . 11.2 Meromorphic potentials 11.3 Generalizations . 11.4

Bibliography

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112 112 113 115 1119

117

Preface One of the most striking development of the last decades in the study of minimal surfaces, constant mean surfaces and harmonic maps is the discovery that many

classical problems in differential geometry - including these examples - are actually integrable systems. This theory grew up mainly after the important discovery of the properties of the Korteweg-de Vries equation in the sixties. After C. Gardner, J. Greene, M. Kruskal et R. Miura [44] showed that this equation could be solved using

the inverse scattering method and P. Lax [62] reinterpreted this method by his famous equation, many other deep observations have been made during the seventies, mainly by the Russian and the Japanese schools. In particular this theory was shown to be strongly connected with methods from algebraic geometry (S. Novikov, V.B. Matveev, I.M. Krichever... ), loop techniques (M. Adler, B. Kostant, W.W. Symes, M.J. Ablowitz...) and Grassmannian manifolds in Hilbert spaces (M. Sato... ). Approximatively during the same period, the twistor theory of R. Penrose, built independentely, was applied successfully by R. Penrose and R.S. Ward for constructing self-dual Yang-Mills connections and four-dimensional self-dual manifolds using complex geometry methods. Then in the eighties it became clear that all these methods share the same roots and that other instances of integrable systems should exist, in particular in differential geometry. This led K. Uhlenbeck [82] to describe harmonic maps on a two-sphere, with values in U(n) using families of curvature free connections depending on a complex "spectral" parameter. At the same period N. Hitchin [54] investigated

finite type tori into SU(2) starting from similar methods. Such formulations were already proposed in the seventies by K. Pohlmeyer [69], V.E. ZhakarovA.V. Mikhailov [91] and V.E. Zhakarov-A.B. Shabat [92]. A catalysor of these developments was the construction by H. Wente in 1984 of an immersed constant mean curvature torus in ]R3, which had the effect of removing an old inhibition due to the fact that people believed that such tori should not exist. Many results then followed quickly and we have now a very rich and fruitful theory for constructing constant mean curvature surfaces and harmonic maps of surfaces with values into symmetric manifolds using integrable systems methods. All that seems apparentely a new theory, but many features of the "completely integrable behaviour" of constant mean surfaces have been guessed by geometers of the nineteenth century. Namely the existence of associated families of such surfaces by O. Bonnet, the study of special surfaces with planar curvature lines by A. Enneper and his students and the various Backlund transformations discovered and studied by A.V. Biicklund, L. Bianchi, S. Lie, G. Darboux, E. Goursat and J. Clairin (see [71] for details and references).

This Monograph is intended to give an introduction to this old and new theory from the point of view of differential geometry. For that reason, it has

8

Preface

seemed more natural for me to introduce the existence of families of curvature free connections for harmonic maps starting from the associated family of constant mean surfaces. Note however that this is not the historical way the theory actually developed and contemporary people were initially more inspired by the example of the KdV equation and the twistor theory. We also presented here some basic exposition of the twistor theory for harmonic maps, which was initiated to my knowledge by E. Calabi. Indeed this theory was an important

stimulation for the integrable system theory, in particular in the work of K. Uhlenbeck, and shares some similarities with the integrable systems theory (for a complete exposition, see [231). 1 also made an effort to present the beautiful result of U. Pinkall and I. Sterling concerning constant mean curvature tori (Chapter 9) in the framework of loop groups theory, in order to show how this result connects with the rest of theory, which uses loop groups. The present text is just an introduction and is far from being complete. We may recommand as parallel lectures the books [46] and [43] plus of course ... reading the cited papers. These Notes come from a lecture that I gave at the Eidgenossische Technische Hochschule Zurich during Spring 1999. Most parts of the text were written and typed by R. Moser. I wish here to thank the ETH Zurich and more particularly

Prof. M. Struwe for his hospitality and to thank R. Moser for his very nice work.

I also want to thank Joseph Dorfineister and Pascal Romon for their valuable comments on this text.

1

Introduction: Surfaces with prescribed mean curvature

Curvature

For a curve I' in a plane, and for any point on this curve, there exists a circle (or a straight line) which is the best approximation at this point of the curve up to third order. The inverse of its radius k = R (or k = 0 if the best approximation is a straight line) is called the curvature of the curve at the given point.

For a surface E in R3 and a point m E E, we consider the 1-parameter family of affine planes which contain the straight line passing through m and being normal to E at m. Each of these planes locally intersects the surface along a planar curve containing in. We may label these planes by choosing one, say Po, and for 9 E R/27rZ naming by PO the image of Po by a rotation of angle 9 around the normal line to E at in. Then the curvature k(O) of PO fl E at m is of the form i2 + 2 2 cos(2(9 - Oo)), where kl and k2 are two numbers, called the principal curvatures of E at m. They are the extremal curvatures, achieved at two extremal positions of Po which are orthogonal (the principal directions). The average H = (k1 + k2) of kl and k2 is called the mean curvature of E at in, their product K2 = klk2 the Gauss curvature. If kl = k2, i. e. if the curvature k(9) is a constant, then the point in is called umbilic.

Experiments with soap

We are going to describe some experiments, carried out by J. Plateau, which may give some physical motivation for what is to follow. Imagine a piece of wire, bent in the shape of a closed curve r, that is dipped

into a solution of water and soap, such that it becomes spanned with a soap film. This film will take the form of a surface Eo with boundary 0EO = r, having least area among all surfaces with the same boundary. Computing the Euler-Lagrange equation of this variation problem, we see that Eo satisfies H = 0,

(1.1)

where H is the mean curvature of Eo as described above. Such a surface is called a minimal surface. Suppose now that there is some device that allows to have different pressure on either side of the soap film in question. Then instead of (1.1), the equation

H=C

(1.2)

10

1. Introduction: Surfaces with prescribed mean curvature

will hold for E0, where C is a constant different from 0 (depending on the difference of the pressures applied). In this case we call Eo a constant mean curvature (CMC) surface.

First and second fundamental form Let E be a piece of a surface, embedded in R3, which is diffeomorphic to the unit disk D2 = {z = x + iy E C : Izi < 1). Let X: D2 --' JR3 be a parametrization of E. Then we can define the following.

Definition 1.1 The first fundamental form of an embedding X: D2 - R3 is the quadratic form given by the matrix

ax aX

ax IZ dx

8x I

t3y

z

Ox ' ay >

1

ay

,

which depends on z E Dz.

= IdX(e)12. Now fix an orientation For = E R2, we compute of E by assuming that (e , ) is an oriented basis of Tx()E. Consider

u=

x Y x ax K

BX ' I

where x denotes the vector product in ]R3. The map u: D2 - S2 thus given is called the Gauss map of X. Definition 1.2 The second fundamental form of X is the quadratic form given by

a2x asay u

0)

Cuaway

+

>

8y

u

With the notions of the first and second fundamental form, we are able to define the principal curvatures of a surface in a more convenient way than before.

Definition 1.3 The principal curvatures k1, k2 are the eigenvalues of II with respect to I, i. e. the solutions of

det(II - Al) = 0.

1. Introduction: Surfaces with prescribed mean curvature

11

By a multiplication of II - AI with I-' from the right it is easily verified that the mean curvature and the Gauss curvature can be computed from I and II in the following way:

H = I tr(II I-' ),

K = det(II)/ det(I).

(1.3)

Conformal coordinates

To simplify things, we will use conformal coordinates. Choose E like above. The following is a well-known result.

Theorem 1.1 There exists an embedding map X : D2 --+ R3, such that

i) X(D2) = E, and ii) X is conformal, i. e.

12-IW12-

\8x'

lax

ay0.

(1.4)

Note that (1.4) means that the vectors az and have at each point the same length and are perpendicular to each other. This implies that there is a function w: D2 -+ R, such that

ex _

ax _

8x

+J

= ewe1,

= e e2,

and (e1, e2) establish an orthonormal basis of TX(Z)E for each z E D2. Thus we may write

I=e2m (1 1\

II=elm( h"

f

1

h22

h12

Then k1 and k2 are the solutions of det ( ( h11

\\ h

12

fi r 0

h21

11

))

`

h 22 )

0

L e. the eigenvalues of the matrix h11

h21

h12

h22

Consequently,

H = h11 + h22 2

h21 1

K = det h12

h22 J

'

1. Introduction: Surfaces with prescribed mean curvature

12

We see that, for conformal X,

2H = h11 + h22 = e-2-(AX, u), or

(OX, u) = 2e2i''H. Moreover, we have the following.

Lemma 1.1 If X is conformal, then

ax ay. ax oxl ax, This can easily be verified by differentiating (1.4). The details are left to the reader. It allows, however, the following conclusions.

Corollary 1.1 The map X satisfies

OX = (OX, u)u = 2He2i'u = 2H

ax

ax

ax x ay ,

as soon as it is conformal.

Apply this to the equation (1.2) in the form

H=C=Ho. We see that it is equivalent to

OX = 2Ho

Ox Ox ax x ay

In particular this implies that minimal surfaces are always images of harmonic and conformal parametrizations. This leads to the problem of finding maps X: D2

11 P3, such that

X is conformal, i. e.

/ax

o=laxl2-

- 2i ax , - e) I2

AX=0.

1. Introduction: Surfaces with prescribed mean curvature

13

Weierstrass representation Using the notation of complex differentiation, a

1

a

az = 2 (ax

a

a

a a a x + i ay)

1

- i ay) ' az - 2

a2

4 azaz'

we find that our problem can be expressed in the form

aX)2

= 0,

( az a (az) OX

= 0.

az

Write f = 2'9x: D2 -+ C3. We want to solve f 2 = 0,

02

= 0.

This can be done explicitly by the representation

f=

i(w2 - 1) ,(W2 + 1) iw

h,

where w, h: D2 - C are holomorphic functions. (The function w might be meromorphic.) Eventually, this gives us Weierstrass representation of a solution of our problem:

X(z) = Re

I

f

f(C)dC

.z

where zo is an arbitrarily chosen point in D2. Completely integrable systems The Weierstrass representation for minimal surfaces seems to be a miracle, since it describes all solutions of a nonlinear geometrical problem by a very simple algebraic construction. We shall see in these lectures more sophisticated miracles, occuring in various geometrical situations. People called them completely integrable systems. Classically, they are nonlinear equations on which mathematicians and physicists discovered unusual properties. Some of these properties are

existence of solitons. This terminology comes from evolution problems and the most famous example is the Korteweg-de Vries (or KdV) equation

14

1. Introduction: Surfaces with prescribed mean curvature at + 6u a2 + a = 0, modelling water waves in a flat shallow channel. The story started in 1834 with the experimental observation of a solitary wave travelling along such a channel along a very long distance by J.S. Russel [73). A model equation was derived in 1895 by D.J. Korteweg and G. de Vries [591. Solitons are solutions of nonlinear partial differential equations which are localised in space, whose profile is not dispersed after a long period of time and which resist to interactions with other solitons 1. Thus it is a smooth field, solution of some partial differential equation, which behaves like a particle. Backlund transformations. A baby example is the following: start from

a harmonic function f of two real variables x and y. We may write the Laplace equation for f as d (- 2L dx + a dy) = 0, which implies that, if we work on a simply connected domain, there exists a function g such that

- ay dx + a dy = dg. Then g is another harmonic function, namely the conjugate function off (i.e. f +ig is a holomorphic function of x+iy). Such transformations, producing a solution of some partial differential equation starting from another solution work also in nonlinear situations. by a nonlinear change of variable, the problem reduces to solving linear equations. Weierstrass representation does it obviously, reducing the minimal surface equation to the Cauchy-Riemann system. We shall meet the same situation, but involving a much more complicated "change of variable". a Hamiltonian structure. Solutions are spanned by the Hamiltonian flows of commuting functions on a symplectic (or Poisson) manifold. This is described by Liouville's theorem. Infinitely many symmetries. Infinite dimensional Lie groups acts on the set of solutions of these systems.

The theory of completely integrable systems is not a pragmatic point of view, in the sense that it does not really provide a method (like the analytic approach based on functional analysis) where, starting from some qualitative intuition, one works (often hardly) to construct the tools one needs for proving (or disproving) what one believe to be true (existence, regularity,...). Indeed working on completely integrable systems is rather based on a contemplation of some very exceptional equations which hide a Platonic structure: although these equations do not look trivial a priori, we shall discover that they are elementary, once we understand how they are encoded in the language of symplectic geometry, Lie groups and algebraic geometry. It will turn out that this contemplation is fruitful and leads to many results. 1for the KdV equation, the simplest solution involving the soliton behaviour is the onesoliton u(x, t) = 2a2sech2(a(x - 4a2t)), for a > 0

2

From minimal surfaces and CMC surfaces to harmonic maps

Let Il be an open subset of C and X: fZ - R3 a conformal parametrization of a surface E. We use all the notation form the previous section. In particular, we have for the mean curvature H =

h11 + h22

2

if we write for the second fundamental form hll

h21

h12

h22

II = e2"

It is clear that II is symmetric. So, for a fixed H, there exist functions a, b:11-- R

such that h11

h12

h2l

_

H + a

b

h22

-

b

H - a

(2.1)

For convenience, we denote coordinates on C alternatively (x, y) or (x1, x2), where x = xl, y x2. Because of the fact

\

\

which is a consequence of the orthogonality of u to II in yet another form:

II = - C/ au, aX

ax' axe

/

and a , we can write

i,i=1,2

(2.2)

Note furthermore that (u, 9'9 ) = 0 for i = 1, 2 (because of Jul = 1).

The map X provides a moving frame on the surface E, i. e. three vector fields that constitute a orthonormal and oriented basis of R3 at each point of E. It consists of the vector fields e1

Thus

au

_

_w

ax = e

ax _ "ay, ax u=el xe2. a -wax, e2=e

0'X ay

/ au aX \ _w / au el + e `ax ax ' ax J

e2,

Using (2.1) and (2.2), we finally obtain

au au ax' ay

__ _

aX aX

H+a

b

ax ' ay

b

H-a

i = 1, 2.

2. From minimal surfaces and CMC surfaces to harmonic maps

16

2.1

Minimal surfaces

Suppose now that E is a minimal surface. Then

- \a , ay/

(Ox,Oy/

\ b -a

Comparison with antiholomorphic functions

A function f : Q - C is defined to be antiholomorphic, if Of

= Of

Oz

Ox

iOf _ 0. Oy

In other terms, (ax

(b a)

ay)

,

if a denotes the real part and b the imaginary part of . This bears some resemblance to (2.4). We will see that this is no coincidence. As a matter of fact, u can be seen as an antiholomorphic function in a certain way. Another point of view on antiholomorphic functions comes form the following (equivalent) definition.

Definition 2.1 A map f : St - C is antiholomorphic if and only if i) it is weakly conformal, i. e. I af12

efl2-2i(af,of Oy

Ox

Ox Oy

0,

and ii) it reverses orientation, i. e.

detl Ox ,ay Dug ( Ox i=1

\

j

9

a

ay

y

2

)

Thus we deduce that Du = 0 on U \ C. Now let a -'dx + aul dy. We remark that da = Du'dx A dy, hence da = 0 on U \C.

(4.3)

C={zEU: aZ=0}.

(4.4)

But we also have

We claim that actually da vanishes at any point zo E U. suppose that zo is interior to C. Then by (4.4) there exists a ball B(zo, r) C C, on which a = 0 and thus da = 0 on B(zo, r).

4. Working with the Hopf differential

40

suppose that zo is not interior to C. Then there exists a sequence ((n)nEN of points in U \ C which converges to zo. By (4.3), dat = 0, and since u is C2, da is continuous and thus da., = 0.

Hence or is closed on U. Thus we can locally integrate a = dg, for some real valued function g. It turns out that ul + ig is a non-constant holomorphic function of z and thus its critical set, C, is composed of isolated points. Corollary 4.1 Let u : S2 - S2 be a weakly conformal map of class C2. Then either u is constant,

or u = Q.ovoPm Oru = Q.olloPm, where Pm : S2 -p cUfool and Q, : C U {oo} -' S2 are stereographic and inverse stereographic projections respectively and v : C U {oo} - C U {oo} is a (meromorphic) rational function of z. Proof. Using Sard's theorem, we can select a point s E S2 which is a regular

value of u, then we let P, : S2 -+ C U {oo} be a stereographic projection mapping s to oo. Similarly we choose some point m such that u(m) 0 s and we let Qm : C U {oo} -+ S2 be an inverse stereographic projection such that Qm(oo) = m. Then v := P, o u o Qm is weakly conformal from C \ {al, ... , ap}

to C, where (al, ... , ap) is the counter-image of s by u o Qm. Lemma 4.1 above proves that v is either constant, or holomorphic, or antiholomorphic. We exclude the constant case and without loss of generality, we shall assume that v is holomorphic. Then, since s is a regular value of u, each point ak is a simple

pole of v and v is a meromorphic function on C which tends to a constant P,(u(m)) at infinity. Let {b1, . . . , by} be the zeroes of v and denote dl E N* the multiplicity of each b1. Then, choosing a radius R > 0 such that a1,. .. , ap, b1,. .. , by are contained in B(0, R) and computing 0 = fOB(o,R) I using residue formula, we show that E l d, = Ek=i 1 = p. Lastly we apply Liouville's theorem to the function

z v -- f (z)

(z - a,)...(z - ap)

= (z - bl)d" ... (z - bq)d9 v(z) '

to deduce that f = c = P,(u(m)) everywhere. This terminates the proof with _ d _dy Qa=P, 1,Pm=Q;' andv(z)=c zb-ai...z-ar

.

5

The Gauss-Codazzi condition

We have seen at the end of the previous chapter, that given any conformal CMC immersion X: Il - 1R3 without umbilic points, with Gauss map u:11- S2, and with H 54 0, we can find a new parametrization Y = 2HX o 0-1, such that

I _- e2w (

1

0

1)

e"' 1

sink w 0 0 cosh w

Moreover, as we will see later, w satisfies the compatibility condition

Ow +sinhwcosh w = 0.

(5.1)

There is a converse result to that. Any solution of (5.1) on a simply connected domain gives rise to a "normalized" CMC immersion. We will prove that later, by considering a more general kind of inverse problem as follows. Given

I=e2"'(1

and II=e' (Hb a

01)

Hb a

where H is a constant and a, b, w: S2 --+ R are certain functions, is there a CMC immersion that has I and II as its fundamental forms?

Gauss-Codazzi condition as a zero-curvature condition

We want to find necessary and sufficient conditions such that the question above has an affirmative answer. Suppose first that X is a CMC immersion corresponding to I and II as given above. Introduce the moving frame (el, e2, e3): SZ - SO(3, R), where dX = e"'(eldx + e2dy), and u = e3 = el x e2 is the Gauss map of X. Write 3

det

=

E A; e,j, i=1,2,3, j=1 3

dX = E A44. j=1

That is, the 1-forms Ai and Ai are the components of de; and dX, respectively, with respect to the basis (el, e2, e3) of 1R3. Remarks

i) We know already that dX = e'°(eldx + e2dy), so Al

A2 A3

1=e"'

dx dy 0

5. The Gauss-Codazzi condition

42

ii) The fact that (ea, eb) = aab implies that the matrix with components AQ is skew symmetric, i. e. Ab + Ab = 0 for 1 < a, b < 3. iii) Consider the (4 x 4)-matrix

F-_

el

e2

e3 X

0

0

0

1

Note that the part R = (el, e2, e3) of this matrix is in SO(3,R). Define furthermore the matrix

A=

0

A2

Al

A2 Ai

0

A

A2

0

0

0 0

3

Al A2

A3 0

that has 1-forms as component. Then we can compute

dF = FA.

(5.2)

We may think of F as an element of SO(3, R) x R3, and analogously of A as being in the corresponding Lie algebra so(3, R) ® R3. Note that SO(3, R) oc R3 acts on R3 in the following way. For any t; E R3,

F

(1) =

(R+X)

Thus SO(3, R) x R3 may be seen as the group of affine isometries of R3.

Lemma 5.1 Assume that X is a CMC immersion such that

I=ez"(10 0)

b Il=e2' ( H+a b H-a

1

and

f := 4

'

(6,U)2 = 4H(a - ib)e2i''.

Then

A-

1

0 *dw

-*dw -Ai e"

Al

0 A2

0

0

-A2 e" dy d27 0

0

0

0

'

(5.3)

where

e-"

(5.4) A2 = e"H dy + e-" Re(i f dz), 4H and *: T`R2 -- T*R2 denotes the Hodge operator. In particular *dw = - Ty- dx+

Ai = e"H dx +

37 dy.

4H

R.e(f dz),

5. The Gauss-Codazzi condition

43

3 and A2 3 need to be computed, the rest Prof. Only the components A2 Al,

then follows from the definition of A. Since this can be done quite analogously for Ai and A2, we will confine ourselves to computing one of them. We have

Al2 =

(-ej' a ---

ax

2 / dx +

- -(,e2)dY ay

ax \ ae2

,/

el dx +

ay ' e2

dy.

\ ae l

But since

/

0=d2X =d(e"(eldx+e2dy))=e`" I

e2- +

Oft _j 7X

aw aeI -elf/ - ay

dandy'

we see that

Thus indeed

Al

=--dx+ y

-IW

xdy=*dw.

Now compute

Ai = (del, e3) _ - (du, el) _

-e_'

du,aXax

\J = e"((H + a) dx + 6 dy)

(recall (2.2)). This can be expressed in terms of f by

Ai = e`"H dx + 4H Re(f dz),

0

which concludes the proof.

Now, take the exterior derivative of (5.2). This yields

0 = d2F = d(FA) = FdA + dF A A = F(dA + A A A). (See Section 7.1 for more details on this notation.) Since F is clearly invertible, we obtain the necessary condition

dA+AAA=0

(5.5)

for X to be a CMC immersion. Remark Think of A as a connection form, then dA + A A A is the corresponding curvature 2-form. Thus (5.5) is a zero curvature condition.

5. The Gauss-Codazzi condition

44

Converse result

We have obtained (5.5) as a consequence of (5.2). The following lemma now states a kind of converse result. The proof will be given in the appendix of this chapter. Lemma 5.2 Let S2 be a simply connected open subset of R n and xo E 0. Let A E CI (S2, T`R n ® gl(n, R)), where gl(n, R) is the Lie algebra of the linear Lie group GL(n, R) (i. e. gl(n, R) consists of all (n x n)-matrices with real components). Write A = 1 Aa dx°. Assume that

dA+AAA= aV'W. i=1

Consider the expressions rla0 = (a°u, 0u) _ (0u, a0u),

a,#? 0,

in some local coordinate z E S2 on S. Note that these are no tensors. If we have a holomorphic change of variables 0: SZ2 -' f11, where f11, S22 are open subsets of C, then rhI (for instance) transforms as follows by 0: (82(u 0 0), 0(u ° 0)) = ((02u, Ou) 0

0)(.90)1

+ ((8u., au) ° 0)(020)(00)

So we have in general 0'(700) 36 (71-0 0

However, equality holds up to expressions containing 17Q,0, with a'+/3' < a+Q. We have seen this for X21; in general, it follows by induction. Furthermore, the following holds.

6. Elementary twistor theory for harmonic maps

53

Proposition 6.1 Let U: S2 -+ S" be harmonic. Then, in all holomorphic local coordinates z, and for all a, /3 > 0, such that a + Q > 1, we have 17.0 = 0. Definition 6.1 We say that u: S2

S" is isotropic, if and only if 17,,Q = 0 for all

a,,3 > 0 satisfying a +,3 >- 1. Proof of Proposition 6.1. We will apply induction on a +,3. The claim is clear for a + Q = 1, since 7701 = 7710 = (au, u) = 0.

Assume now that -y E N is given, and that dap = 0 for all a,,3 such that

1 1, such that (a''''u, ... , Mru)(z) is of rank r. By isotropy, we have

=0, 1 0

-

Ak

+

8. Construction of finite type solutions

84

where o = (to)t + (&b is the splitting according to t ® C = t ® bR, shows that we have the Lie algebra splitting AeC = Agr ®A+

C

(8.1)

Or

Let r: Agc7 -+ Agr

and

j: AgC 7 -' A R0C

be the two accompanying projections.

Remark Agr is the Lie algebra of Asr. One can also check easily that A+ gc, is a loop Lie algebra: we shall see in paragraph 8.4 the associated loop Lie group. This will turn to be crucial in the Chapter 11.

8.3

The algorithm

Now we are ready to construct harmonic maps. First choose an odd number d E N', and set d

Adgr =

JA H a) E Agr : Ca =

k=-d

k \k

This is no longer a Lie algebra. However, it is a finite dimensional vector space. Define the two vector fields XI, X2: Adgr - Ag by (8.2)

X2(G) = a,r(i)1d-lea)] _ Note that

f

r(.\d-ll;a) =

dt + (6,,cc-SlA

11;-d,

A-l -d + 06-d)t - Ail;-d.

We will use the notation tt a, (S,\) = r(.1d-lSA), a2(G) = r(iAd-lea),

/

tt

l3l(Sa)

t = j(Ad-ISA),

Lemma 8.1

i) The vector fields XI, X2 are tangent to Adgr. ii) Moreover, their flows preserve the quantity

2

,

dl(A)

is negative definite, the flows So, for 0 = SO(n + 1) or SU(n), since of Xl and X2 stay in spheres and thus exist for all times.

8. Construction of finite type solutions

85

Proof i) We first have to check that XI and X2 take values in AgT. But note that since d - 1 is even, r is well defined and we have r(ad-ISA) E AgT for CA E AgT. Thus the same holds for XI(CA), X2 (CA) by the Lie algebra property

of AgT. To show that XI and X2 actually take values in AdgT, we use the observation (8.3). Taking the Lie bracket of one of the expressions on the right hand side of (8.3) and CA yields only powers of A between -d and d. Another way to prove this is to take into account that 01 (CA), 02 (CA) E A'

gc

for CA E AdgT. Hence the Lie bracket of CA and /31(Ca) or f32(£,\) contains only

powers of A no less than -d. Since XI(CA) and X2(CA) are real, we have even XI (CA), X2 (CA) E AdgT.

ii) This property is easy to check by a direct computation.

O

Lemma 8.2 The vector fields XI and X2 commute, i. e. [XI, X2] = 0. Proof. Since a1 and a2 are linear, we have [XI, X2](CA) =

[XI ' A,

X1 '

-t [X2 ' CA, a1(. (G)] - [CA, X2

-

(bA/)],

I \CA)] CC

KA, a2(I* A, a1(SA)])]

[CA,al([CA,a2(CA)])]-

Now use the Jacobi identity for Lie algebras to conclude [XI, X2](CA) = [CA, KA],

where KA = a2([G,a1(SA)]) - a1([SA,a2(SA)]) + [a,(CA),a2( ,,)]

We substitute a1(CA) = \d-ICA - #I(CA) and a2(G) = iAd-ICA - Q2(CA) and obtain tt / ,,)] / K,\ = -a2([bA,Ql( i)1d/-I[CA,Q1(CA)]+Q2([CA,QI(CA)])+\d-I[CA,02(A)] _

-

Q2(eA)])

- \d-I[CA,Q2(CA)]

Ql`SA)]

+ [31(CA),132(G)]

= Q2([A,QI(A)]) - Q1([A,Q2(A)]) + WI(A),Q2(A)].

A - 02(CA)]

8. Construction of finite type solutions

86

Here we use the fact that AgT and A R gC are Lie algebras: since the ranges of al, a2 and of 31, /32 are AgT and A n gC, respectively, we see from these two representations of K,,, that Ka E AgT fl A A9C = {O}.

0

Thus [X1, X2] = 0.

Now, because of Lemma 8.2, we can simultaneously integrate Xl and X2 to construct a map rla: R2 - Adg" such that

8x

'rla

ay

= XI (77,0, = X2(77A) -

We denote as := r(ad-lr7adz) and /3a := j(Ad-1,%dz). Then we have the Lax equation dr1, = [rfa, r(ad-Ir1a dx)] = [rja, aa], as can easily be seen from the definition of rla. Moreover,

(8.4)

as = A-In-ddz + 01-ddz)t + An-ddz, so we have a representation of as according to our usual decomposition. But we have also the following. Lemma 8.3 The I -forms as and /3a satisfy

daa +2[a.,Aa.\]=0,

dfa-2[0aAQa]=0. Hence Lemma 5.2 and Proposition 7.1 may be applied to prove the existence

of an Fa: fl - 0, such that dFa = Faaa, and such that ua = [F,,] is harmonic. Proof. Recall /3a = j(Ad-Itrya dz). We know that

= 0, dta - [Qa, rla] = 0.

d7ja + [aa, rla]

We may write /3a = Ad-Ii,, dz - aa. Hence daa + 2 [aa A aa] + dQa -

2

[Qa n Qa] _ -Jid- I (drla + [aa, rla]) (az) dz A dz = 0.

8. Construction of finite type solutions

87

Since we have

daa + 2 [aa A ax] E Ag, and

doa - 2

A,3,\] EAR 9c

this means that both terms vanish because Ag, fl AKgcr = {0}.

We have thus indeed constructed harmonic maps by integrating a pair of vector fields on the finite dimensional vector spaces Adg,. Such maps are named finite type harmonic maps.

8.4 Some further properties of finite type solutions i) The algorithm developed above does not only create aa, but also a 1-form /3 E T'Q ® AgC, with the property

dpa-

1[QAApa]=0.

This is-up to the sign-the property requested in lemma 5.2. So in this case the same procedure provides a map BA: Q - Arc, such that dBA = 0.\B.\,

and it is unique up to a prescribed value at some point zo E Q. We may e. g. choose BA(0) = 16. Similarly, we may assume that FA(0) = 16. (Assume 0 E S2.)

In fact, we have even pa E T' St ®AR gc, that is, 00

QA=>QkAk, QOE11R. k=0

Hence, if we define the loop group A++ MA

OC r

= {[A'-' ga] E AOC : ga = >gkAk, 9o E `Bx}, k>O

where AC = A.s$jq , we can check easily that the Lie algebra of A+ OC T is

A+ 9C. Thus we obtain Ba E A++ OC

8. Construction of finite type solutions

88

ii) Consider once more the equation drla + [aa,lla] = 0, where as = FT 1 dFA. It implies that Fa-'d(Fa11AFa 1)Fa = drla + [aa, rla] = 0.

So 77,\: = Fa1l,, Fa ' is a constant in Ag,..

iii) Furthermore, we have Ad- l rla dz = a,\ + i3a,

a fact that can also be expressed in the form Ad-1 1)a dz

1 dz = F,\ (F;- 1dFa +dB,B_ 1)F_ 1 = = d(FABA).(FABA)-'.

But this means that FABA =

ead-1(8.5)

O

iv) Write 17a in the form 11a=

e-ad-'n"= 'la ead-I

= Ba 'F;-'. 17.\ F\B,, = Ba

We know that 17,\ E Adgr, i. e. d

11A _ E 11k '1k.

k=-d On the other hand, since Ba E A*+AQST, it has the representation

Ba =

BkAk. k>O

Hence in the expression B,1riABA, only powers AA; of A with k > -d appear.

But the reality condition 7IAE Ag, implies then that in fact 1IAE Adg,.

Constant mean curvature tori are of finite type

9

Recall that CMC surfaces in R3 correspond to harmonic maps into S2 by the Gauss map, which are neither holomorphic, nor antiholomorphic. We shall see now that for CMC tori, all such harmonic maps are of finite type, a result of U. Pinkall and I. Sterling. This result can be generalized to harmonic maps from torus into Lie groups [21] or more generally into symmetric spaces [21], [22].

9.1

The result

Theorem 9.1 (681 All CMC immersions T2 - R3 are of finite type. Sketch of the proof (details are in the Appendix) i) Any immersion of the torus T2 in R3 defines a unique conformal structure on T2 and so can be seen as a conformal immersion of R2/aZ + bZ, where (a, b) is some basis of R2. Such an immersion can be lifted into a conformal one X : R2 -- R3, periodic with period vectors a and b. Now the Hopf differential of the immersion has the form f(z)(dz)2, where f is holomorphic and periodic, so is constant by Liouville theorem. Would

f be equal to zero, then X would be a totally umbilic immersion, so its image would be a subset of a round sphere Sz in R3 (see the proof of Hopf's Theorem in Chapter 4). But this is not possible 3. So we are left with the situation where f is a constant different from 0. After suitable normalizations we see that all we need to do is to work with conformal CMC immersions X: R2 -+ R3, where X is periodic (with

respect to a lattice aZ + bZ), such that H = 2, and

f =la I

2-I

I2-2z\

=-1

for the Gauss map u: R2 - S2. ii) For any A E S1 C C*, construct a "normalized framing" e2,a

0

ua Xa) 0

1

J

: R2 -* SO(3) oc R3

3assume that it would be the case: endow R2/aZ + bZ with g, the pull-back of the metric on SX by X. Then on the one hand the Gauss curvature Kg of g is positive everywhere, but on the other hand the Gauss-Bonnet formula implies fTs K9dvol9 = 0, a contradiction

9. Constant mean curvature tori are of finite type

90

of X, such that

A,\ =Fa'dF,, =

2

0 0

sinhw

-icoshw

0

0

+*dw

A

+2

- sinhw e' icoshw -iew

0

0

0

-1

1

0

0

0

0

0

0 0 0 0

0 0

0 0

0 0 0 0

(9.1)

0

0

- sinhw

e'

0

0 i cosh w 0

-icoshw

iew

0 0

0 0

sinhw 0

-*dw

0 dw

0

- sinh w dx,, - cosh w dy,,

e' dx,, e' dy,,

0

0

0

0

sinhwdxa coshwdya 0

dz

0

'

where xA = Z(A-lz + Xi) and y,\ = 2i(A-lz - \z). (Cf. Chapter 5.) We have then the associated family of CMC immersions XA with mean curvature H = z and with f = -A. The function w: R2 -+ R satisfies Ow + cosh w sinh w = 0.

(9.2)

The map FA may be seen as an application with values in the loop group ASO(3) x 1R3. Moreover let

P:=

1

0

0

1

0 0

0

0

-1

0

0

0

-1

0

and

z:

M(4, IR)

M

0 0 I'

-+ M(4, R) '-- AdPM = PMP-1.

Then z is an automorphism of ASO(3) x R3. We also remark that -r(AA) = A_A,

so that AA has his coefficients in Aso(3) ®R3 :_ {A u-+ t;, E Aso(3) ®1R3/A E S', 7-(t

9. Constant mean curvature tori are of finite type

91

This is obviously a sub-Lie algebra of Aso(3) ® R3, and the Lie algebra of

G,, E ASO(3) a R3/A E S', rr(G,,) = G_,,}.

ASO(3) a R3

We deduce that if we choose F,,(0) E ASO(3) oc Rr (for instance the iden-

tity), then Fa takes its values in ASO(3) a R. We shall denote E the set ASO(3) v R such that (9.1) is satisfied, for some of maps F,, : 1R2 function w : R2 - IIt which is a solution of (9.2) (Fa isn't required to be periodic here).

iii) Study deformations of the given F,, E E that remain within E. Suppose that the surface Xa is thereby transformed into X,, + eO,ua + o(E).

(9.3)

What conditions are to be required of t/ia such that this is still a CMC surface with H = Z? Of course the surface given by (9.3) is no longer conformaly parametrized. This problem may however be avoided by considering instead the deformation X,, + E(tlel,,\ + tae2,A + 1/1),u),) + o(c)

for suitable ti,, t2. This amounts to deforming F,, into F,, + eF,,T,, + o(e) E E, where 0

ti A

TA =

t3

A

0

t2 a

0 t2 A

0

t3 ,, t3 A

t1

d

tA1

0

to 0 1

and t3 Note that T,, is completely determined by 0,\ up to some constant (see in the Appendix). One of the compatibility conditions on 0a that we are looking for turns out to be £

+bacosh 2w=0,

1

(9.4)

miraculously (!) the linearization of (9.3) (see Theorem 9.2 in the Appendix). iv) By this deformation, w is changed into W + EW + O(E).

One can compute that 1

9. Constant mean curvature tori are of finite type

92

where Xa is some function constructed starting from 0A by the following algorithm: we let 4a be a solution of a0a

I

'OX

ox

=

4Jiw:

=

sinh2w tG

X.\

:= 4

,

(9.5)

2

az

and we set

a,

(A2 a2)2

- aw: Oa

(09

(9 . 6)

The existence of solutions to system (9.5) is guaranteed because 4(iPaw.)Z +

sink 2w (ox

= 0,

2

consequence of (9.2) and (9.4). If Xa is constructed from tPa by means of (9.5) and (9.6) we shall write t'A X.\. An important property is that if t/ia X.\, then XA is also a solution of (9.4). Note that the solution XA of (9.5) and (9.6) is unique up to some constant (which may depend on a) times w..

v) There is now an easy way to construct an infinite sequence of formal complex solutions of (9.4). Namely: we start from tPa°) = 0 and then we build recursively a sequence of solutions of (9.4) such that

(t/

nEN

t/,jn+l), Vn E N. It is possible to choose these maps in the form IVin) _ \2n-2P(n) [w], where each P(n)[w] is a polynomial in wzf wZ;, w;t;, etc ... homogeneous in yoz- of degree 2n- 1. These functions can in principle be explicitely computed, the sequence begins by

,(n)

+G(°)

7Pa)

= a-2P(°)[w] = 0, = A°P(1)[w] = wz,

(3)

[w] =

lp(4)

A4P(3)[w]

2 ' ),

= 42a4(w:z::: -

10w:z=w2

- 10w2zw: +6:

etc.

vi) After some algebraic manipulations, we construct an infinite family of infinitesimal deformations T(n'b) E TF,,E, where n E N, b E C, characterized by n

n

t(n,b),3

=

bE k=1

ka-2kp(n-k+l)[w]

+

bEkA2kP(n-k+1)[w],

k=1

9. Constant mean curvature tori are of finite type

93

and the deformation rate of w induced by T("'b) is w(n,b)

(bp(n+l) [w]

=

4

+ bP("+1) [w])

.

In particular this expression does not depend on A. vii) This is the only analysis part in this process. Assume everything is periodic, i. e. we are working on a torus. Then the linear equation (9.4), which is an eigenvalue problem for the Laplace operator, has a finite dimensional space of solutions. In particular, there exists an integer N, such that

ranks{P(1)[w],...,P(N)[w]} = ranks{P(1)[w],...,p(N+1)[wJ} = N. viii) We now choose (b1, ... , bN+1) E CN+1 \ {O} (in particular bN+1

0) such

that b1 p(1)[w] + ...+ bN+1p(N+1)[w] = 0 on R2.

(9.8)

Then we let Tao,b,)

+ ... +TAN'bN+I) It is an infinitesimal deformation in TFAE which does not vanish. Indeed the T'A :=

family (A-2P(1)[w], ... , EN,

+Ta1,b2)

kA-2kP(N-k+1)[w])

is linearly independant

and so to 3 = q,* := q,(l,b2) + ... + T(N'bN+,) does not vanish.

ix) Think and interpret : Let

Fa+eFATT+o(e)Ee be the infinitesimal deformation of Fa induced given by T,*. The deformation rate of w induced by Ta is wa°'b1) + + t:J(N,bN+') (see (9.7)). But

because of (9.8) this is just zero: the surface is not deformed at all, but just moved around in R3. So FAT.\ = DAFA

(9.9)

for some constant D.% in the Lie algebra of ASO(3) a R3. x) We conclude that Ta = Fa 1 DAFa satisfies

dTa+[A,,TaJ=0. If we write

then we have

dTa + [Aa, Ta J = 0. But by construction, this includes only a finite number of powers of A (i. e.

Ta, Aa are polynomial in A). Hence we have a finite type representation of X, as claimed.

9. Constant mean curvature tori are of finite type

94

9.2

Appendix

In this Appendix we expose the technical details of the proof of Theorem 8.1. We shall use the notations Alto

{a '-- to E IR/t_A = to},

{A'--- to E R/t_A = -tA}.

AR1

Study of the deformation of FA Given some extended frame FA E E we want to study infinitesimal deformations FE,A := FA + EFATA + o(E) where TA is a smooth map from R2 to ASO(3) x IYT and such that FE,A still belongs to E. Thus TA may be viewed as an element of TF,,E, the set of tangent vectors WE at FA. We shall denote 0

TA =

t2 A t A

to

to

t2 A

t3 , A 0

0

0

0

t1 2 A

0

tl A 0

to

and sometimes drop the A when there is no ambiguity. The question is to characterize all such TA's. In view of (9.1) the answer is simple: one considers any one-parameter deformation of w, (wE)EE(_EO,EO) with wo = w such that for any e, wE is a solution of Ow,, + sinh wE cosh wE = 0.

Then

A E'A '

0

- *dwE

-sinhwEdxA

e"°dxA

*dwE

0

- cosh w4 dyA

el, d y,\

0

0

0

0

sinhw,dxj coshw(dyA 0

0

(9 10) .

is integrable, i.e. dAE,A + AE,A A A,,A = 0 and integration of the equation dFE,A = FE,AAE,A leads to the family of deformations. Notice that if one set wE := w + Ew + o(E), then cw is a solution of the linearized sinh-Cordon equation

Ow +w cosh 2w = 0.

A crucial point - which is remarked by U. Pinkall and I. Sterling in [681 - is that one may also construct a TA in TFAE knowing its component to (modulo some constant), a map into Alto.

9. Constant mean curvature tori are of finite type

95

Theorem 9.2 a) Let T,, E TFAE, then the component to of TA is a solution of the linearized sink-Gondon equation

/ to+tocosh2w=0. b) Let

a : R2

(9.11)

AR0. Then the overdetermined linear system

4aa a

a0A

=

doa

- . sink w

az

(9.12)

2

has a solution if and only if V)A is a solution of the linearized sink-Gordon equation (9.11).

c) (A partial converse to a)) Let

A

: R2 -+ AR0 be a solution of (9.11).

such that to = A if and only there exists a

Then there exists some TA E solution 0A of (9.12), such that z

T 1 A, A) := (a az - az A)

z

+ a- az z

- a- lawa - 2lA (9.13) 1

is a function which does not depend on A. The proof of Theorem 9.2 relies on the following Lemma. Lemma 9.1 Let TA E TFAE then

the component to of TA satisfies the linearized sink-Gordon equation (9.11)

there exists a map 0A : R2 - AR1 (where AR, := JA '--- t,, E R/t_A = -tA}), which is a solution of (9.12), with ipa =\ta, such that to + itA = e"' 12A

tia+it2A=em (\ a t21 'A

-- i

2

a20a az2

-

4A)z o

a - 0A 1

e-"' I

az2

w = n(IPA, OA)-

(9.14)

,

8 - 2a) az

,

(9.15)

(9.16)

(9.17)

The proof of Lemma 9.1 occupies the end of this section. We shall first prove Theorem 9.2 assuming that Lemma 9.1 is true.

9. Constant mean curvature Lori are of finite type

96

Proof of Theorem 9.2. The property in a) is a restatement of part of Lemma 9.1. To prove b), we need to write down the integrability condition of system (9.12), i. e. that

d 14Awz a dz -

asin22w Ad-

I=0

-

(Here we use the subscript w., for 0, etc... ). Using wZZ + s sinh 2w = 0, this simplifies into

/

Aw,z 1 4 z a 19

+ V i a cosh 2w'1 &Z- A dz = 0,

-z-

which is clearly equivalent to (9.11). Lastly for proving c) let us consider some i/ia, which is a solution of (9.11). Then Let us assume first that there exists a Ta E TFAe such that to

the infinitesimal deformation w of w induced by Ta needs to be independent of A. But because of (9.17), in Lemma 9.1, it means exactly that n(o.\, 0.\) is independent of A, for a suitable choice of 4a. Conversely if there exists some solution 0,\ of (9.12), such that I (th, 0,0 is independent of A, then formulas (9.14), (9.15) and (9.16) defines a map TA. A long computation (which basically uses the material of the proof of Lemma 9.1) 0 shows that actually Ta E TFE. Proof of Lemma 9.1 Step 1: Writing the relations satisfied by Ta One defines FE,a := Fa + FATA + o(e) E e and one writes that Ff.a dFE, a = AE,a,

9.18)

where A,,a has been defined in (9.10). The left hand side of (9.18) is

F-'dF(a = Aa +e(dT\ +

+o(e),

where dTa + [Aa, TA] = 0

dti _t2 sinhwdx,, +t3coshwdya dti-t2*dw+t1coshwdya 0

dt2 - t2 sinhwdxa + t 1 coshwdya 0

dtz+t1*dw+t2sinhwdxa 0

0

dt'-t2*dw-t3sinhwdx, -e"'t2dyA dt2+t'*dw-t3coshwdya-e"'tidxa dt3+t sinhwdxa+t2coshwdya-e"'(tidxa+tzdya)

0

0

dt3+t2*dw+t2coshwdya

dt3-t1*dw+tIsinhwdxa

9. Constant mean curvature tori are of finite type

97

Whereas the right hand side of (9.18) is

Aa +,E

0

- * dth

*dw

0

-wcoshwdxa we'dxa

-ii sinh wdya

c;,e"dye,

0 0

0 0

wcoshwdxa wsinhwdya 0

0

+ O(E).

Thus we are led to the following system of equations.

dt' -

- t3 sinh wdxa - e"t2dya = we"dxa, dt2 + t' * dw - t3 cosh wdya + e't2dxa = we"dya, t2 * dw

(9.19) (9.20)

dti - t2 *&A) - t2 coshwdya = wcosh wdxa,

(9.21)

dt2 + t * dw + t2 sinh wdxa = c;j sinh wdya,

(9.22)

1

dt2 - t2 sinhwdxa + ti coshwdya + *d = 0,

dt3 + t' sinh wdx), + t2 cosh wdya - e"(t1dxa + t2dya) = 0.

(9.23) (9.24)

Step 2: Useful relations Relation (9.24) allows to express ti and t2 in terms of t', t2 and t3: 3

e

e

a + t'sinhw a

(9.25)

3

+ t2

e

w

a

Furthermore, relations (9.19), (9.20) and (9.21) can be replaced by their

respective projections along 0 := a8 +)-' j and 0 := iA - i) ' &:

a

i a

+ t2 8y - t3 sinh w = we",

8t'

8w = t2e t2 8xa Oia 0t2

8xa 2

ay-,\

+ t'

- t' fla 8w

axa

(9.27)

-4e",

(9.28)

- t3 cosh w = we",

(9.29)

_ , cosh w,

(9.30)

' +t3

8xa

_

(9.26)

ava

9. Constant mean curvature tori are of finite type

98

3

aw t3 a= t2 cosh w, xa aya 1

at3e

- t;

axa

aya

(

-4sinhw.

.

(9.32)

= w si n h w,

aya + t1

9 31 )

(

9 33 ) .

Lastly, two further relations can be deduced from the four first ones of the previous series: substracting (9.26) - (9.29) gives 8(e-Wtl)

- 8(e "t2) _ -t se -2"

,

( 9. 34 )

and (9.27) + (9.28) gives

a(e-"t') aya

+

a(e-" t2) ax.\

= 0.

(9.35)

Step 3: We show that t3 is a solution of the linearized sink-Gordon equation Suming (9.30) + (9.33) gives we2w

= a(e"'t1) + a(e°'t2) axa

aya

which, using (9.25) to eliminate ti and t2, leads to

wee" = Ot3 +

a(t' sinh w) + a(t2 cosh w) aya axa

Now (9.26) + (9.29) gives CJe2w =

1

(O(ewt') + a(e"'t2) _ t3e2,,,) axa aya /I

These two relations implies by eliminating cwe2w

a(a t')

e1, =

Qt3 + 2

2 C

_ a(e-" t2)

a

Oya

/

And by using (9.34) one gets Ot3 + t3 cosh w = 0.

(9.36)

9. Constant mean curvature tori are of finite type

99

Step .¢: Equations on t 1 and t2 We sum/ - sinhw (9.26) + cosh w\(9.29) to obtain

- t3 sinh w I + cosh w I

- sinh w I a I + t2

+t1

and substracting e" (9.30) - e"(9.33) gives 3

ew

23

+t3l 52J1

'

- t3 cosh wJ

awl-ew(atl+t3aw,=-f.1.

2axA

/

A

Now we sum both equations and we use (9.25) in order to eliminate ti and t2. It leads to 2

a2

coshw - 2

W\

a1

sinh w - 2t2

ax,\ 82t3

-

+

(

cosh w + 2t1

A)2

Ow oxA

sinh w

+2---2= t3. aw 0t3

92t3

(OXA)2

c .1

C

gXA axA

v at3 A

A

A computation using (9.34) gives a(e-mtl)

ax)

a(e-"t2)

+

eyA

- 2w -e

a

a

at3)

-20

5X A \

OXA /

A

3

-2w

(e8Y,\

(9.37)

Similarly the combination - sinh w (9.27) - /cosh w (9.28) gives - sinh w I

-A

- t2 5x-,\

I - cosh w I

A

- tl

= t2,

and the sum ew (9.31) + ew (9.32) gives 3 -e" (at1 - t2

- r1\ 9x,\ - t1 aXA ) 3 2

OW

) W\ We sum these two relations and use (9.25) to obtain A

2

- aa -

_ -t2.

1

cosh w t3

axA O YA

v

+ aw

sinh w + t2 at3

+ aw

cosh w + t 1 at3

v

sinh w

=0.

a XA atA

a XA a XA

Then a computation using (9.35) gives

a(e wtl)

A-

a(e wt2) = e2e2 OXA

a 8XA

e -2w

0t3 OYA)

+ a OYA

e -2o

t3

(9.38)

a-A / J

9. Constant mean curvature tori are of finite type

100

We can conclude this step by the following equations which summarize relations (9.34), (9.35), (9.37) and (9.38).

8(e-"(tl + it2)) =

2

8zA

'

8(e-"(tl + it2))

where oia = w and We shall pose 2

3

8za

- 4 8w 8t3

82t3

8za

-

(9 39)

8ZA 8za'

(8ZA)2

e-2" 0 '

(9.40)

2

=A-la

- e-"(tl + it2)

. e-"(tl + it2) = 2

3

8za

- 0,\.

(9.41)

Then relations (9.39) and (9.40) allows to express 4a in terms of t3 (also using (9.36)): e46a 09Z,\

'90,\

=2

82t3 (8z,,)2 a2t3

=2

&. 8t3 - aza a (e-"(tl + it2)) = 4 8za aza

(9.42)

-

(9.43)

a (e-"(tl + it2)) = -sinhwt3.

8za 8za 8z-.\8za Step 5: Computation of cv Suming (9.26) + (9.29) we obtain e2"(2w + t3) =

2

8(e"(tl + it2)) + a(e"(tl + it2)) 8za

8za

And using identities like

8(e"(tl + it2)) = e2" 8(e-"(tl + it2)) 8za

8za

+ 2e 2" 8w (e-"(tl 8z,\

we get

W=-2+I +

ew

Ca

(e-"(tl +it2))

(e-"(tl - it2)) +

8tz

1

2

+ it2)),

it2))

1 8(e-"(tl - it2)) 2

8x

Now we use (9.41) in order to obtain a formula which relates w in function of t3 and 0a. Simplifications using (9.42) and (9.43) occur and give t3

02t3

- 2 + I\ ((OZa)2

We recognize the relation 9.2 by formula (9.13).

Ow

aZa

)

( (92t3

Ow

OX J

= fl(t3, 0a) where fl has been defined in Theorem

9. Constant mean curvature tori are of finite type

101

Step 6: Conclusion The map ¢a which was introduced through the relation (9.41) is actually completely determined from t/ia := to by relations (9.42) and (9.43), up to a constant. Then the relation (9.41) implies easily relation (9.14). And using (9.25) one deduces (9.15). Lastly one gets from (9.27) and (9.28) that 2O(e"'(t' e-2"' a(e"'t') _ 8(e"t2) \ I + it2)) 2 ( ayA = -e'IM t2 , aza axa s from which we deduce (9.16).

A Biicklund transformation of solutions of the linearized sinh-Gordon equation

In the previous subsection we have seen that for any ipa : R2 - ARo which is a solution of Ozpa + a cosh 2w = 0,

(9.44)

there exists a solution 0x : R2 -. ARI of a4

Ua oft

= 4X8 ft,\ stinh =

-

2

(9.45) 0x

Obviously 4x is unique up to a constant in ARI.

It will be useful to extend our setting to maps 0a : R2 -p ACo := {A '--+ to E C/t_a = t,\} which are solutions of (9.44). Then there exist solutions 0a : R2 --- AC1 := {A' -+ tX E C/t_a = -t,\} of (9.45). Definition 9.1 Let ika : R2 -- + ACo be a solution of (9.44) and Xx : R2 -. ACp. We write zpa

Xa if and only if there exists a solution 4

: R2 - ACI of

(9.45) such that Xa = 4 (X If zpa

2

(O z)2 a2 'A

A8Bzw4 1

XO, the set of maps XX such that zpa = Xa is exactly {X0 +CAwZ/C,, E

ACo } .

This definition is motivated by Lemma 9.1: indeed relation (9.17) can be restated as

2zGa,

for zpA = t3 and zp,,

X,, 4. Proposition 9.1 below leads to another charateri-

zation of Xa. 4 similarly relation (9.16) can be written as t2',.\

=4Xa-4

9. Constant mean curvature tori are of finite type

102

Lemma 9.2 If zya : R2 -p ACo is a solution of (9.44) and zlia = Xa, then Xa is also a solution of (9.44). Proof. It can be checked through a direct computation. Alternatively it follows from Lemma 9.1, since cw and to are both solutions of (9.44) and because of 0 (9.17). Lastly it is also a consequence of the Proposition below. Proposition 9.1 Let V)a : R2 -+ ACo is a solution of (9.44). Then the set of

maps Xa : R2 - AC0 for which there exists two maps aa, ba : it2 -+ AC such that

8

as

e

cosh w -i sinh w i sinh w cosh w i sinh w cosh w

8z

-isinhw

ba

i

ba

as

coshes

coincides with the set of Xa :

_ Ana

l

-i sinh 2w )

\

=

cosh 2w J i sinh 2w cosh 2w

2 -+ ACo such that z/ia

(9.46)

)

Xa.

Proof. We shall denote

( wa)

(

i sinh w

cosh w) (

(

(

i sinh w

cosh w

b,\

)

and

)

-

)(

ba

Step 1: Uniqueness of Xa modulo w,z

Since the system (9.46) is linear in 0a and Xa, it suffices to study the solutions of the system (9.46) for z/ia = 0. This reads 8

8 8z

(

= 0

V-\

0 cosh 2w

-i sinh 2w

wa i sinh 2w cosh 2w

v,\

) (wa)

_

- X-1 Xa

i sinh 2w cosh 2w

The first equation means that va and wa are holomorphic. The second equation gives

) (i sinh 2w) ('

(8zua

l az -

- Xa 1

2zvx

l cosh 2w

i sinh 2w cosh 2w

+

Oz

+ 2xwZwa

)(

cosh 2w

-i sinh 2w

9. Constant mean curvature tori are of finite type Since

U cosh 2w

,

-isinh2w

( i sinh 2w

t

cosh 2w

8w.\ 8x

)

103

l is a basis, this is equivalent to

- 2iwzva +

= 0

8Z- + 2iwzwa

=

0.

Assume that wa 36 0. Then since va and wj, are holomorphic, the second equation would imply that wZ is meromorphic. This is however impossible (excepted in the case where w - 0) because (w=) = - s sinh 2w 36 0. Thus we necessarily have wA - 0 and the second relation implies * = 0. Hence va is equal to a constant Ca E AC. Now the first equation gives Xa = 2iACAwZ. X Step 2: Existence of a solution to system (9.46) as soon as ?lia

We let 0a to be the solution of (9.45) such that X = 4 (\2 820

- a ¢a) and

we set W'\

4A

&Z

Then using (9.45) and (9.44) we obtain

8

Xa (

i sinh 2w cosh 2w

i sinh 2w cosh 2w

) ( wx )

va

(wa

=

We then find that

8

cosh 2w

8z

-i sinh 2w

- -4,\_1

A282OX

(

_

v',

)

(8z)2

i sinh 2w cosh 2w

0

and the conclusion follows.

Corollary 9.1 If /a : R2 -+ ACo and X : R2 (9.44), then 0a = X,, if and only if 3 T\.

ACo are two solutions of

Proof. The system (9.46) is invariant by complex conjugation provided one changes (ip.\ XA) into (-5F\, (and (aA, bA) (aa, ba)), so the result fol-

0

lows.

An algorithm to produce solutions to (9.44)

An obvious solution to (9.44) is wz. By Lemma 9.2 all maps X,, such that XA are also solutions of (9.44). A computation shows that such of the form

wZ

4,\2(W,

:: - 2wZ) + CAw2.

are

9. Constant mean curvature tori are of finite type

104

We may repeat this computation, finding that 2L,,3)

42A4(wzzzzz

- 10wzzzwZ - 10wZZw2 +6wZ),

etc... We can hence generate a unique sequence (Oan))tEN defined by ?

(O)

_ -2P(O)[w] := 0 P(1)[w] :_ Wz

Vn E N,

titan)

,P (n+1)

_'\2n-2P(n)[w] where P(n)[w] is a polynomial in Vn E N, O(n) C[wz, wz2, ...] homogeneous in of degree 2n - 1

The last assumption means that

aa'w Qj) ((OZ)ai

#'

/

...

8a,w 13,

((8z)aj)

,

where Cn (a l , ... , aj, Q1 i ... , Qj) = 0 if a1,01 + + aj,Oj 34 2n + 1. It can be deduced from an algebraic recursion formula which is proved in [68], Proposition

3.1, p. 149. An immediate consequence of Corollary 9.1 is that 'tan+1)

J1n), Vn E N.

Algebraic construction of a series of infinitesimal deformations in TFAE We wish to construct families of solutions zJ' : R2 - Alto of (9.44) such that there exist 0a : R2 -' AC solving (9.45) and such that St(t/ia, a) is independent of ) . 4 (23 - AWOa), this amounts to find pairs (tk.\, Xa) By denoting Xa ik,\ of solutions of (9.44) such that Xa and Xa + a Xa - ipa does not depend a z on X We look for maps t p,\ of the form n

E A-2kakP(n-k+1)[w] + [>2kQkP(n-k+1)[w]' kk=O

L

k=O

9. Constant mean curvature tori are of finite type

105

where ao, ..., an are complex constants to be determined. Note that this ansatz garantees that 1a is real valued. The maps Xa such that t/ Xa are n

n

X,\ = CAP(1)[w] + A2 E a-2kak p(n-k+2)[w) + X2 r` 2kak p(n-k)[wJ, k=0

l

where Ca E AC and so 1

1 (Xa + Xa

n-1

n

E A-2k+2 ak p(n-k+2) [w] + E X2k+2akP(n-k)[w] + CaP(1)(w]

4 n

k=O

Lk=O

A2k-2T-k+2)[w]

k=O

n-1 + > A-2k-2akP(n . k)[w] + CaP(1)[w] k=O

n X-2kakp(n-k+1) [wJ

-2 >2

n

- 2 r A2kak p(n-k+1) [w] k=O

k=O

- 2an)P(1)[,] +C,\P(1)[w] +,\-2n+2(an + an-2 - 2an-1)P(2)[w] L,\_2n(a,1_i

4

+a-2(a2 + ao - 2a1)P(n) [w1 + X-2a0P(n+2) [w) +(a1 - 2ao)P(n+1)[w] + (a1 - 2a-o) p(n+ 1) [W]

+.\2(a2 + a.0 - 2a1)P(n)[w] +.\2ap(n+2)[w]

+.12n-2(an-2 + an - n_1)P(2)[w] + A2n(an-1 - 2an)P(1)[w] +CaP(1)[w], aZ

.

It turns out that this quantity is independent of A if and only if we choose

ak=kb, VkENand Ca=(n+1)bA-2n+c for some constants b, c E C. We will choose c = 0. Then we are left with 4 (Xa + Xa - 2t(ia) = 4 (bp(n+1)[w] + hP(n+1)[w]) Sin genera) for c # 0,

(x + 4

-2i'A) = 4 (bP(n+1)[w)+6P(n+1)[wJ+cP(1)[w] +"cP(1)[w))

.

9. Constant mean curvature tori are of finite type

106

We conclude: for any it E N and b E C we define the neaps 41(n'b)

:= b> kA-2k p(n-k+1)[

n ]

+bEkA2kP(h+-k+l)[w],

k=1

k=1

Then there exists a unique snap an'b) such that (9.45) holds and such that IP(n.b),4,(n.b))

= 4 (bP(n+1)jw] +

and so in particular is independent of A. By Theorem 9.2 there exists a unique E T,.- E such that t(n.b).3 = *(n,b) and W(n.b) = fg p(n,b)' 4,(n,b)). T("") is defined by the formulas in Lemma 9.1. Examples a) For W(A'b) = 0, where b E C, we have (O'b) _ _ LA- I t

(O'b)

4

= lxdZ

and SZ(q,(D'b), 4 (O.b))

t(O, ) ' 1

b) For

q,al'b)

_

(bw, + & ) ,

= t (bu - boo .

= A-2bwz + A2bwZ,

where b E C, we have

A -:+2,\-I tx,,z -

.6)

2

and

w(I.b) _ t(t b),1 2.a

2bA-2w2

4 (1.b)) =

A31) cosh 2w,

+ 4b(ws,z-- - w3),

b(wzss - w3) + b(wzsa - wi)+

= ibA-2wZ + ib(wzzz - wz3) - ib(w-.:z - w3) - ib A2wz. 2

2

10 Wente tori Historically the first immersed constant mean curvature tori were constructed by H. Wente in 1984 [86] by analytical methods. A short time later U. Abresch simplified this construction. He remarked that Wente tori should possess planar curvature lines and thus studied all CMC surfaces with planar curvature lines. It leads to an overdetermined system of equations which can be solved by quadratures using elliptic integrals. And U. Abresch showed that some of the obtained immersed surfaces do close up, giving CMC tori [1] (see also [87]). A similar treatment was given in [84], [85] by R. Walter who remarked that each curvature line of these tori is either planar or spherical. Moreover, as pointed out in [66] special surfaces (including constant mean curvature ones) with planar or spherical curvature lines were already investigated by A. Enneper [37], [38], [39] and his students [63], [15], [83], (see also [27]), [28], [61], [3], in the last century. Explicit formulas were then produced.

10.1 CMC surfaces with planar curvature lines We again look at conformal CMC inmmersions X : R2 - R3 such that H = 1 /2 and such that their Gauss map u : R2 --0 S2 satisfies

Then the curvature lines are the image of the curves [x'-- X (x, y)] (small curvature lines) and [y a- X(x, y)] (large curvature lines). Let us focus at small curvature lines as an instance. Their Fhenet framing (tl, nl, 91) is defined by

=

nl

I-I8t1 11 'Ox

bl

ax

= E1 xn1.

The curvature and the torsion along these lines are defined by rcl := a

8x

,

rl :_ -e

ax ,

nl

/

108

10. Wente tori

Computing these datas, one finds that

tl

= et

(,,U, ) 2

e-"

fGl

ay

+ sinh2 w

Kln1

8w = e-" ( -a y e2+sinhwe3

K161

= e -" (-sinhwe2 -

gi2 rl

82w = e-3" Csinhw axay -

e3 coshw8axw 88yw)

The small curvature lines are planar if and only if the torsion rl vanishes identically. This is clearly equivalent to the condition 2

sinhw

wOW

- cosh

OW = 0.

(10.1)

Similarly the large curvature lines are planar if and only if cosh w& W

- sinhw

= 0.

(10.2)

U. Abresch showed that choosing one the two above conditions leads to an overdetermined system of equations which can be solved explicitely using elliptic integrals. This is based on the observation that for instance (10.1) can be restated as 8x (;17nhw Oy) 8y (si hw a 0' which implies that one may find two functions f and g of one real variable such

that 1

OW

I

Ow

sinhw 8x -f (x), sinhw 8y -g(y). Thus the geometrical hypothesis (10.1) leads actually to a separation of variables. Using these new variables together with the equation Ow + cosh w sinh w = 0,

one can then show by a tedious computation that f and g are solutions of

(f')2=f4+(1+a2-,02)f2+a2, (g )2 = g4 + (1 - a2 + /32)g2 + Q2, for some constants a and 6. Thus f and g can be given using elliptic integrals. A similar conclusion holds if one assumes (10.2) instead of (10.1). However only CMC immersions with planar small curvature lines can close up into immersed compact tori.

10. Wente tori

109

10.2 A system of commuting ordinary equations In a subsequent work [2], U. Abresch proved that the above solutions arise from commuting flows. Namely we consider R6 with the coordinates c

s

U:=

U1

U2 U3 U4

l

and we define on R6 the two vector fields sue

sui

I

XI

(U)

Cui

CU2

%F2

su3 - C8

2

CU4

X2(U) :_

V`

CU4

2

-8u3 - C8

-8ul - U2U4

8U2 + U1U4

-CU2 + U2U3

-CU1 - U1U3

One can check that XI and X3 commute by a direct computation. Moreover the quantities C2 - 82 and s2 + u + u2 + u3 + u4 are preserved by X 1 and X2 so their flows remain in a compactisubset of R6 and thus are defined for all time. So let us take any solution U : R2 R6 of

8U = Xi (U),

and

= X2(U)

(10.3)

with initial conditions (at (x, y) = 0) such that c2(0)

- 82(0) = 1.

(10.4)

Then since the quantity c2 - 82 is preserved by the flow, we can pose c = cosh w

and s = sinhw, where w is a function on R2. Equations (10.3) can then be rewritten as wa

8x =

1

i72

(u, - 2U2),

- i coshwu4), (a )2 = 4 (sinhwu3

iz = -coshw sinhw.

(10.5) (10.6) (10.7)

We see hence that w is a solution of the sinh-Gordon equation. So we can associate conformal CMC immersions to each solution of (10.3) with the initial condition (10.4). These immersions turn to enjoy two properties:

10. Wente tori

110

they contains all conformal CMC immersions with planar small curvature lines or planar large curvature lines (see [2]) and in particular "Wente" tori (actually (10.1) is equivalent in choosing a2 = 0 in (10.9) below) they are of finite type, as we will see in the next paragraph

10.3

Recovering a finite type solution

We analyze the conformal CMC immersions obtained by integrating (10.3) with the initial condition (10.4). We can first observe (as in [2]) that the following quantities are conserved along the flow of Xl and X2:

a1 := cu3 - 2 (ui - u2),

(10.8)

a2 := su4 - u1u2. (10.9) This is proved by a straightforward computation. We take a solution of (10.3)

with c2 - s2 = 1, so that we may pose c = coshw and s = sinhw. Then all higher derivatives of w can be expressed in terms of algebraic functions of U: the first relations are precisely (10.5), (10.6) and (10.7). The next one consists in computing 7"37'. Using first (10.6) and (10.3) (a

=8

)3

=

8

i9 x 1

-i

) (8Zl3 - icil4) .9y

(ul - iu2)(1 + 2cu3 - i2su4).

And now, by (10.8) and (10.9), "13W

(az)

=

3

1 W2

(ul - iu2)(1 + 2al - i2a2 + (ul - iu2)2)

1 + 2a - i2a2 w 4

z

+2

I

wz I 3

where we have used (10.5) in the last line. We thus conclude that 4.1

2

03W

(0z)3

_

raw\ 3 2 1\

8z

11

_ J2(1 + 2al - i2a2) 8w =

8Z

,

.12(1 + 2a1 - i2a2)w,t7 which is e., with the notations of Chapter 9, W'.' 0. In the light of the analysis of Chapter 9, it just means that w is a solution of finite type. In particular the relevant deformation T of the surface (such that property (9.9) holds) can be chosen so that i.

equivalent to w,,

ta'3

=

-2wz + '\2Gl1 .

10. Wente tori

10.4

111

Spectral curves

Any finite type solution is characterized by the existence of a map 77a : IR2 Adg, which is a stationary solution of (8.4): dpi + [aa, pa] = 0. Assuming that ,p, and as are matrices we can make sense of the characteristic determinant P(A,,u) := det(tA - Al).

Then equation (8.4) just implies that this polynomial does not depend on z. By compaetifying the complex algebraic curve of C2 of equation P(A, µ) = 0, we obtain a Riemann surface called the spectral curve of the finite type solution. Lastly it is possible to use tools from algebraic geometry (the Abel-Jacobi integrals and the Jacobi torus of the spectral curve, see [32], [19]) to produce expressions of the conformal CMC immersions in terms of theta functions. This has been carried out by A. Bobenko for surfaces in R3, S3 and H3 [11], [12] (see also [13]).

In particular Wente tori correspond to genus 2 spectral curves invariant by a complex involution [13].

Weierstrass type representations

11

The theory for finite type solutions developped in Chapter 8 can be generalized in order to represent all harmonic maps from a simply connected surface to symmetric spaces like the sphere S2. This has been developped by J. Dorfineister, F. Pedit and H.Y. Wu and leads to a Weierstrass type representation [30]. ead_`17AZ

The main idea is to try to generalize property (8.5) FaBa = which was found for all finite type solutions (see paragraph 8.4), by substituting the right hand side of (8.5) by a suitable holomorphic function of z with values in a complexified loop group. One then needs to find a way to split this holomorphic function into a product of the form FaBa, i.e. we need a non linear analog of the Lie algebra splitting (8.1). Such a splitting does exist and can actually be interpreted as an extension to loop groups of the Iwasawa decomposition described in paragraph 8.1. We will present here briefly this theory. We will be concerned with harmonic

maps u from a simply connected domain 12 C R2 into a symmetric manifold

s'3/A. We assume that there exists an automorphism r : 0 -- 0 which is an involution (r2 = 11) such that, if Or := {g E 0 : r(g) = g}, (fr)o C . C a,.. 11.1

Loop groups decompositions

We use here the notations AO, AOc concerning loop groups and twisted loop groups introduced in paragraph 7.2. We also will need the loop group A+ OC defined in paragraph 8.4. We assume that . is a compact semi-simple group, so that the following Iwasawa decomposition holds: .q xB.R

.

gb

(g,b)

is a diffeomorphism. We summarize by AC = A.BR this property. Then the first tool is an extension of this decomposition to the loop group AOT, namely Lemma 11.1 Assume that 0 is a compact semisimple Lie group. Then the mapping

AO, x ABA OC - ABC, (ga, ba)

'-'

gaba

is a diffeomorphism. We denote by AOC = AO,.A

,q

OC this property.

11. Weierstrass type representations

113

We also introduce the subgroups

A+Oc := {[A - a] E AOT extending holomorphically in the disk D2},

A OC := {[A'- a] E AOC extending holomorphically in C U {oo}\D2 and We then have the following.

Lemma 11.2 Assume that 0 is a semisimple Lie group. Then there exists a dense open subset C of the connected component of the identity of AOC, called the big cell, such that the mapping

A* OC x A+Oc -+

C

is a diffeomorphism. We denote by C = At 0c7 A+Oc 7 this property.

These two results are showed in [30]. The proofs are based on similar decomposition results for non twisted loop groups proved in [70].

11.2

Solutions in terms of holomorphic data

Extended lifts of harmonic maps are in correspondance with holomorphic datas as defined below. We first denote A-l,oogC

l;k'\k

E

k=-1

is the Definition 11.1 The set of holomorphic potentials, denoted set of holomorphic 1-forms on fl with values in A_1,,,,,gc. So any form IA., in has the expression 00 JIa = E 00 ftkAk = k k(z)Akdz,

k=-1

k=-1

where Vz E f2, rk -1 lk(z).\k E A-1,oogC Lemma 11.3 Let F,, : ft -+ Ae7 be the extended lift of a harmonic map u : fZ -- O/11 and assume that ft is contractible. Then

there exist a holomorphic map H,,

:

f2 - AOC and a map B,,

:

fl

A++ OC such that F,, = H,, B,, .

the Maurer-Cartan form µa := (Ha)-1dH,, is a holomorphic potential.

11. Weierstrass type representations

114

Proof. (see [301 for details) The existence of Ha and Ba relies on solving the equation 0 = a(FAB

1)

= Fa

(a,\ (') - BA

) (BA)-1,

1

which is equivalent to

/

1

= Ba(ao + Aa1 +'\2 a2) 1 YJ

,

with the constraint that Ba takes values in A+ Oc.\The existence of a solution is first obtained locally, then we can glue local solutions into a global one. This proves the first assertion. Now we write (Ha)-1dHa = B,\ (a,\ - Ba 1dBa)Ba 1,

and using the fact that Ba takes values in A++ OC and that z '- HA(z) is holomorphic, we deduce that µa := Ha 1dHa has the desired properties. Conversely any holomorphic potential in 11_1,,o(Q) produces harmonic maps as follows. po Theorem 11.1 Assume that 0 is compact semi-simple. Let µa E a point in 1 and Ha a constant in AOC. Then there exists a unique holomorphic map Ha : S2 -y AOC, such that dHa = H,\µ,\ and HA(po) = H.

we can apply the loop groups decomposition AOC = AO,.A A95C to HA(z)

for all value of z. It follows that there exist two maps Fa : SZ - As, and Ba : 11 -- A+

OC such that HA(z) = FA(z)BA(z),

`dz E Q.

Then Fa is an extended lift of a harmonic map SZ - 01A. Proof. Since µa = t:adz, with Pok = 0, it follows easily that dµ,\ + µa A µa = 0, hence the existence and the uniqueness of Ha. Using now Lemma 11.1 in order to perform the decomposition Ha = FABA, we obtain (11.1) (FA)-1dFa = Baµa(BA)-1 - dB.(BA)-1. Now using the fact that µa E 7{_1,.(ft) and Ba takes value in A+ Oc, it is easy to check that the right hand side of (11.1) has the form Ek _1 akAk. But (11.1) implies also that this quantity should be real, i.e. a 1-form with coefficients in + &0 + &1a and moreover AC,. Hence as := F-'dFA reduces to as Lastly a Taylor expansion in A of (11.1) proves that &o is real and &1 &_1 is a (1,0)-form, which ensures the result by Chapter 7.

11. Weierstrass type representations

11.3

115

Meromorphic potentials

The holomorphic potentials constructed in Lemma 11.3 are far from being unique. Moreover they involve in general infinitely many holomorphic maps. These defects can be mended, provided we allow meromorphic potentials and under some hypotheses on 0. We define A-19C := {[a -0, ] E A9c/Ca =

-is-11.

Definition 11.2 The set of meromorphic potentials, denoted M-1 (11), is the set of meromorphic 1-forms on SZ with coefficients in A_1gC,. So any form µa in M_1(1l) has the expression

µa = Fl-la-1 =

-1a-1(z)dz,

where t;-1(z)a-1 E A_l.oogc,.

Then using the same methods as in [30]; one can prove the following

Theorem 11.2 Assume that 0 is semi-simple. Let F., : 11 - AO, be the extended lift of a harmonic map into 01.tiv. Then there exists a finite subset {a1, ... , ap} of fl such that there exists a holomorphic map Fa : fl \ {a1 i ... , aP} -- A,-,Oc and a map Fa : SZ \ {a1i ... , ap} - A+Oc such that

F,,(z) = Fa (z)Fa (z),

dz E fZ \ {ai, ... , ap}

z I.-# Fa (z) extends to a meromorphic map on fl

the Maurer-Cartan form µa := (Fa)-1dF, of F;- is a meromorphic potential in M_1(fZ). Proof. (see [30] for details) The decomposition F,,(z) = Fa (z)F, (z) is possible as soon as we can prove that F,(z) belongs to the big cell C. Using Lemma 11.3

in the same way as in [30], one can show that this is true for all z, excepted maybe on a finite subset {ai, .... ap} CO. The second property is proved also in [30]. The last one follows easily by writing

µa=Fa

[aa-(Fa)-1dFa}(Fa)-1

which implies on one hand that u,\ is in

once one keep

in mind the fact that F' (z) E But on the other hand F; -(z) E A* OC and thus there is no nonnegative power of A in the Fourier expansion of µ,,. A+Oc.

This implies the conclusion.

0

11. Weierstrass type representations

116

11.4

Generalizations

This theory can be extended to other two-dimensional differential geometric situations. An instance is the Willmore surface problem [881: it consists in looking at surfaces E immersed in R3 which are critical points of the functional

W[EJ := fE H2dA, where H is the mean curvature along E and dA is the area element induced by the first fundamental form of the immersion. These critical points are named Willmore surfaces and they satisfy the order four Euler-Lagrange equation AEH + 2H(H2 - K) = 0. This problem was actually considered at the beginning of the century by G. Thomsen (se [81J, [9]). Willmore surfaces may be thought as an analog of the minimal surface problem in conformal geometry and turns to be completely integrable. A Weierstrass representation theory in the spirit of the above theory was constructed in 1491 (see also [501).

Another instance is the study of Hamiltonian stationary Lagrangian surfaces in a four-dimensional Kidder manifold. They are Lagrangian surfaces which are critical points of the area functional under Hamiltonian flow deformations. The Lagrangian constraint and the restriction on the testing deformation vector fields

lead to an order three Euler-Lagrange equation. In cases where the ambient space is a symmetric Ki hler manifold, it can he shown that this is a completely integrable system and a Weierstrass type representation can also be built [511, [521, [531.

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Frederic Helein Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems

This book intends to give an introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. The presentation is accessible to undergraduate and graduate students in mathematics but will also be useful to researchers. It is among the first textbooks about integrable systems, their interplay with harmonic maps and the use of loop groups, and it presents the theory, for the first time, from the point of view of a differential geometer The most important results are exposed with complete proofs (except for the last two chapters, which require a minimal knowledge from the reader). Some proofs have been completely rewritten with the objective, in particular, to clarify the relation between finite mean curvature tori, Wente tore and the loop group approach - an aspect largely neglected in the literature. The book helps the reader to access the ideas of the theory and to acquire a unified perspective of the subject.

783764 365769

ISBN 3-7643-6576-5

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