WILLMORE SURFACES

JOEL PERSSON

Master’s thesis

2003:E9

Centre for Mathematical Sciences

Mathematics

CE

NT

RU

MSC

IEN

TIA

RU

MM

AT

HE

MA

TIC

AR

UM

Abstract

The main aim of this Master’s dissertation is to investigate the interest-ing connection between minimal surfaces in the 3-dimensional sphere S3 andWillmore surfaces in the Euclidean space E3. Willmore surfaces S in E3 arestationary immersions for the so called Willmore functionalW = ZS H 2 � dA;where H is the mean curvature of S . These can be characterized as the solu-tions of the heavily non-linear partial differential equationDH + 2H (H 2 � K ) = 0;where K is the Gaussian curvature of S . Special cases of Willmore surfaces arethe minimal surfaces with H � 0.

We build up the necessary technical tools to deal with the problem. Weprove that the Willmore functional is conformally invariant. Using calculus ofvariation we deduce the Euler-Lagrange equation for the problem. Finally weshow that the problem actually has a solution on compact oriented surfaces ofany genus g � 0:

It has been my firm intention throughout this work to give references to statedresults. Any statement or proof without references is considered to be too well-knownfor a reference to be given.

1

Acknowledgements

I would like to thank everyone who has helped me with my work and inparticular my supervisor Sigmundur Gudmundsson for his support and guid-ance through this project. I am very grateful for his encouragement during thewhole time as well as interesting discussion on various subjects.

Joel Persson

Contents

Introduction 3

Chapter 1. The Willmore Functional in E3 5

Chapter 2. The Pull-Back Bundle 9

Chapter 3. The Willmore Functional in (M3; g) 151. The Tangent and Normal Bundles 152. The Conformal Invariance 18

Chapter 4. Variational Methods 211. Minimal Submanifolds 212. The Laplace Operator in Vector Bundles 233. The Euler-Lagrange Equation 25

Bibliography 33

1

Introduction

In this Master’s dissertation we study a generalization of minimal surfacescalled Willmore surfaces. For a natural number n 2 N and a smooth compactorientable surface S of genus n, let I(n) be the set of smooth immersionsf : S ! M3 of the surface S into a 3-dimensional Riemannian manifold(M ; g). On I(n) the Willmore functional W : I(n) ! R+0 is defined by

(1) W(f ) = ZS (H 2 + �K )dA;where H : S ! R and �K : S ! R are the mean curvature of S and thesectional curvature in M3 evaluated at df (TS), respectively. The study of thisfunctional was proposed by T.Willmore in [16] from 1965. Critical points ofthe functional W are called Willmore surfaces. In the case M3 = E3 that isR3 with the Euclidean metric the subject was first considered by G.Thomsonin [11] from 1923 and W.Blaschke in [1] from 1929. They also noticed thatW is invariant under conformal transformations on E3. This was later provedexplicitly by J. H. White in [14]. One way to characterize the critical pointsof W is to calculate its first variational formula. This leads to the non-linearpartial differential equation

(2) DH + 2H (H 2 � K ) = 0:This was actually known to G.Thomson and W.Blaschke. A more generalizedversion was considered by J.Weiner in [13] allowing a more general receivingspace then E3 as well as surfaces with boundary. Since W is invariant underconformal transformations of E3 and by a result of Lawson in [7] stating thatthere are minimal surfaces of arbitrary genus embedded in S3, we can concludefrom (2) that there are compact embedded Willmore surfaces in E3 of arbitrarygenus.

3

CHAPTER 1

The Willmore Functional in E3

In this chapter we will by E3 denote the 3-dimensional Euclidean space i.e.R3 equipped with the standard Euclidean metric.

PROPOSITION 1.1. Let f : S ! E3 be an immersion of a compact surfaceS into E3. Then the Willmore functional of f satisfies

(3) W(f ) � 4pwith equality if and only if S is embedded as the standard round sphere S2 in E3.

PROOF. [17] Let S+0 be the part of S with non-negative Gaussian cur-

vature. Taking an affine hyperplane not intersecting S and translate it in itsnormal direction towards the surface, any of the first points p 2 S touch-ing the plane has non-negative curvature. This means that the Gauss mapN : S ! S2 restricted to S+

0 is surjective and henceZS+0

KdA � 4p:We have

H 2 � K = 1

4(k1 � k2)2 � 0

and we find ZS H 2dA � ZS+0

H 2dA � ZS+0

KdA � 4p:We also see that equality is obtained if and only if k1 = k2 for each point p.This is only the case when S is embedded as a sphere [15]. �

WILLMORE CONJECTURE 1.2. [16] Let f : T 2 ! E3 be an immersionof the 2-dimensional torus T 2 into E3. Then the Willmore functional of f satisfiesW(f ) � 2p2:

This conjecture is still open but has been proved in several special cases.Equality is actually obtained for the torus in E3 with generating circles of radii

of the ratio 1 :p

2 We will now consider the case when the T 2 is a ”tube”embedded in E3. For this we will need Fenchel’s theorem.

5

6 1. THE WILLMORE FUNCTIONAL IN E3

THEOREM 1.3. Let g be a simple closed curve in E3 with curvature k. ThenZg jkjds � 2pwith equality if and only if g is a plane convex curve.

PROOF. [3]. Take a tube S of radius r around g given by

x(s; v) = g(s) + r(n(s) cos v + b(s) sin v) s 2 [0; l] and v 2 [0; 2p];where n and b is the normal and bi-normal to the curve and l is the length ofthe curve. By calculating the coefficients of the first and second fundamentalforms we find

K = LN �M2

EG � F 2= �k � cosv

r(1� rkcos v)

and

dA = pEG � F 2dsdv = r(1� rkcos v)dsdv:

So for k = 0 we have K = 0 and also K � 0 for v 2 [p=2; 3p=2] Integratingover S+

0 yieldsZS+0

KdA � � Zg jkjds

Z 3p=2p=2

cos vdv = 2

Zg jkjds

and by Proposition (1.1) we haveZg jkjds � 2p:Left is to prove that equality is obtained if and only if g is a plane convex curve.For every fixed s 2 [0; l] we get a circle on the tube which the Gauss-map mapsbijectively onto a great circle Gs on S2. Let G+

s be the closed half-circle given bythe points of Gs which correspond to points on the original circle on the tubewith positive Gaussian curvature. Assuming that g is a plane convex curvethen all semicircles G+

s have the same end points p; q for every s 2 [0; l]. Byconvexity of g we have G+

si\ Gsj

+ = fp; qg. Hence in this case the area of theimage of the Gauss normal map is actually 4p andZg jkjds = 2p:To prove the converse suppose

Rg jkjds = 2p thenRS+

0

K = 4p. All semicircles

must have the same end points p and q, or there are two distinct great circlesGsi and Gsj arbitrarily close to each other with corresponding semicircles withdistinct endpoints p and q that intersects in two antipodal points. One of thesemust then correspond to two points with a positive Gaussian curvature. Thuswe have two points on the tube with positive Gaussian curvature which maps

1. THE WILLMORE FUNCTIONAL IN E3 7

to the same point on S2. Since the Gauss normal map is locally a diffeomor-phism at each point and also each point of S2 is the image of some point ofS+

0 we get ZS+0

KdA > 4pgiving a contradiction.The intersection of the bi-normal b of g with S gives the points of zero Gauss-ian curvature. Thus the bi-normal must be parallel to the vector p � q and glies in a plane normal to p� q. For convexity note thatZu k � ds � 2pholds for any closed curve u with positive orientation number thus in our casek = jkj. �

FIGURE 1. Torus of ratio of the radii 1 :p

2.

PROPOSITION 1.4. [10] Let S be a torus in E3 given by moving a circle inE3 around a smooth closed curve g in such a way that the normal of the circle isparallel to the normal of the normal plane of the curve at each point and the centerof the circle always is a point on g. ThenZS H 2dA � 2p;with equality if and only if g is a circle and ratio of the radii is 1 :

p2.

8 1. THE WILLMORE FUNCTIONAL IN E3

PROOF. [17] S can be considered as tube with radius r as in the proofabove. The mean curvature H is given by

H = EN + GL� 2FM

2(EG � F 2);

with the coefficients from the fundamental forms of the surface S . In our casewe get

H = � (1� 2rkcos v)

2r(1� rkcos v)and ZS H 2 = Z l

0

Z 2p0

(1� 2rkcos v)2

4r(1� rkcos v)dvds:

Evaluating the inner integral yieldsZS H 2 = p Z l

0

1

2rp

1� r2k2ds = p

2

Z l

0

jkjjkrjp1� r2k2ds:

Now jkrjp1� r2k2 attains its minimum 1=2 when kr = 1=p2.We are left with ZS H 2 � p Z l

0

jkjds:By Fenchel’s theorem we finally getZS H 2dA � 2p2

with equality if and only if kr = 1=p2 i.e. when S is the anchor ring with

radii of ratio 1=p2. �

CHAPTER 2

The Pull-Back Bundle

Throughout this chapter we assume �M and M to be smooth manifolds ofdimension �m and m, respectively. Let f : �M ! M be a smooth map from �Mto M . By f �(TM ) we denote the pull-back bundle

f �(TM ) = f(p; v)jp 2 �M ; v 2 Tf (p)Mgover �M via f . Then f �(TM ) is a smooth m-dimensional vector bundle whichfiber at a point p 2 �M is the vector space Tf (p)M . The following diagramillustrates the situation.

f �(TM ) TMp�??y ??yp�M f���! MHere p� is the projection map of the pull-back bundle f �(TM ) given by p� :(p; v) 7! p and p is the projection map of the tangent bundle TM of M . Notethat a vector field X 2 C1(TM ) induces a section X � 2 C1(f �(TM )) of thepull-back bundle given by

X �p = Xf (p)

for each p 2 �M . This means that a local frame feig for TM induces a localframe fe�i g for the pull-back bundle f �(TM ).

DEFINITION 2.1. A smooth vector bundle (E ;M ; p) is said to be Rie-mannian if there exists a smooth map

h : C1(E) C1(E) ! C1(M )

and a connection rE : C1(TM )� C1(E) ! C1(E)

on E which is compatible with h. That is

X (h(y;f)) = h(rEXy;f) + h(y;rE

Xf)

for any vector field X 2 C1(TM ) on M and smooth sections y;f 2 C1(E)of the bundle E . The map h is called a metric on the bundle E . The curvatureR : C1(TM )�C1(TM )�C1(E) ! C1(E) of the connectionrE is definedby

9

10 2. THE PULL-BACK BUNDLE

R(X ; Y )y = rEXrE

Yy�rEYrE

Xy�rE[X ;Y ]y:

A Riemannian metric g on a smooth manifold M induces the Levi-Civitaconnection r on the tangent bundle (TM ;M ; p) of M . By the fundamentaltheorem of Riemannian geometry we know that g and r turn the tangentbundle TM into a Riemannian vector bundle over M since we for any vectorfields X ; Y ;Z 2 C1(TM ) have that

X (g(Y ;X )) = g(rX Y ;Z )+ g(Y ;rX Z ):We will now use g and r on M to make the pull-back bundle f �(TM ) into aRiemannian vector bundle over �M .

PROPOSITION 2.2. Let (M ; g) be a Riemannian manifold and f : �M !M be a smooth map from �M to M. Then the Levi-Civita connectionr on (M ; g)induces a unique connection r� on the pull-back bundle f �(TM ) defined byr��Xf� = (rdf (�X )f)�for �X 2 C1(T �M ) and f 2 C1(TM ).

PROOF. It is clear that r� is a connection. To prove uniqueness considera local orthogonal frame feig for TM . Then given a local section �f of f �(TM )we have �f =P

i aie�i for some local functions ai. Thenr��X �f = r��X (

Xi

aie�i )= X

i

�X (ai)e�i +X

i

ai(r��X e�i ):To show that the expression above is independent of the choice of frame feigconsider another frame ffjg for TM . We have�f =X

j

bjf�

j and f �j =Xi

xije�i

for some functions bj; xij locally defined on �M . Thenr��X (X

j

bjf�

j )= Xj

(�X (bj)f�

j + bjr��X f �j )= Xj

f�X (bj)X

i

xije�i + bjr��X (

Xi

xije�i )g= X

i

nXj

f�X (bj)xije�i + bj

�X (xij)e�i + (bjxij)r��X e�i go

2. THE PULL-BACK BUNDLE 11= Xi

f�X (X

j

bjxij)e�i + air��X e�i g= r��X (aie

�i ):

Thus the connection r� is unique and independent of the choice of the localframe. �

PROPOSITION 2.3. Let (M ; g) be a Riemannian manifold and further letf : �M ! M be a smooth map from �M to M. Then the pull-back connection r�on the pull-back bundle f �(TM ) satisfiesr��X(df (�Y ))

� �r��Y(df (�X ))� = (df ([�X ; �Y ]))�:

PROOF. We define T : C1(T �M )� C1(T �M ) ! C1(f �(TM )) by

T (�X ; �Y ) = r��X(df (�Y ))� �r��Y(df (�X ))� � (df ([�X ; �Y ]))�:

Let a 2 C1( �M ) be a function on �M , then

T (a�X ; �Y )= r�a�X(df (�Y ))� �r��Y(df (a�X ))� � �df (ra�X �Y �r�Ya�X )

��= ar��X(df (�Y ))� � ar��Y(df (�X ))� � �Y (a)(df (�X ))���df (ar�X �Y � (�Y (a)�X + ar�Y �X ))��= ar��X(df (�Y ))� � ar��Y(df (�X ))� � �Y (a)(df (�X ))�+�Y (a)(df (�X ))� � a(df ([�X ; �Y ]))�= aT (�X ; �Y ):

By skew-symmetry of T we also have T (�X ; a�Y ) = aT (�X ; �Y ) and hence T isa tensor field. Let p 2 �M be an arbitrary point and consider around p andf (p) 2 M two local systems of coordinates fxig and fyjg. Since T is tensorialit is enough to show that T (�=�xi ; �=�xj) = 0. We have

df (��xi

) =Xk

�fk�xi

� ��yk

so

T (��xi

; ��xj

)= Xk

fr���xi

( �fk�xj( ��yk

)�) �r���xj

( �fk�xi( ��yk

)�)g= Xk

f �2fk�xi�xj

� (��yk

)� + �fk�xj

� r���xi

( ��yk)�

12 2. THE PULL-BACK BUNDLE� �2fk�xj�xi

� (��yk

)� � �fk�xi

� r���xj

( ��yk)�g= X

r;k f�fk�xj

�fr�xi

�r��yr

��yk

�� � �fr�xi

�fk�xj

�r��yk

��yr

��g= Xr;k �fk�xj

�fr�xi

� ��yr

; ��yk

��= 0:We have shown thatr��X(df (�Y ))

� �r��Y(df (�X ))� = (df ([�X ; �Y ]))�: �

DEFINITION 2.4. Let (M ; g) be a Riemannian manifold and f : �M ! Mbe a smooth map from �M to M . Then the pull-back metric

g� : C1(f �(TM )) C1(f �(TM )) ! C1( �M )

on the pull-back bundle f �(TM ) is defined by

g�(X �; Y �) = g(X ; Y );where X ; Y 2 C1(TM ).

THEOREM 2.5. Let (M ; g) be a Riemannian manifold andf : �M ! M be a smooth map from �M to M. Then the pull-back bundle(f �(TM ); �M ; p�) equipped with the metric g� and the connection r� is a Rie-mannian vector bundle.

PROOF. Let �X 2 C1(T �M ) be a vector field on �M and further let �f; �y 2C1(f �(TM )) be sections of f �(TM ). Let feig be a local orthonormal framefor TM then fe�i g is a local orthonormal frame for f �(TM ). We have�f =X

i

aie�i and �y =X

i

bie�i

for some functions ai; bi 2 C1( �M ). By the definition of g� and r� it is clearthat

g�(r��X e�i ; e�j ) + g�(e�i ;r��X e�j ) = �X (g�(e�i ; e�j )) = 0:Then

g�(r��X �f; �y) + g�(�f;r��X �y)= g�(r��X (X

i

aie�i );X

j

bje�j ) + g�(X

i

aie�i ;r��X (

Xj

bje�j ))= X

i

�X (ai)bi +Xi;j aibj � g�(r��X e�i ; e�j )

2. THE PULL-BACK BUNDLE 13+Xi

ai�X (bi) +X

i;j aibj � g�(e�i ;r��X e�j )= �X (X

i

aibi)= �X (g�(�f; �y)): �

CHAPTER 3

The Willmore Functional in (M3; g)

In this chapter we will consider the special case of the pull-back bundlewhen f is an immersion. Also we will show that the Willmore functional isconformal invariant.

1. The Tangent and Normal Bundles

From now on we will assume throughout this chapter that the map f :�M ! M is an immersion. Then f is locally a diffeomorphism onto its im-age. This means that given a vector field �X 2 C1(T �M ) on �M and a section�f 2 C1(f �(TM )) of f �(TM ) we can, at each point p 2 �M , find a neigh-bourhood U of p, a neighbourhood V in M of f (p) and local vector fieldsX ;f 2 C1(TV ) such that

df (�Xp) = Xf (p) and �fp = f�f (p);for every p 2 U . We will call X and f local extensions of �X and �f, respectively.

The pull-back metric g� splits the vector space f �(TM )p at each pointp 2 �M into the direct sum

(f �(TM ))p = Tp�M � Np

�M ;where Tp

�M is identified with the image df (Tp�M ) in Tf (p)M and Np

�M withthe normal space

(df (Tp�M ))?

with respect to g in Tf (p)M . This gives a splitting of the pull-back bundlef �(TM )

f �(TM ) = T �M � N �M ;where T �M is the tangent bundle of �M and N �M the normal bundle of f ( �M )in M . We obtain a metric�g : C1(T �M ) C1(T �M ) ! C1( �M )

on the tangent bundle and a metric

g : C1(N �M ) C1(N �M ) ! C1( �M )

on the normal bundle by restricting g� to T �M and N �M , respectively.

15

16 3. THE WILLMORE FUNCTIONAL IN (M 3; g)

PROPOSITION 3.1. Let (M ; g) be a Riemannian manifold and f : �M !M be an immersion of �M into M. Then�r : C1(T �M )� C1(T �M ) ! C1(T �M )

defined by �r�X �Y = (rX Y )>is the Levi-Civita connection of ( �M ; �g) Here X ; Y are some local extensions of �Xand �Y .

PROOF. We show that �r preserves the metric �g . Let �X ; �Y ; �Z 2 C1(T �M )and let X ; Y ;Z denote some local extensions. Then we have�X (�g(�Y ; �Z )) = X (g(Y ;Z ))= g(rX Y ;Z )+ g(Y ;rX Z )= �g((rX Y )T ; �Z ) + �g(�Y ; (rX Z )T )= �g( �r�X �Y ; �Z ) + �g(�Y ; �r�X �Z ):So the connection �r is metric. Furthermore�r�X �Y � �r�Y �X � [�X ; �Y ] = (rX Y )> � (rY X )> � [X ; Y ]>= (rX Y �rY X � [X ; Y ])> = 0;which shows that �r is torsion-free. �

PROPOSITION 3.2. Let N �M be the normal bundle of f ( �M ) in M. Let�X 2 C1(T �M ) be a vector field on �M, �f 2 C1(N �M ) be a section of N �M andlet X and f be some local extensions. Then

D : C1(T �M )� C1(N �M ) ! C1(N �M )

defined byD�X �y = (rXy)?

is a connection on N �M which preserves the metric g.

PROOF. Let �X 2 C1(T �M ) be a vector field on �M and �y; �f 2 C1(N �M )be two sections on N �M . Let X ;y;f be some local extensions. Then we have�X (g(�y; �f)) = X (g(y;f))= g(rXy;f)+ g(y;rXf)= g((rXy)?; �f) + g(�y; (rXf)?)= g(D�X �y; �f) + g(�y;D�X �f):

1. THE TANGENT AND NORMAL BUNDLES 17�PROPOSITION 3.3. Let (M ; g) be a Riemannian manifold and f : �M !

M be an immersion of �M into M. Further let �X ; �Y 2 C1(T �M ) be two vectorfields on �M and denote by X ; Y some local extensions of �X ; �Y . Then the secondfundamental form of f

B : C1(T �M ) C1(T �M ) ! C1(N �M )

defined by

B(�X ; �Y ) = (rX Y )?is a symmetric tensor field.

PROOF. We first show B(�X ; �Y ) = B(�Y ; �X ):B(�X ; �Y )� B(�Y ; �X ) = (rX Y )? � (rY X )?= (rX Y )? � (rX Y )? + [X ; Y ]?= 0

Since obviously B(m�X ; �Y ) = mB(�X ; �Y ) for any m 2 C1( �M ) we haveB(�X ; m�Y ) = mB(�X ; �Y ) by the symmetry of B and thus B is tensorial in both itsarguments. �

DEFINITION 3.4. Let (M ; g) be a Riemannian manifold and f : �M !M be an immersion of �M into M . Then the mean curvature vector fieldH 2 C1(N �M) of the immersion f : �M ! M is defined byH = 1�m trace B:

To the second fundamental form B and a section �f 2 C1(N �M ) we asso-ciate the corresponding shape-operator A�f : T �M ! T �M given by�g(A�f(�X ); �Y ) = g(B(�X ; �Y ); �f)

for any vector fields �X ; �Y 2 C1(T �M ).

PROPOSITION 3.5. Let �X 2 C1(T �M ) be a vector field on �M, �f 2C1(N �M) be a section of N �M and X ;f some local extension �X and �f. Then

A�f(�X ) = �(rXf)>:PROOF. Let �Y 2 C1(T �M ) be a vector field on �M and Y a local extension

of �Y . Then �g(A�f(�X ); �Y ) = g(B(�X ; �Y ); �f)= g((rX Y )?; �f)= g(rX Y ;f)

18 3. THE WILLMORE FUNCTIONAL IN (M 3; g)= X (g(Y ;f))� g(Y ;rXf)= g(Y ;�rXf): �2. The Conformal Invariance

In this section we will prove that the Willmore functional is invariant underconformal changes of the Riemannian metric g on M .

PROPOSITION 3.6. Let M be a smooth manifold and g, gl be two Riemann-ian metrics on M such that

gl = l2 � g

for some positive real valued function l : M ! R+ . Denote by r and rl theLevi-Civita connections on TM induced by the metrics g and gl. Thenr andrlare related by

(4) rlX Y �rX Y = X (logl)Y + Y (logl)X � g(X ; Y )grad(logl):

PROOF. Let X ; Y 2 C1(TM ) be two vector fields on M . Further let feigbe a local frame on M which is orthonormal with respect to gl. Thenrl

X Y �rX Y = Xi

(gl(rlX Y ; ei)ei � gl(rX Y ; ei)ei)= X

i

(gl(rlX Y ; ei)ei � l2g(rX Y ; ei)ei):

By the definition of the Levi-Civita connection we have

gl(rlX Y ;Z )� l2g(rX Y ;Z )= 1

2fX (l2g(Y ;Z )) + Y (l2g(Z ;X ))� Z (l2g(X ; Y ))+l2g(Z ; [X ; Y ]) + l2g(Y ; [Z ;X ])� l2g(X ; [Y ;Z ])�(l2X (g(Y ;Z ))+ l2Y (g(Z ;X ))� l2Z (g(X ; Y ))+l2g(Z ; [X ; Y ]) + l2g(Y ; [Z ;X ])� l2g(X ; [Y ;Z ]))g= 1

2fX (l2)g(Y ;Z ) + Y (l2)g(Z ;X )� Z (l2)g(X ; Y )g:

Thus we haverlX Y �rX Y= X

i

ngl(rl

X Y ; ei)ei � l2g(rX Y ; ei)ei

o= 1

2

X fX (l2)g(Y ; ei)ei + Y (l2)g(ei;X )ei � g(X ; Y )ei(l2)eig

2. THE CONFORMAL INVARIANCE 19= X (logl)Y + Y (logl)X � g(X ; Y )grad(logl):And so

(5) rlX Y �rX Y = X (logl)Y + Y (logl)X � g(X ; Y )grad(logl) �

After the necessary preparations we can now prove the following theorem.

THEOREM 3.7. [14] Let S be a surface and f : S ! (M ; g) be an im-mersion of S into a 3-dimensional Riemannian manifold. Then the Willmorefunctional ZS (H 2 + �K )dA

is conformal invariant. Here �K is the sectional curvature in M evaluated atdf (TS).

PROOF. [2] As above let gl be a Riemannian metric on M such that

gl = l2 � g

for some positive real valued function l : M ! R+ . Denote by r and rlthe Levi-Civita connection on TM induced by the metrics g and gl. Let �g and�gl be the pull-back metrics on S induced by f and further let �r and �rl bethe Levi-Civita connections on S induced by f . Let �X ; �Y 2 C1(TS) be twovector field on S and denote by B and Bl the second fundamental form onS with respect to the different metrics �g and �gl. Then by the definition ofsecond fundamental form and (4) we have

(6) Bl(�X ; �Y )� B(�X ; �Y ) = �g(�X ; �Y )(grad(logl))?:Let �f 2 C1(NS) be a local section of the normal bundle of S of unit lengthwith respect to g. Then from (6) and the duality between B and A�f and Bland Al�f, respectively, we obtain

(7) �g(Al�f(�X ); �Y ) = �g(A�f(�Y ); �X ) + �g(�X ; �Y )g((grad(logl))?; �f):Since A�f is a symmetric endomorphism we can find a local orthonormal framefE1; E2g of S which are in the principal direction of A�f with respect to �g.Then fl�1Eig is a local orthonormal frame in the principal directions ofAl�f with respect to the metric �gl by (6). We denote by (k1(�f); k2(�f)) and(kl1(�f); kl2(�f)) the corresponding principal curvature of A�f and Al�f, respec-tively. We have, by (7)

(8) kli (�f) = ki(�f) + g((grad(logl))?; �f):

20 3. THE WILLMORE FUNCTIONAL IN (M 3; g)

Since �f 2 NS is a local normal section of unit length with respect to g weknow that �fl = l�1�f is a local normal section of unit length with respect togl. From the fact that

Al�f = lAl�fl(8) and the Gauss equation we findl(kl1(�fl))� kl2(�fl) = k1(�f)� k2(�f)l2(kl1(�fl)� kl2(�fl))2 = (k1(�f)� k2(�f))2l2(H 2l � Kl + �Kl) = H 2 � K + �K :We notice

dAl =pElGl � (Fl)2dxdy = l2

pEG � F 2dxdy = l2dA

which implies

(H 2l � Kl + �Kl)dAl = (H 2 � K + �K )dA:If we now apply the Gauss-Bonnet theorem we getZS (H 2l + �Kl)dAl � 2pq(S) = ZS (H 2l � Kl + �Kl)dAl= ZS (H � K + �K )dA= ZS (H 2 + �K )dA� 2pq(S)

which leads to ZS (H 2 + �K )dA = ZS (H 2l + �Kl)dAl: �

CHAPTER 4

Variational Methods

In this chapter we calculate the Euler-Lagrange equation for W . For thesecalculation we need the first variational formula for the volume element. Thisalso give some motivation for the notation of minimal surface for surfaces ofconstant zero mean-curvature.

1. Minimal Submanifolds

Let �M be a smooth �m-dimensional manifold and (M ; g) a smooth m-dimensional Riemannian manifold. Denote by I the interval (�e; e). LetF : �M � I ! M be a differentiable mapping such that for each t 2 I the mapft : �M ! M given by

ft : x 7! F (x; t)

is an immersion and F (x; 0) = f0(x) = f (x). Then F is said to be a variationof f . We will by V denote the section of the pull-back bundle f �(TM ) givenby

V (p) = (dF (��tj(p;0)))

�:For each t 2 I denote by dA(t) the volume element induced by ft and letdA(0) = dA.

PROPOSITION 4.1. Let �M be a smooth �m-dimensional manifold and letF : I � �M : M be a smooth variation of the immersion f : �M ! M as above.

d

dtdA(t)jt=0 = ��m � g(H ;V ?)dA + div(V >)dA:

PROOF. For a point p 2 �M let f�e1; � � � ;�e�mg be a local orthonormal framefor T �M with respect to the metric �g0 such that for at p we have

( �r�ei�ej)p = 0:We have the corresponding 1-forms wi given by wi(�X ) = �g(�ei; �X ) constitutinga local frame for the co-tangent bundle T �M�. Thus the metric at a given timet is given by �gt = �mX

i;j=1

ri;j(t)wi wj;21

22 4. VARIATIONAL METHODS

where ri;j(t) = g(dft(�ei); dft(�ej)). By the definition of the volume form we have

dA(t) =pr(t) � w1 ^ : : : ^ w�m =pr(t) � dA;where r(t) = det(ri;j(t)). So in order to determine �=�t(dA(t))jt=0 we must

calculate �=�t(pr(t))jt=0. By lemma (4.2) we have

d

dtdA(t)jt=0 = 1

2

mXk=1

drkk

dt(0) � dA = 1

2trace(r0i;j(0)) � dA:

We can locally extend �=�t;�e1; : : : ;�e�m to a local frame in T ( �M�I ) and denotethe images of this frame under dF by V ; e1; : : : ; e�m. Hence locally at p we haverkk(t) = gt(ek; ek)

and

1

2

d

dtrkk(t) = 1

2V (g(ek; ek))= g(rV ek; ek)= ek(g(V ; ek))� g(V ;rek

ek):Here we used the fact that [ek;V ] = 0. By construction we have (rek

ek)> = 0

at p so �mXk=1

g(V ;rekek) = �m � g(H ;V ?)

Hence at p

1

2

�mXk=1

drkk

dt(0) = ��m � g(H ;V ?) + �mX

k=1

�ek(�g(V >;�ek))= ��m � g(H ;V ?) + div(V >): �LEMMA 4.2. Let B(t) = ((bij(t))) be a C1 family of m � m matrices with

B(0) = I . Thend

dtdB(t)jt=0 = trace(B0(0)):

PROOF. For each fixed t we can consider B(t) to be a linear mappingB(t) : Rm ! Rm with respect to the canonical basis of Rm which we willdenote by fe1; : : : ; emg. From linear algebra we know that

det(B(t)) = W (B(t)e1; � � � ;B(t)em);

2. THE LAPLACE OPERATOR IN VECTOR BUNDLES 23

where W : Rm ! R is the unique linear function given by the condition thatW is alternating and W (e1; : : : ; em) = 1. We then have

d

dt(det(B(t)))jt=0 = mX

k=1

W (B(0)e1; : : : ;B0(0)ek; : : : ;B(0)em)= trace(B0(0)): �2. The Laplace Operator in Vector Bundles

Let (E ;M ; p) be a vector bundle with connection rE . Then(Hom(TM ; E);M ; p) is defined as the vector bundle which fiber at each pointp 2 M consists of the space of homomorphisms from TpM to Ep. We state thefollowing well know propositions and leave out the proofs which are straightforward calculations.

PROPOSITION 4.3. Let (E ;M ; p) be a vector bundle with connection rE .The connection rE on E induces a connectionrH : C1(TM )� C1(Hom(TM ; E)) ! C1(Hom(TM ; E))

on Hom(TM ; E) defined byrHX f : Y 7! rE

X (f(Y ))� f(rX Y );where f 2 C1(Hom(TM ; E)) and X ; Y 2 C1(TM ).

PROPOSITION 4.4. Let E be a Riemannian vector bundle with metric h andHom(TM ; E) the corresponding homomorphism bundle then h induces a metricH on Hom(TM ; E) given by

H (f;f) =Xi

h(f(ei);f(ei));where f;f 2 C1(Hom(TM ; E)) and feig is a local orthogonal frame for thetangent bundle TM of M.

For a section y 2 C1(E) of the vector bundle E we have the section(rEy) 2 C1(Hom(TM ; E)) given byrEy : X 7! rE

Xyfor X 2 C1(TM ).

With use of the connection rH on the homeomorphism bundleHom(TM ; E) we define the mappingrH

X ;Yy : C1(TM ) � C1(TM ) � C1(E) ! C1(E)

by rHX ;Yy = (rH

X (rEy))(Y ):

24 4. VARIATIONAL METHODS

Fixing a y 2 C1(E), the mapping

(X ; Y ) 7! rHX ;Yy

is a C1(E)-valued bilinear form.

DEFINITION 4.5. Let (M ; g) be a Riemannian manifold and(E ;M ; p) be a Riemannian vector bundle over M with connection rE . Wedefine the Laplacian of the section y 2 C1(E), denoted by DEy, byDEy = trace

�(X ; Y ) 7! rH

X ;Yy�where rH is the induced connection on the homeomorphism bundleHom(TM ; E) and rH

X ;Yy = (rHX (rEy))(Y ):

PROPOSITION 4.6. Let (M ; g) be a Riemannian manifold and(E ;M ; p) be a Riemannian vector bundle over M with connection rE . Thengiven a local orthogonal frame for the tangent bundle feig such that (rei ej)p = 0we have

(DEy)p =Xi

(rEei(rE

eiy))p

for any y 2 C1(E)

PROOF.

(DEy)p = Xi

(rHei

(rEy)(ei))p= Xi

(rEei((rEy)(ei)))p �X

i

((rEy)(reiei))p:Here the second term vanishes since (rei ej)p = 0 and we get

(DEy)p =Xi

(rEei(rE

eiy))p: �

PROPOSITION 4.7. Let (M ; g) be a compact Riemannian manifold with-out boundary and (E ;M ; p) be a Riemannian vector bundle with metric h andconnection rE . Then the Laplacian DE satisfiesZ

M

h(DEy;f)dVM = � ZM

H (rEy;rEf)dVM = ZM

h(y;DEf)dVM ;where y;f 2 C1(E) are sections of the vector bundle E.

3. THE EULER-LAGRANGE EQUATION 25

PROOF. By Proposition (4.6) we have

h(DEy;f) = Xi

h(rEei(rE

eiy);f)= X

i

ei(h(rEeiy;f))�X

i

h(reiy;reif)= div(X )� H (rEy;rEf);where X 2 C1(TM ) is defined by

X =Xi

h(rEeiy;f)ei

and div(X) is the divergence of the X and H is the metric on Hom(TM ; E).Since M is compact and without boundary we have by Green’s formula in [12]Z

M

h(DEy;f)dVM = � ZM

H (rEy;rEf)dVM : �Let (SM ;M ; p) be the sub-bundle of Hom(TM ;TM ) which fiber at each

point p 2 M consists of the vector space of symmetric endomorphisms. LetA : NM ! SM be the bundle homomorphism given by

A : f 7! Affor f 2 C1(NM ). Since we have metrics on SM and NM we can define thetranspose tA of A. Then tA : SM ! NM is given by

g(tA(s);f) = H (s;A(f))

for s 2 C1(SM ) and f 2 C1(NM ). We define ~A : NM ! NM by~A = tA Æ A:3. The Euler-Lagrange Equation

Let f : �M ! M be a immersion and F : �M � I ! M be a variation of f .It follows from Theorem (2.5) that the pull-back bundle

(F �(TM ); �M � I ; p)

over �M � I is a Riemannian vector bundle with the pull-back metric g� andpull-back connection r� induced by F . According to Proposition (2.3) wehave

(9) r��X(dF �Y )� �r��Y(dF �X )� = (dF [�X ; �Y ])�

26 4. VARIATIONAL METHODS

where �X ; �Y 2 C1(T ( �M � I )). If R is the curvature tensor on M we have

(10) (R(dF (�X ); dF (�Y ))f)� = r��Xr��Yf�r��Yr��Xf�r�[�X ;�Y ]f

where �X ; �Y 2 C1(T ( �M � I )) and f 2 C1F �(M ). The pull-back bundleF �(TM ) splits into the direct sum

F �(TM ) = T �N

where T(p;t) = dFt(Tp�M ) and N(p;t) is the orthogonal complement with respect

to g�. We have the connection �r on T and D on N by taking the projectionof r� onto T and N , respectively. Also we have the metric �g on T and g onN simply by restricting g�.

We want to characterize the critical point of the Willmore functional Wand therefore we compute the Euler-Lagrange equation for W . Following themethod used by Weiner in [13] we consider��t

W(Ft)j(t=0) = ��t

ZS (H 2t + �Kt)dAj(t=0)

where S = �M is a surface and M is m-dimensional Riemannian manifold.First we compute �=�t(H 2). Let v = �=�t then we have

v(W(ft))jt=0 = ZS v((H 2t + �Kt)dA)jt=0= ZS (v(H 2

t + �Kt)dAt + (H 2t + �Kt)v(dAt))jt=0

Let V 2 C1(f �(TM )) let be given by

V (p) = df (v)(p):and let H 2 C1(N ) be the mean curvature of �Mt at p. We have

v(H 2) = 2g(DvH ; H ):Let Ei where i 2 (1; 2) and Ej where j 2 (3; : : : ;m) be frames at (p; 0) in Tand N respectively. Parallel transport Ei and Ej along f(p; t)j � e < t < eg inT and N . That is �rvEi = 0 and DvEj = 0 at (p; t). Further parallel transportEi in T and Ej in N along all geodesics through p in Mt . Since Ei is tangentialfor each i 2 (1; 2), we can find local vector fields fe1; e2g on �M � I such thatdFt(ei) = Ei. By the definition of the mean curvature H we haveH = 1

2

2Xi=1

DeiEi

thus

DvH = 1

2

Xi

DvDeiEi = 1

2

Xi

Dvr�eiEi:

3. THE EULER-LAGRANGE EQUATION 27

We have from (10)

1

2

Xi

(R(V ; Ei)Ei)� = 1

2(X

i

r�vr�

eiEi �r�

eir�

v Ei +r�[v;ei]

Ei)

and hence

(11)1

2(X

i

(R(V ; Ei)Ei)�)? = 1

2

Xi

Dvr�eiEi �Deir�

v Ei + D[v;ei]Ei:So from (9) and (11) we have

DvH= 1

2f(X

i

(R(V ; Ei)Ei)�)? + Deir�

v Ei + D[v;ei]Eig= 1

2f(X

i

(R(V ; Ei)Ei)�)? + Deir�

eiV+DeidF [v; ei] + D[v;ei]Eig:

Since [v; ei] is tangential we have by definition of the second fundamental formB

D[v;ei]Ei = B([v; ei]; ei) = DeidF [v; ei] = �Dei�reiV

where the last equality come from (9) and the fact that �r�=�tEi = 0 at (p; t).We obtain

DvH = 1

2((X

i

(R(V ; Ei)Ei)�)? + Deir�

eiV � 2Dei

�rei V )= 1

2((X

i

(R(V ; Ei)Ei)�)? + DeiDeiV � Dei

�rei V ):Then

DvH= 1

2f(X

i

(R(V ?; Ei)Ei)�)? + DeiDeiV

? � Dei�rei V

?+(X

i

(R(V >; Ei)Ei)�)? + DeiDeiV

> � Dei�reiV

>g= 1

2f(X

i

(R(V ?; Ei)Ei)�)? +DN V ? �X

i

B(ei; �rei V?)+(

Xi

(R(V >; Ei)Ei)�)? +X

i

(DeiB(ei;V >)�Dei�rei V

>)g

28 4. VARIATIONAL METHODS= 1

2f(X

i

(R(V ?; Ei)Ei)�)? +DN V ? �X

i

B(ei;AV?(ei))+(X

i

(R(V >; Ei)Ei)�)? +X

i

(DeiB(ei;V >)�Dei�rei V

>)g:We notice

2Xi=1

B(ei;AV?(ei)) = mXj=3

2Xi=1

g(B(ei;AV?(ei)); Ej)Ej= mXj=3

2Xi

�g(AEj (ei);AV?(ei))Ej= mXj=3

H (AEj ;AV?)Ej= mXj=3

g(tA(AV?); Ej)Ej= tA Æ A(V ?) = eA(V ?)

and at (p; 0)

(R(V >; Ei)Ei)�)? + DeiB(ei;V >)� Dei

�reiV>= (�R(Ei;V >)Ei)

�)? + DeiDV>Ei �Dei�rV>Ei � D[ei;V>]Ei= �Deir�

V>Ei + DV>r�eiEi + D[ei;V>]Ei+DeiDV>Ei �Dei

�rV>Ei �D[ei ;V>]Ei= DV>DeiEi:So we find

DvH = 1

2f(

2Xi=1

R(V ?; Ei)Ei)? +DN �M V ? + eA(V ?)g+ DV>H :

Restricted to the case when S is a surface without boundary and M has con-stant constant sectional curvature �K we have at (p; 0)

vf(H 2 + �K )dAg= v(H 2) � dA + (H 2 + �K ) � v(dA)= 2g(D�=�tH ; H ) � dA + (H 2 + �K ) � ��t(dA)= g(2 �K V ? +DN �M V ? + eA(V ?) + DV>H ; H ) � dA+(H 2 + �K )(�2g(V ?; H )) � dA + (H 2 + �K ) � div(V >) � dA

3. THE EULER-LAGRANGE EQUATION 29= g(DN �M V ? � 2H 2H + eA(V ?); H ) � dA+V >H 2 � dA + (H 2 + �K ) � div(V >) � dA:Since eA is symmetric with respect to g , S has no boundary and

V >H 2 � dA + (H 2 + �K ) � div(V >) � dA = div((H 2 + �K )V >) � dA

we have by Proposition (4.7) and Green’s theorem

vfZS (H 2 + �K )dA(t)gjt=0 = ZS g(DNH � 2H 2H + eA(H );V ?)dA:Hence we can conclude:

THEOREM 4.8. [13] Let f : S ! (M ; g) be a smooth immersion from acompact orientable surface S without boundary to a Riemannian manifold (M ; g)of constant sectional curvature �K . Then f is a stationary point of the Willmorefunctional W if and only if

(12) DNH � 2H 2H + eA(H ) = 0:Sometimes the condition that a surface satisfies (12) is taken as the defini-

tion of a Willmore surface.

FIGURE 1. Example of a Willmore surface in R3 obtained at [5].

In the case when M is a 3-dimensional manifold we have the following corol-lary.

COROLLARY 4.9. Let (M3; g) be a 3-dimensional Riemannian manifold ofconstant sectional curvature �K and S be a compact orientable surface withoutboundary. Then a smooth immersion f : S ! M is a stationary point of theWillmore functional W(f ) if and only if

(13) DH + 2H (H 2 � G) = 0;where G is the determinant of the second fundamental form B.

30 4. VARIATIONAL METHODS

PROOF. The normal space is spanned by a vector h and we have H = H �hand DNH = DH �h. Let fe1; e2g be a local orthonormal frame for the tangentbundle TS . We then haveeA(H ) = X

i

B(ei;AV?(ei))= Xi

B(ei;Xj

�g(AH (ei); ej); ej)= Xi;j �g(AH (ei); ej)B(ei; ej)= Xi;j g(B(ei; ej); H )B(ei ; ej)= Xi;j H (B(ei; ej)

2) � h= H (B(e1; e1)2 + 2B(e1; e2)2 + B(e2; e2)2)= H ((2H )2 + 2(jB(e1; e2)j2 � jB(e1; e1)jjB(e2; e2)j)= (4H 3 � 2GH ) � hand the theorem follows. �

THEOREM 4.10. Let S be a minimal surface in S3 and further let s :S3 � fng ! R3 be the stereo graphic projection of S3 into R3 from the north pole.Then the image s(S) is a Willmore surface in E3.

PROOF. By Corollary (4.9) a minimal surface in S3 is also a Willmoresurface S3. By Theorem (3.7) we know that W is invariant under conformaltransformation of the metric in E3. Hence taking the stereo graphic projectionfrom S3 to E3, which is a conformal transformation, yields a Willmore surfacein E3. �

COROLLARY 4.11. Let g be a natural number, then there exist an embeddedcompact Willmore surface S in E3 of genus g.

PROOF. By a result of Lawson in [7] we know there exists compact em-bedded minimal surfaces in S3 of arbitrary genus. Hence by Theorem (4.10)taking the stereo graphic projection of these surfaces gives the desired surfacesin E3. �

In the article [7] from 1985 U.Pinkall has shown that there exists Willmoresurfaces in E3 not obtainable by stereo graphic projection of minimal surfacesin S3.

3. THE EULER-LAGRANGE EQUATION 31

FIGURE 2. A Willmore surface of positive genus obtained at [5].

Bibliography

[1] Blaschke, W. Vorlesungen �uber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativittstheorie.

III, Leipzig, Springer, (1929).

[2] Chen, B-Y. Some Conformal Invariants of Submanifolds and Their Application, Boll. Univ. Mat. Ital. (4) 10

(1974), 380-385.

[3] Carmo, M. P. do Differential Geometry of Curves and Surfaces New Yersey, Prentice Hall, (1976).

[4] Eels, J. & Lemaire, L. Selected topics in harmonic maps,CBMS Regional Conf. Ser. Vol 50, Amer. Math. Soc.

(1983).

[5] Hellner, U. http://www.gang.umass.edu/gallery/willmore/wilmore0101.html (2002).

[6] Kusner, R. Conformal Geometry and Complete Minimal Surfaces, Bull. Amer. Math. Soc. 17 (1987), 291-295.

[7] Lawson, H.B JR. Complete minimal surfaces in S3, 92 (1970), 335-374.

[8] Lawson, H.B JR. Lectures on Minimal Submanifold Volume 1, (1977).

[9] Pinkall, U. Hopf tori in S3, Invent. Math. 81 (1985), 379-386.

[10] Shiohama, K. & Takagi, R. A characterization of a standard torus in E3, J. Differential Geometry 4 (1970),

477-485.

[11] Thomsen, G. Uber Konforme Geometrie, I:Grundlagen der konformen Flachentheorie, Hamb. Math. Abh. 3

(1923), 31-56.

[12] Urakawa, H. Calculus of Variations and Harmonic Maps, Translated from the 1990 Japanese original by the

author, Amer. Math. Soc. (1993).

[13] Weiner, J. L. On the Problem of Chen. Willmore. et al. , Indiana Univ. Math. Jour. 27 (1978), 19-35.

[14] White, J. H. A Global Invariant of Conformal Mappings in Space, Proc. Amer. Math. Soc. 38 (1973), 162-164.

[15] Willmore, T. J. An introduction to differential geometry, Oxford University Press, (1959).

[16] Willmore, T. J. Note on embedded surfaces, Ann. Stiint. Univ. Al. I. Cuza, Iasi. Sect. I a Matt. (N.S.) 11B (1965),

493-496.

[17] Willmore, T. J. Riemannian Geometry, The Clarendon Press, Oxford University Press, New York, (1993).

33

Master’s Theses in Mathematical Sciences 2003:E9

ISSN 1404-6342

LUNFMA-3022-2003

Mathematics

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