surfaces of constant mean curvature2.2. the weierstrass representation formula 10 chapter 3. cmc...

SURFACES OF CONSTANT MEAN

CURVATURE

CARL JOHAN LEJDFORSMaster’s thesis2003:E11

Centre for Mathematical SciencesMathematics

CE

NT

RU

MSC

IEN

TIA

RU

MM

AT

HE

MA

TIC

AR

UM

i

Abstract

The aim of this Master’s dissertation is to give a survey of some basic resultsregarding surfaces S of constant mean curvature (CMC) in R3 . Such surfacesare often called soap bubbles since a soap film in equilibrium between tworegions is characterized by having constant mean curvature. The surface areaof these surfaces is critical under volume-preserving deformations.

CMC surfaces may also be characterized by the fact that their Gauss mapN : S ! S2 is harmonic i.e. it satisfiest(N ) = 0;where t(N ) is the tension field of N , generalizing the classical Laplacian. Thisis a non-linear system of partial differential equations. It was proved in the1990s that this system has global solutions on compact surfaces of any genusg � 0.

In this dissertation we study necessary and sufficient conditions for a sur-face to have CMC. We study the minimal case (characterized by mean curva-ture H � 0) and the well-known Weierstrass representation for such surfaces.Also CMC surfaces with rotational symmetry are considered and a generaliza-tion of the Weierstrass representation to surfaces of non-zero constant meancurvature is presented. Finally we show that the only compact embeddedCMC surfaces in R3 are spheres.

It has been my intention throughout this work to give references to the statedresults and credit to the work of others. The only part of this Master’s dissertationwhich I claim is my own is the elementary proof of a special case of Ruh-Vilmstheorem for surfaces in R3 given in Theorem 4.1.

iii

Acknowledgements

I wish to thank my supervisor, Sigmundur Gudmundsson, for his time, knowl-edge and patience. In particular, I wish to express my gratitude for him inspir-ing me to study the wonderful subject of geometry.

Carl Johan Lejdfors

Contents

Short History 1

Chapter 1. Some basic surface theory 31.1. Notation 31.2. Isothermal coordinates 51.3. The tension field 7

Chapter 2. Minimal surfaces 92.1. Conformality of the Gauss map 92.2. The Weierstrass representation formula 10

Chapter 3. CMC surfaces of revolution 153.1. Kenmotsu’s solution 153.2. Delaunay’s construction 17

Chapter 4. CMC surfaces 234.1. Harmonicity of the Gauss map 234.2. Kenmotsu’s representation formula 24

Chapter 5. Compact CMC surfaces 33Recent developments 35

Appendix A. Harmonic maps 37

Bibliography 41

v

Short History

In 1841 Delaunay characterized in [1] a class of surfaces in Euclidean spacewhich he described explicitly as surfaces of revolution of roulettes of the conics.These surfaces are the catenoids, unduloids, nodoids and right circular cylinders.Today they are known as the surfaces of Delaunay and are the first non-trivialexamples of surfaces having constant mean curvature, the sphere being thetrivial case.

In an appended note to Delaunay’s paper M. Sturm characterized thesesurfaces variationally as the extremals of surfaces of rotation having fixed vol-ume while maximizing lateral area. Using this characterization the followingtheorem was obtained:

THEOREM (Delaunay’s theorem). The complete immersed surfaces of revo-lution in R3 having constant mean curvature are exactly those obtained by rotatingabout their axis the roulettes of the conics.

These surfaces where also recognized by Plateau using soap film experi-ments. In 1853 J. H. Jellet showed in [2] that if S is a compact star-shapedsurface in R3 having constant mean curvature then it is the standard sphere. Ahundred years later, in 1956, H. Hopf conjectured that this, in fact, holds forall compact immersions:

CONJECTURE (Hopf ’s conjecture). Let S be an immersion of an oriented,compact hypersurface with constant mean curvature H 6= 0 in Rn . Then S mustbe the standard embedded (n� 1)-sphere.

Hopf proved the conjecture in [3] for the case of immersions of S2 into R3

having constant mean curvature and a few years later A. D. Alexandrov showedthe conjecture to hold for any embedded hypersurface in Rn , see [4]. It waswidely believed that this conjecture was true until 1982 when Wu-Yi Hsiangconstructed a counterexample in R4 . Two years later Wente constructed in [5]an immersion of the torus T 2 in R3 having constant mean curvature.

Wente’s construction has been thoroughly studied but has only been ableto create surfaces having genus g = 1. A different method for construct-ing surfaces in R3 having constant mean curvature of any genus g � 3 waspresented in 1987 by N. Kapouleas [6]. A proof of the fact that there existCMC-immersions of compact surfaces of any genus was published in [8] in1995 by the same author.

1

CHAPTER 1

Some basic surface theory

In this chapter we introduce the notation to be used in this text. We alsointroduce some basic results concerning isothermal coordinates and the tensionfield of the Gauss map of a surface in R3 .

1.1. Notation

DEFINITION 1.1. A non-empty subset S on R3 is said to be a regularsurface if for each point p 2 S there exists an open neighborhood U in Saround p and a bijective map f = (x; y) : U � S ! R2 such that its inverseX : f(U ) ! U

i. is a homeomorphism,ii. is a differentiable map,

iii. (Xx � Xy)(q) 6= 0 for all q 2 f(U ).

The functions x; y are called local coordinates around p. The map X is called alocal parametrization of S around p.

Let S be a regular surface in R3 and p 2 S be an arbitrary point. Bya tangent vector to S , at the point p, we mean the tangent vector a0(0) of adifferentiable parametrized curve a : (�e; e) ! S with a(0) = p. The set oftangent vectors of S at a point p 2 S is called the tangent space of S at p 2 Sand is denoted by TpS . A local parametrization X determines a basis�

Xx;Xy

of TpS , called the basis associated with X .

On the tangent plane we have the usual induced metric from the ambientspace R3 with the associated quadratic form Ip : TpS ! R called the firstfundamental form of S at p 2 S . Given a local parametrization X of S and aparametrized curve a(t) = X (x(t); y(t)) for t 2 (�e; e) with p = a(0) we havethe following form

Ip(a0(0)) = Xxx

0 + Xyy0;Xxx

0 + Xyy0�= hXx;Xxip (x0)2 + 2

Xx;Xy

�p

x0y0 + Xy;Xy

�p

(y0)2= E(x0)2 + 2Fx0y0 + G(y0)2; (1.1)

where the values of E , F and G are computed for t = 0.By condition (iii) of the definition of a regular surface (1.1) we have, given

a local parametrization X of a surface S in R3 at a point p 2 S that the map

3

4 1. SOME BASIC SURFACE THEORY

N : S ! S2 defined by

N (p) = Xx � Xy��Xx � Xy

�� (p) (1.2)

is well defined. This map is known as the Gauss map of S . The quadratic formIIp defined in TpS by

IIp(v) = � dNp(v); v�

is called the second fundamental form of S at p. Given a local parametrizationX on S at a point p 2 S and, as above, letting a be a parametrized curve suchthat a(t) = X (x(t); y(t)) for t 2 (�e; e) with p = a(0) = X (x(0); y(0)), we get

dNp(a0) = N 0(x(t); y(t)) = Nxx0 + Nyy

0Since hN ;N i = 1 we must have that Nx;Ny 2 TpS and hence

Nx = a11Xx + a21Xy;Ny = a12Xx + a22Xy; (1.3)

for some functions aij. We find that

IIp(a0) = � dNp(a0); a0� = � Nxx0 + Nyy

0;Xxx0 + Xvx

0�= e(x0)2 + 2fx0y0 + g(y0)2; (1.4)

where

e = �hNx;Xxi = hN ;Xxxi ;f = � Ny;Xx

� = N ;Xxy

� = N ;Xyx

� = � Nx;Xy

� ;g = � Ny;Xy

� = N ;Xyy

� :Using the terms from equations (1.1), (1.3) and (1.4) we arrive at

a11 = fF � eG

EG � F 2; a12 = gF � fG

EG � F 2;

a21 = eF � fE

EG � F 2; a22 = fF � gE

EG � F 2;

known as the Weingarten equations.Continuing by using that fXx;Xyg is a basis for TpS and that N is orthog-

onal to both Xx and Xy we have that fXx;Xy;Ng is a basis for R3 . We findthat

Xxx = G 111Xx + G 2

11Xy + eN ;Xyx = G 1

12Xx + G 212Xy + fN ;

Xxy = G 121Xx + G 2

21Xy + fN ;Xyy = G 1

22Xx + G 222Xy + gN : (1.5)

The G kij are known as the Christoffel symbols and are invariant under isometries

(i.e. can be computed from the first fundamental form alone). Using that

(Xxx)y � (Xxy)x = 0;

1.2. ISOTHERMAL COORDINATES 5

(Xyy)x � (Xxy)y = 0;Nxy � Nyx = 0;

we find that

ey � fx = eG 112 + f (G 2

12 � G 111)� gG 2

11;fy � gx = eG 1

22 + f (G 222 � G 1

12)� gG 212: (1.6)

These equalities are known as the Mainardi-Codazzi equations.

DEFINITION 1.2. Let S be a surface in R3 and p 2 S an arbitrary point.Let dNp : TpS ! TpS be the differential of the Gauss map. Then the deter-minant of dNp is called the Gaussian curvature K of S at p. The negative halfof the trace of dNp is called the mean curvature H of S at p.

In terms of the first and second fundamental forms K and H are given by

K = eg � f 2

EG � F 2; (1.7)

H = eG � 2fF + gE

2(EG � F 2): (1.8)

1.2. Isothermal coordinates

In this section we introduce the notion of isothermal coordinates which isa useful tool in differential geometry.

DEFINITION 1.3. Let S be a surface in R3 . Then local coordinates(x; y) : U � S ! R2 on S are said to be isothermal if there exists a strictlypositive function, called the dilation, l : U � S ! R such that

E = hXx;Xxi = l2 = Xy;Xy

� = G; F = 0:We have the following result regarding existence of isothermal coordinates

on an arbitrary surface in R2 .

THEOREM 1.4. Let S be a differentiable surface in R3 and p 2 S be a pointon S . Then there exists an open neighborhood U of p and isothermal coordinates(x; y) : U � S ! R2 around p.

This was proved in the analytic case by Gauss. For a complete proof in thegeneral case please see [9]. Having chosen isothermal coordinates the meancurvature simplifies

H = eG + gE � 2fF

2(EG � F 2)= e + g

2l2:

The Christoffel symbols similarly simplifyG 111 = G 2

12 = �G 122 = 1

2l2

�l2�x;�G 2

11 = G 112 = G 2

22 = 1

2l2

�l2�y: (1.9)


Using this we get the following form of the Mainardi-Codazzi equations (1.6):

ey � fx = (e + g)G 222;

gx � fy = (e + g)G 111: (1.10)

The Weingarten relations (eq. 1.3) reduce to

a11 = � el2; a21 = a12 = � fl2

; a22 = � gl2: (1.11)

Suppose S is a surface in R3 and f = (x; y) : U � S ! R2 are localisothermal coordinates on S . We may then consider the local coordinates(x; y) as a complex-valued map z = x + iy : U � S ! C . The inverseX : z(U ) � C ! S can then be considered as map from an open subset z(U )in C into S i.e. a local parametrization of S . We then have

Xz = 1

2

�Xx � iXy

� :The complex notation for surfaces in R3 has many advantages which we willbe useful in chapters 2 and 3.

Letting h�; �i be the usual inner product in R3 and let (�; �) be the complexbilinear extension of h�; �i in C 3 we have the following result.

PROPOSITION 1.5. Let S be a surface in R3 and let z = x+iy : U � S !C be local isothermal coordinates on S . Then the inverse X : z(U ) � C ! S ofz is conformal i.e. satisfies

4 (Xz;Xz) = jXxj2 � ��Xy

��2 � 2iXx;Xy

� = 0;4 (X�z;X�z) = jXxj2 � ��Xy

��2 + 2iXx;Xy

� = 0: (1.12)

Conversely, if z = x+ iy are local coordinates on S satisfying equation (1.12) thenthey are isothermal.

PROOF. The first statement follows by a direct computation. The reverseimplication follows by considering real and imaginary parts of equation (1.12).�

PROPOSITION 1.6. Let S be a surface in R3 and z = x+iy : U � S ! Cbe local isothermal coordinates on S with dilation l. Then the inverse X : f(U ) �C ! S of z satisfies

4X�zz = Xxx + Xyy = 2l2HN ;where N : S ! S2 is the Gauss map of S .

PROOF. By a direct computation using the differentiated form of equation(1.12) we have

4X�zz = 2l2[(X�zz;Xz) Xz + (X�zz;X�z) X�z] + 4 (X�zz;N ) N=

Xxx + Xyy;N�

N = 4(e + g)N= 2l2HN

1.3. THE TENSION FIELD 7

This immediately gives our sought result. �1.3. The tension field

In this section we give an explicit formula for the tension field (see ap-pendix A) for maps from a surface in R3 into S2 in terms of local isothermalcoordinates.

PROPOSITION 1.7. Let S be a surface in R3 and f : S ! S2 be a mapinto the unit sphere S2 in R3 . If (x; y) : U � S ! R2 are local isothermalcoordinates on S then the tension field t(f) of f is locally given byt(f) = 1l2

�Df�T

i.e. as the tangential part of the classical LaplacianD = �2�x2+ �2�y2

in R2 .

PROOF. By the definition of the tension field of a smooth map f : S ! S2

we have t(f) = 2Xk=1

�r�ek

df(ek)� df(rekek)� ;

where r� is the pull-back connection on the pull-back bundle f�1TS2 over Svia f.

Let p 2 S be an arbitrary point and (x; y) : U � S ! R2 be isothermalcoordinates around p. We then haverek

ek = �ekek + l2

2

�2ek(

1l2)ek � g(ek; ek) grad

1l2

�= �ekek + l2

2

�2ek(

1l2)ek � grad

1l2

�for k = 1; 2. Then using the definition of the gradient grad we obtainre1e1 +re2e2 = �e1e1 + �e2e2 + l2

�e1(

1l2)e1 + e2(

1l2)e2 � grad

1l2

�= r 1l ��x

�1l ��x

�+r 1l ��y

�1l ��y

�= 1l ��x

�1l� ��x

+ 1l ��y

�1l� ��y

This implies that

df(re1e1 +re2e2) = 1l ��x

�1l� �f�x

+ 1l ��y

�1l� �f�y

(1.13)


The other term is given byr�e1

df(e1) = �1l ��x

�1l �f�x

��T = 1l � ��x

�1l� �f�x

+ 1l �2f�x2

�T

(1.14)r�e2

df(e2) = �1l ��y

�1l �f�y

��T = 1l � ��y

�1l� �f�y

+ 1l �2f�y2

�T

(1.15)

It follows by equations (1.13), (1.14) and (1.15) thatt(f) = r�e1

df(e1) +r�e2

df(e2)� df(re1e1)� df(re2e2)= 1l2

��2f�x2+ �2f�y2

�T = 1l2

�Df�T �Harmonic maps generalize the concept of harmonic functions well known

from complex analysis. A harmonic map is one for which the tension fieldvanishes everywhere and, as stated in Appendix A, arises as a critical point of acertain variational problem.

THEOREM 1.8. Let S be a surface in R3 and f : S ! S2 be a map intothe unit sphere S2 in R3 . If f is conformal then it is harmonic.

PROOF. Let p 2 S be an arbitrary point and (x; y) : U � S ! R2 belocal isothermal coordinates around p. Then the conformality of f means thatfx;fy

� = 0 and hfx;fxi = fy;fy

� :By differentiating we then obtainfxx;fy

� = � fyx;fx

� ; fyy;fx

� = � fxy;fy

� ;hfxx;fxi = fyx;fy

� ; fyy;fy

� = fxy;fx

� ;and therefore fxx + fyy;fx

� = fyx;fyy

�� fyx;fyy

� = 0;fxx + fyy;fy

� = � fyx;fxx

�+ fxy;fxx

� = 0:These relations imply that t(f) = 1l2

�Df�T = 0: �

CHAPTER 2

Minimal surfaces

In this chapter we introduce some results concerning minimal surfaces. Wealso prove the famous Weierstrass representation for minimal surfaces.

DEFINITION 2.1. A surface S in R3 is said to be minimal if its meancurvature H satisfies H � 0.

2.1. Conformality of the Gauss map

PROPOSITION 2.2. Let S be a minimal surface in R3 . Then the Gauss mapN : S ! S2 of S is conformal.

PROOF. Let p 2 S be an arbitrary point on S and (x; y) be local isother-mal coordinates around p. Then it follows by

H = e + g

2l2= 0

and equation (1.11) that

Nx = 1l2

�eXx + fXy

� ; Ny = 1l2

�fXx � eXy

� :This implies that

Nx;Ny

� = 0 and hNx;Nxi = Ny;Ny

� = e2 + f 2l2� 0:

Hence N is conformal. �A partial reverse implication of the previous theorem is obtained via the

following.

PROPOSITION 2.3. Let S be a real analytic surface in R3 and N : S ! S2

be a Gauss map of S . If N is conformal then S is either minimal or part of asphere.

PROOF. For local isothermal coordinates (x; y) on S we have

0 = Nx;Ny

� = �1l2

�eXx + fXy

� ; 1l2

�fXx + gXy

��= 1l4

�ef hXx ;Xxi+ fg

Xy;Xy

��= fl2

�e + g

� = 2fH :Let p 2 S be a point. Suppose H (p) 6= 0 then there exists an open neighbor-hood V � S around p such that f jV= 0. For every point in q 2 V we have,

9

10 2. MINIMAL SURFACES

by equation (1.11), that Nx and Ny are parallel with Xx and Xy, respectively.

By conformality we have that jNxj = ��Ny

�� and, since (x; y) are isothermal, thatjXxj = ��Xy

��. Hence q is umbilical i.e. the principal curvatures coincide. Letk = k1 = k2, where k1 and k2 are the principal curvatures. DifferentiatingNx = �kXx and Ny = �kXy gives

(kXx)y = (Nx)y = (Ny)x = (kXy)x

and since Xx and Xy are linearly independent we must have kx = ky = 0 so kis constant. If k = 0 then H = 0 which contradicts the assumption. Hencek 6= 0 and

N = �kX + a

where a is a constant vector. Then X is a local parametrization for a spherehaving radius 1=k centered at a=k sincekX � 1

kak2 = k1

kNk2 = 1

k2:

Thus by real analyticity S is either minimal or part of a sphere. �2.2. The Weierstrass representation formula

The Weierstrass representation formula was first presented by Karl Weier-strass in [10]. It states that given two holomorphic functions defined on somesimply connected subset of C there exists an associated minimal surface. Thissurface is unique up to motions.

THEOREM 2.4. Let S be a surface in R3 and f = x+ iy : U � S ! C belocal isothermal coordinates on S . Suppose U is an open simply connected subsetof S . If X : z(U ) � C ! S is the inverse of z then S is minimal if and only ifthe derivative Xz : z(U ) ! C 3 is holomorphic.

PROOF. This is a direct consequence of Proposition 1.6 and the fact thata map f : U � C ! C is holomorphic if and only if f�z = 0. �

Integration gives us the following corollary.

COROLLARY 2.5. Let S be a surface in R3 and f = x + iy : U � S ! Cbe local isothermal coordinates on S . Suppose U is an open simply connected subsetof S . Then the inverse X : z(U ) � C ! S of z is given by

X (z) = 2 Re

Z z

z0

Xz(z)dz + C ; (2.1)

where C is some constant vector in R3 .

PROOF. We have

Xzdz = 1

2

�(Xx � iXy)(dx + idy)

�= 1

2

�Xxdx + Xydy + i(Xxdy � Xydx

� ;X�zd�z = 1

2

�Xxdx + Xydy � i(Xxdy � Xydx

� :

2.2. THE WEIERSTRASS REPRESENTATION FORMULA 11

Integrating dX = Xzdz + X�zd�z = 2 Re Xzdz gives us our sought relation. �This corollary gives us the famous Weierstrass representation for minimal

surfaces.

THEOREM 2.6 (Weierstrass Representation). Let V be an open simply con-nected subset of C . Suppose f : V ! C is holomorphic on V , g : V ! C ismeromorphic on V and the product fg2 is holomorphic on V . Then X : V ! R3

defined by

X (z) = Re

Z z

z0

Xz(z)dz; (2.2)

whereXz(z) = f (z)(1� g(z)2; i(1+ g(z)2); 2g(z))

is a minimal surface.

PROOF. Using the above results the only thing we need to show is thatequation (2.2) define isothermal coordinates. This, however, follows by directcomputation using Proposition 1.5. �

Examples of minimal surfaces are the surfaces of Sherk (Fig. 2.1) and Cata-lan (Fig. 2.2).

FIGURE 2.1. Sherk’s minimal surface. (f ; g) = ( 21�z4 ; z)

An interesting observation is the fact that the Gauss map of minimal sur-faces generated using Theorem 2.6 can be identified with the complex valuedfunction g.

PROPOSITION 2.7. Let S be a minimal surface in R3 given by the Weier-strass representation

X (z) = Re

Z z

z0

f (z)(1� g(z)2; i(1+ g(z)2); 2g(z))dz:Then the Gauss map N : S ! S2 of S may be identified, via stereographicprojection s, with g.

12 2. MINIMAL SURFACES

FIGURE 2.2. Catalan’s minimal surface. (f ; g) = �i�

1z � 1

z3

� ; z�

PROOF. Let G = s�1 Æ g then

G = (2 Re g; 2 Im g; g2 � 1)

1 + g2:

Now the real and imaginary parts of�f (1� g2); if (1 + g2); 2fg

�represent two orthogonal tangent vectors on S in R3 . Then�

fg

2

�1

g� g; i

2(1

g+ g); 1

� ;G

� = fg

2

�g

g� g�g + jgj2 � 1

� = 0

and since jGj = 1 it is clear that G is a Gauss map for S . �We can conclude that the above examples (Figures 2.1, 2.2) have bijective

Gauss map i.e. for every point p 2 S2 there exists only one point on S havingthat point as a normal. Amongst the Enneper surfaces, defined by (f ; g) =(1; zn) for every n 2 N , only the case of n = 1 satisfies this property (seeFigures 2.3, 2.4 and 2.5).

FIGURE 2.3. First order Enneper surface. (f ; g) = (1; z)

2.2. THE WEIERSTRASS REPRESENTATION FORMULA 13

FIGURE 2.4. Second order Enneper surface. (f ; g) = (1; z2)

FIGURE 2.5. Third order Enneper surface. (f ; g) = (1; z3)

CHAPTER 3

CMC surfaces of revolution

In this chapter we study complete CMC surfaces with rotational symme-try. We present Kenmotsu’s modern solution given in [11] to the problem offinding all such surfaces. Furthermore we describe the classical construction ofthe same due to Delaunay, see [1].

DEFINITION 3.1. A surface S in R3 is said to have constant mean curva-ture (CMC) if and only if there exists a c 2 R such that H � c.

3.1. Kenmotsu’s solution

Let f : I � R ! R2 with f(s) = �x(s); y(s)

�be a parametrization of some

regular planar C2 curve. Assume that f is an arclength parametrization andthat 0 is contained in the open interval I . Let S be the surface of revolutionin R3 defined by

(s; �) 7! (x(s); y(s) cos �; y(s) sin �); s 2 I ; 0 � � � 2p:Then the first and second fundamental forms are given by

Ip = ds2 + y2d�2;IIp = (x00y0 � x0y00)ds2 + x0yd�2:

Assuming y(s) > 0 for s 2 I we have, by definition of H , that

2Hy � x0 � x00yy0 + x0yy00 = 0; s 2 I : (3.1)

Multiplying by x0 and y0, respectively, and simplifying using the fact that

(x0)2 + (y0)2 = 1 and x0x00 + y0y00 = 0; s 2 I ;we obtain

2Hyx0 + (yy0)0 � 1 = 0 and 2Hyy0 � (yx0)0 = 0:Setting Z (s) = y(s)y0(s)+iy(s)x0(s) and combining these equations the followingfirst order complex linear differential equation is obtained

Z 0 � 2iHZ � 1 = 0; s 2 I : (3.2)

Restricting our attention to the case of H being constant we have:If H = 0 then the solution is given by

Z (s) = s + C = s + c1 + ic2

15

16 3. CMC SURFACES OF REVOLUTION

for some C = c1 + ic2 2 C . This gives us

y(s) = jZ (s)j =q(s + c1)2 + c2

2;x0(s) = Im Z

y= c2p

(s + c1)2 + c22

: (3.3)

By integrating x we obtain

x = c2 arcsinh

�s + c1

c2

�hence s + c1 = sinh

�x

c2

�c2:

Substituting into equation (3.3) we obtain

y =q(s + c1)2 + c2

2 =ssinh2

�x

c2

�c2

2 + c22 = c2 cosh

�x

c2

� :It is clear that this is a parametrization of a catenary.

If H 6= 0 then

Z (s) = �1

2iH

�1� e�2iHs

�+ C

�e2iHs= 1

2iH

�(1+ 2iHC )� e�2iHs

�e2iHs= Bei(2Hs+�) � 1

2iH; (3.4)

where Bei� = 1 + 2iHC for some B; � 2 R and C 2 C is an arbitraryconstant. Using the fact that y(s) > 0 we have by translation of the arclengthand by restricting our attention to H > 0

y(s) = jZ j = 1

2H

p1+ B2 + 2B sin 2Hs;

x0(s) = Im Z

y= 1 + B sin 2Hsp

1 + B2 + 2B sin 2Hs:

Hence the solution to equation (3.4) is the one-parameter family of sur-faces of revolution having constant mean curvature H given byf(s; H ;B) = �Z s

0

1 + B sin 2Htp1 + B2 + 2B sin 2Ht

dt;1

2H

p1+ B2 + 2B sin 2Hs

�(3.5)

for any B 2 R and H > 0.Studying f for varying B (see Fig. 3.1) we find that f(s; H ; 0) is a generat-

ing curve for a right circular cylinder and f(s; H ; 1) is a generating curve for asequence of continuous half-circles centered on the x-axis. For 0 < B < 1 thefunction x(s) increases monotonously whereas in the case of B > 1 it does not.

3.2. DELAUNAY’S CONSTRUCTION 17

THEOREM 3.2 (Delaunay’s theorem). Any complete surface of revolutionwith constant mean curvature is either a sphere, a catenoid or a surface whosegenerating curve is given by f(s; H ;B) for some B 2 R.

PROOF. Let H 2 R be given and let fH (s) be a generating curve parametrizedby arclength for a complete surface of revolution having constant mean cur-vature H . By uniqueness of solution of (3.2) we have fH (s) = f(s; H ;B) forsome B 2 R. �

FIGURE 3.1. Solutions for H = 0:5 and B = 0; 0:5; 1; 1:53.2. Delaunay’s construction

The surfaces of Delaunay are constructed by rolling a conic ` along astraight line in the plane and taking the trace of the focus F . This is calleda roulette of the conic `. This trace then describes a planar curve which is ro-tated about the axis along which it was rolled. This gives a surface of revolutionhaving constant mean curvature. The construction presented here is based onthe article [12] by J. Eells.

3.2.1. ` is a parabola. Let ` be a parabola given by ` : t 7! (t; at2) forsome a which we take to be strictly positive. Let F be the focus and A bethe vertex of ` (see Figure 3.2). Let K be a point on ` and denote by P theintersection of the tangent line of ` at K with the horizontal axis. By solvingthe line equation for the tangent at K we find that for K = `(t) = (t; at2) thenP = (t=2; 0). This implies that PK = OP. And since \FOP = \PKF we

P

K

O

A

F

FIGURE 3.2. ` is parabola


FIGURE 3.3. Catenary

also have \OPF = \KPF = p2. By definition of the trigonometric functions

we have

FA = FP cos\AFP = FP cos\PFK :Now let FP denote the x-axis along which our parabola rolls. Then the

ordinate of F in this system of coordinates is given by PF . Denote this by y.We have

cos\PFK = dx

ds:

where s is the arclength of the locus of F . This is equivalent to

dx

ds= cos a; (3.6)

where a denotes the angle made by the tangent of F with the x-axis. We thenarrive at

c = ydx

ds= y

dxdsq�

dxds

�2 + � dyds

�2= yq

1 + � dydx

�2

or, equivalently,

dy

dx=r

y2 � c2

c2: (3.7)

The solution to this differential equation is given by

y = c

2

�ex=c + e�x=c

� = c coshx

c

which is a catenary (Fig. 3.3). The corresponding surface of revolution is thecatenoid (Fig. 3.4). The Gauss map of the locus of F into S1 is given by x 7! ax

where

cos ax = dx

ds= c

y

showing that the Gauss map is injective onto an open semicircle.


FIGURE 3.4. Catenoid

3.2.2. ` is an ellipse. Let F and F 0 be foci of ` and O its center. Takea point K on ` and let P and P 0 be the points on the tangent at K closest toF and F 0, respectively (Fig. 3.5). As above, letting PK be the x-axis and PF(P 0F 0) the ordinate y (y0). Let T and T 0 denote the intersection with the x-axisof the tangent of the locus of F and F 0, respectively.

We have \FKP = \F 0KP 0. Also the tangent of the locus of F (F 0) isorthogonal to FK (FL0) and hence \KFT = \KF 0T 0 = p

2. This gives us

y

FK= sin\FKP = cos\FTP = dx

ds;

y0F 0K = sin\F 0KP 0 = cos\F 0T 0P 0 = dx

ds:

From the characterization of the ellipse, FK + F 0K = 2a for some a > 0, andthe pedal equation, PF � P 0F 0 = b2 for some b > 0, we find

y + y0 = 2adx

dsand yy0 = b2 so

T

P·

PK

F·F O

FIGURE 3.5. ` is ellipse


FIGURE 3.6. Undulary, H = 0:5, B = 0:5.

y2 � 2aydx

ds+ b2 = 0:

Taking a � b we get the following cases (when the angle is obtuse and acute,respectively)

y2 � 2aydx

ds+ b2 = 0: (3.8)

A solution to this problem is given in Section 3.1

x(s) = Z s

0

1 + B sin 2Htp1 + B2 + 2B sin 2Ht

dt;y(s) = 1

2H

p1 + B2 + 2B sin 2Hs; (3.9)

where H = 12a

and B = �q1� b2

4H 2 . The locus of either foci is called

the undulary (Fig. 3.6). The corresponding surfaces is called the unduloid(Fig. 3.7). The Gauss map of the undulary is given by x 7! ax where

cos ax = dx

ds= �y2 + b2

2ay:

FIGURE 3.7. Unduloid, H = 0:5, B = 0:5.


3.2.3. ` is a hyperbola. We proceed as in the case of the ellipse but in-stead use the characterization FK � F 0K = 2a > 0 of the hyperbola andthe pedal equation PF � P 0F 0 = �b2 (Fig. 3.8). We arrive at the differentialequation

y2 � 2aydx

ds� b2 = 0:

P·

P

K

F·F O

FIGURE 3.8. Hyperbola

This differential equation can be solved in the same manner as for the ellipsewith the exception that B in equation (3.9) is given by

B = �r1 + b2

4H 2:

Here the two loci fit together to form the curve known as the nodary (Fig. 3.9)and the corresponding surface is called the nodoid (Fig. 3.10) The Gauss mapof the nodary is given by x 7! ax where

cos ax = �y2 � b2

2ay:

This map has no extreme points and is clearly surjective.

FIGURE 3.9. Nodary, H = 0:5, B = 1:5.


FIGURE 3.10. Nodoid, H = 0:5, B = 1:5.

CHAPTER 4

CMC surfaces

The main aim of this chapter is to give a new elementary proof of a specialcase of the Ruh-Vilms’ theorem. We also present the Kenmotsu representationformula for CMC surfaces with H 6= 0.

4.1. Harmonicity of the Gauss map

Let (M ; g) be an orientable m-dimensional Riemannian manifold, i : M !Rm+p be an isometric immersion and N : M ! Gop(Rm+p) be the associated

Gauss map, mapping x 2 M to the oriented normal space of i(M ) at i(x).Then Ruh-Vilms’ theorem presented in [13] states that the tension field t(N )of N satisfies t(N ) = �mrH ;where rH is the covariant derivative of the mean curvature vector field H .This implies that the Gauss map N is harmonic if and only if the mean cur-vature vector field H is parallel. For surfaces in R3 this is equivalent to thesurface having constant mean curvature.

THEOREM 4.1. Let S be an oriented surface in R3 . Then S has constantmean curvature if and only if its Gauss map N : S ! S2 is harmonic.

PROOF. We prove that the following equationt(N ) = �2 grad H

holds. Our sought result then follows trivially.Let p 2 S be an arbitrary point and (x; y) : U � S ! R2 be isothermal

coordinates around p. Then we have

(Nxx)T = hNxx;XxiXx + Nxx;Xy

�Xyl2= 1l2

�(�ex � hNx;Xxxi)Xx + (�fx � Nx;Xyx

�)Xy

�= 1l2

�(�ex � hNx;Xxxi)Xx + (�ey + (e + g)G 2

22 � Nx;Xyx

�)Xy

� :This follows by using�ex = ��x

hNx;Xxi = hNxx;Xxi+ hNx;Xxxi ;�fx = ��x

Nx;Xy

� = Nxx;Xy

�+ Nx;Xyx

� ;23

24 4. CMC SURFACES

and the Mainardi-Codazzi equations (1.10). Similarly we have

(Nyy)T = 1l2

�(�gx + (e + g)G 1

11 � Ny;Xxy

�)Xy + (�gy � Ny;Xyy

�)Xy

� :By using the Christoffel relations (1.9) we findhNx;Xxxi =

a11Xx + a21Xy;G 111Xx + G 2

11Xy

�= l2�a11G 1

11 + a21G 211

� = � �eG 111 � f G 2

22

�Nx;Xyx

� = l2�a11G 1

12 + a21G 212

� = � �f G 111 + eG 2

22

�Ny;Xyx

� = l2�a12G 1

21 + a22G 221

� = � �gG 111 + f G 2

22

�Ny;Xyy

� = l2�a12G 1

22 + a22G 222

� = � ��f G 111 + gG 2

22

�Adding and using the above equations we gethDN ;Xxi =

Nxx + Nyy;Xx

�= 1l2

��(ex + gx) + (e + g)G 111 � hNx;Xxxi � Ny;Xxy

��= 1l2

��(ex + gx) + 2(e + g)G 111

� ;DN ;Xy

� = Nxx + Nyy;Xy

�= 1l2

��(ey + gy) + 2(e + g)G 222

� :Hence we have�t(N ) = � 1l2

(DN )T = 1l2

�ex + gxl2

� e + gl4

�l2�x

�Xx+ 1l2

�ey + gyl2

� e + gl4

�l2�y

�Xy= 1l2

�ex + gxl2

+ (e + g)��x

(1l2

)

�Xx+ 1l2

�ey + gyl2

+ (e + g)��y

(1l2

)

�Xy= �

1l ��x(e + g)

1l2+ (e + g)

1l ��x(

1l2)

�1lXx+ �1l ��y

(e + g)1l2+ (e + g)

1l ��y(

1l2)

�1lXy= 2 grad H :

This proves our theorem. �4.2. Kenmotsu’s representation formula

In this section we show a corresponding result to Weierstrass’ representa-tion formula for surfaces having non-zero constant mean curvature.

4.2. KENMOTSU’S REPRESENTATION FORMULA 25

Let S2 be the unit sphere in R3 . Cover S2 by open sets Ui, i = 1; 2 whereU1 = S2 � fng and U2 = S2 � fsg and n and s are the north and south pole,respectively. Let s be the stereographic projection with respect to the northpole n: s(x) = x1 + ix2

1�3

for x = (x1; x2; x3) 2 U1: (4.1)

For a surface S in R3 having Gauss map N : S ! S2 consider the follow-ing composition f : S N�! S2 s�! Cwhich we also call the Gauss map of S . This map is considered as a complexmapping from a 1-dimensional complex manifold S in R3 into the Riemannsphere. Using this notation we have the following theorem presented in [14]due to K. Kenmotsu.

THEOREM 4.2 (Kenmotsu’s representation formula). Let V be an opensimply connected subset of C and H be an arbitrary non-zero real constant. Sup-pose f : V ! C is a harmonic function into the Riemann sphere. If f�z 6= 0 thenX : V ! R3 defined by

X (z) = Re

Z z

z0

Xz(z)dz; (4.2)

with

Xz(z) = (�1)

H (1+ f(z)�f(z))2

�1� f(z)2; i(1+ f(z)2); 2f(z)

� �f��z (z);for z 2 V , is a regular surface having f as a Gauss map and mean curvature H.

First we derive an explicit formula of the tension field of the Gauss mapf : S ! C of an arbitrary surface S .

PROPOSITION 4.3. Let S be a surface in R3 and f : S ! C be a Gaussmap on S . If z = x + iy are local isothermal coordinates with dilation l thent(f) = 4l2

� �2f�z��z � 2�f1 + f�f �f�z

�f��z � : (4.3)

PROOF. Let z = x+ iy be local isothermal coordinates and denote f(z) =u+ iv. Then

g = l2(dx2 + dy2) and h = 4

(1 + u2 + v2)2(du2 + dv2)

which gives us Christoffel symbols on the Riemann sphereG 111 = G 2

12 = �G 122 = �2u

1 + u2 + v2;�G 2

11 = G 112 = G 2

22 = �2v

1 + u2 + v2:

26 4. CMC SURFACES

By the explicit formula for the tension field (eq. A.2) we gett(u) = 1l2

�Du + G 111

��u�x

�2 + ��u�y

�2�+ G 122

��v�x

�2 + ��v�y

�2�+ 2G 112

��u�x

�v�x+ �u�y

�v�y

��;where D is the classical Laplacian. Adding this and the similar formula for t(v)we arrive atl2t(f) = l2

�t(u) + it(v)�= D(u + iv)�� 2

u � iv

1 + u2 + v2

��u�x

�2 � 2i�u�x

�v�x� ��v�x

�2 +��u�y

�2 � 2i�u�y

�v�y� ��v�y

�2�= Df� 8�f

1 + f�f �f�z

�f��z= 4�2f�z��z � 8

�f1+ f�f �f�z

�f��zThis proves our sought formula. �

Next we show that a surface having prescribed mean curvature H satisfiesthe following equation.

LEMMA 4.4. Let S be a surface in R3 having mean curvature H : S ! Rand let f : S ! C be a Gauss map of S . Then

H�f�z

= y�f��z ;where y = 1l2

�(e � g)=2� if

�.

PROOF. We show that the following equation holds�f��z = �H

2

�1 + f�f�2 �(X 1 + iX 2)��z : (4.4)

By direct computation using the Weingarten equations (1.11) together withequation (4.1) we have�f��z = ��z �N 1 + iN 2

1� N 3

� = 1

2l2(1�N 3)2

��X 1x + N 3X 1

x � N 1X 3x

�� i�X 2

x �N 3X 2x + N 2X 3

x

��e��(1�N 3)

�X 2

y � X 2x + iX 2

y + iX 1x

� + (N 1 + iN 2)�

X 3y + iX 3

x

��f

4.2. KENMOTSU’S REPRESENTATION FORMULA 27+��X 2y �N 3X 2

y + N 2X 3y

�� X 1y � N 3X 1

y + N 1X 3y

��g

�:Using the definition

N = 1l2

�X 2

x X 3y � X 3

x X 2y ;X 3

x X 1y � X 1

x X 3y ;X 1

x X 2y � X 2

x X 1y

�(4.5)

and the fact that z = x + iy are isothermal together with equation

(1 + f�f)(1� N 3) = 2 on U1; (4.6)

our sought relation is obtained via�f��z = 1

2l2(1� N 3)2

�(�2)(e + g)

�(X 1 + iX 2)��z �= �2H

(1� N 3)2

�(X 1 + iX 2)��z= �H

2(1 + f�f)2�(X 1 + iX 2)��z :

By similar calculations we have�f�z= �y

2(1+ f�f)2�(X 1 + iX 2)��z : (4.7)

From equations (4.4) and (4.7) we may conclude that

H�f�z

= y�f��z : �The computations carried out in the proof of the previous lemma give us

a way of describing the dilation l in terms of the Gauss map f.

COROLLARY 4.5. Let S be an orientable surface in R3 with mean curvatureH : S ! R and f : S ! C be the Gauss map of S . Let z = x + iy be localisothermal coordinates on S with dilation l. Then��f��z �� = l

2(1 + f�f) jH j ; (4.8)��f�z

�� = l2

(1 + f�f) jyj : (4.9)

PROOF. By equations (4.5) we have

4��X 1�z + iX 2�z ��2 = 2l2(1�N 3)� 4

��X 3�z ��2 ;4��X 3�z ��2 = l2((N 1)2 + (N 2)2)2:

The above formulas gives us

4��X 1�z + iX 2�z ��2 = l2(1� N 3)2: (4.10)

From equations (4.4) and (4.6) we get our first sought result. Equation (4.9)follows from equations (4.10) and (4.7). �

28 4. CMC SURFACES

Next we show how the derivatives of the components of X are related tothe Gauss map.

PROPOSITION 4.6. Let S be an orientable surface in R3 with mean curva-ture H : S ! R and f : S ! C be the Gauss map of S . Let X : V � C ! Sbe a local conformal immersion of S , then the following equations hold on U1:

H�X 1��z = � 1� (�f)2

(1 + f�f)2

�f��z ;H�X 2��z = i

1 + (�f)2

(1 + f�f)2

�f��z ;H�X 3��z = �2

�f(1 + f�f)2

�f��z : (4.11)

PROOF. Let ~s : U2 � S2 ! C denote the stereographic projection withrespect to the south pole s of S2 and put r = ~s Æ N . Then by similar calcula-tions as in the proof of Lemma 4.4 we have�r��z = �H

2

�1 + r�r�2 �(X 1 � iX 2)��z : (4.12)

Using that fr = 1 we may conclude that the following equations hold onU1 \ U2:

0 = �(fr)��z = �H

2

��f+ �r� �(X 1 � iX 2)��z + ��f+ r� �(X 1 + iX 2)��z �By (�f+ r)=(f+ �r) = �f2 we have

H

��(X 1 � iX 2)��z + �f2�(X 1 + iX 2)��z � = 0:Rewriting as

H (1+ �f2)�X 1��z = iH (1� �f2)

�X 2��zand substituting into equation (4.4) we arrive at

H�X 2��z = i

1 + �f2

(1+ f�f)2

�f��z : (4.13)

Similarly, we obtain the first formula of equation (4.11). The third equationfollows from �X 3�z

��X 1��z + i�X 2��z � = l2 f

(1 + f�f)2:

This last equation follows by rewriting equation (4.6) to

4

��X 3�z

��X 1 + iX 2��z �� l2 f(1 + f�f)2

�= 4�X 3�z

��X 1 + iX 2��z �� l2(1�N 3)(N 1 + iN 2):


Expanding using equation (4.5) and separating out real and imaginary parts we

find that both vanish and the equation follows. If �f��z 6= 0 then by equations(4.10) and (4.6) we have

H�X 3�z

= l2 f(1 + f�f)2

1

X 1�z + iX 2�z = l2 f(1+ f�f)2

X 1�z + iX 2�zjX 1�z + iX 2�z j2= 4f(1 + f�f)2

X 1�z + iX 2�z(1� N 3)2

= f�X 1�z + iX 2�z �:This implies the last equation of (4.11) by conjugating and substituting using

equation (4.13) and the similar formula for �X 1��z . If �f��z = 0 then H must be0. �

PROOF OF KENMOTSU’S REPR. FORMULA. We have shown that for anarbitrary surfaceS having constant mean curvature H 6= 0 and complex Gaussmap f : S ! C the following equations hold

ds2 = �2

H (1+ f�f)

��f��z ��2 jdzj2 ; (4.14)�2f�z��z � 2�f1 + f�f �f�z

�f��z = 0: (4.15)

Similarly, given f and H , we may construct a surface S which is locallyparametrized by X using equation (4.2). �

The last thing we need to show is that this representation is unique up toconformal transformations.

THEOREM 4.7. Let S be a simply connected surface in R3 and f : U ! Cand ef : U ! C be smooth mappings satisfying equation (4.15) for some constant

H 6= 0. We define a surface X and eX by Theorem 4.2 i.e.

X = Re

Z z

z0

Xzdz where

Xz = (�1)

H

�1

2

1� f2

(1 + f�f)2; i

1 + f2

(1 + f�f)2; 2

f(1+ f�f)2

��f��zThen the following conditions are equivalent:

i. There exists a holomorphic mapping w = f (z) with f 0(z) 6= 0 on S and

a motion � of R3 such that eX Æ f (z) = � Æ x(z) for z 2 S .ii. There exists a holomorphic mapping w = f (z) with f 0(z) 6= 0 on S

satisfying f(z) = ef Æ f (z) for z 2 S :PROOF. Assume condition i. hold. We may assume � = id. Then Xz =eXw Æ f 0 and X�z = eX~w Æ f 0. We have

1l �Xx + iXy

� = f 0jf 0j 1~l �eX~x + ieX~y�

30 4. CMC SURFACES

and therefore

2N (z) = il2

�Xx + iXy

�� Xx � iXy

� = 2eN Æ f (z):Hence f(z) = ef Æ f (z) and ii. follows from equation (4.4).

Conversely suppose condition ii. hold. Then it follows from Theorem 4.2�X�z= �eX�w Æ f 0(z) that

X (z) = eX (f (z)) + C ;where C 2 R3 is some constant. �

Kenmotsu’s representation formula allows us to, given a harmonic map-ping f : U � C ! C , construct a surface having specified mean curvatureand f as a Gauss map.

EXAMPLE 4.8. Let f : C �f0g ! C be given by f(z) = �1=�z and H =1. The immersion X obtained via Theorem 4.2 is the standard immersion ofthe unit sphere (see fig. 4.1) in R3 :

X (z) = 1

1 + z�z (z + �z; i(z � �z); z�z � 1) ; z 2 C � f0g:

FIGURE 4.1. Unit sphere in R3 .

EXAMPLE 4.9. Let f : C � f0g ! C be given by f(z) = 1=�z2. The byTheorem 4.2 we have

X (z) = Re1

H

Z z

z0

�1

2

1� �z�4

1 + (z�z)�2; i

1 + �z�4

1 + (z�z)�2; 2

�z�2

1 + (z�z)�2

�dz:

The corresponding surface of mean curvature H = 1 is given in figure 4.2.


FIGURE 4.2. Surface with f = 1=�z2 with H = 1.

CHAPTER 5

Compact CMC surfaces

In this chapter we prove a result due to Alexandrov [4] stating that everycompact embedded CMC surface in R3 is a sphere.

THEOREM 5.1 (Alexandrov’s theorem). Let S be a compact embedded sur-face in R3 having constant mean curvature H 6= 0. Then S is a sphere.

PROPOSITION 5.2 (Willmore’s theorem). [15] Let S be a compact surfaceembedded in R3 . Then ZS H 2 dA � 4pwith equality if and only if S is a round sphere.

PROOF. Let K +(p) = maxfK ; 0g then we will show thatZS K + dA � 4p (5.1)

holds for any compact surface in R3 . This follows from the observation thatthe left hand side of the above equation represents the area of the image underthe Gauss map N of the part of S having K � 0. So the only thing we needto prove is that the image of N covers all of S2. Assume that N is orientedoutwards, then the point of S having minimal y-component has a normal(0;�1; 0) and K � 0 (since otherwise it would be a saddle point). But wemay orient our surface however we want (while maintaining K ) and hence wehave shown that there exists a point for every normal direction having K � 0.The surface area of the unit round sphere is 4p implying equation (5.1).

This gives us ZS jK j dA � ZS K + dA � 4p:If K < 0 for any point then K < 0 in a neighborhood of that point and thefirst inequality is strict. Hence equality implies K � 0 everywhere and by theprevious equation and

H 2 = �k1 + k2

2

�2 = k1k2 + �k1 � k2

2

�2 � K

we have proved the first part of our result. If H 2 = K then k1 = k2 and weknow by Proposition 2.3 that S is a round sphere. If S is a round sphere thenH 2 = K proving our second statement. �

33

34 5. COMPACT CMC SURFACES

LEMMA 5.3. [16] Let S be a compact embedded surface in R3 bounding adomain D of volume V . If the mean curvature H of S is positive everywhere, thenZS 1

HdA � 3V (5.2)

Equality holds if and only if S is a round sphere.

PROOF. Let N be the normal of S and for every e 2 [0; 1] let Se be theshell of S defined by Se = fp + eh(p)N j p 2 Sg ;where

h(p) = supfr j the point p is the unique nearest point on S to the

point q at distance r from p along the normal S at pg:The volume of Se is then given by

V = ZS F dA; (5.3)

where

F = Z h(p)

0

j(1� k1t)(1� k2t)j dt = Z h(p)

0

��1� 2Ht + Kt2�� dt

Note that the definition of h we prevents overlap i.e. every point in the interiorof Se lies on a unique normal to S .

For every point q 2 D it lies in the closed shell Se. Let d = dist(q;S)then the open ball Bq(d ) centered at p with radius d satisfies Bq(d ) \ S = ;.But there exists at least one point p 2 S contained in the boundary of Bq(d ).For any q0 on the radius from q to p, if r is the distance from q0 to p then the

closed ball Bq0(d ) is contained in Bq(d ) except for the point p. Hence p is theunique point of S having distance r from q0. By definition of h(p) we musthave d � h(p) and hence q lie on the closed shell.

Since all points of D are covered and the only points covered twice are inthe image of the boundary we have that the volume V of D is exactly equal tothe volume of the shell given by equation (5.3)

The equation1

h(p)� maxfk1(p); k2(p)g (5.4)

hold for every p 2 S and we may conclude that

(1� k1t); (1� k2t)

are non-negative for 0 � t � h(p) and hence

F = Z h(p)

0

j(1� k1t)(1� k2t)j dt = Z h(p)

0

(1� k1t)(1� k2t) dt: (5.5)

By the inequalities of geometric and arithmetic mean

(1� k1t)(1� k2t) � (1� Ht)2 (5.6)

RECENT DEVELOPMENTS 35

and by equation (5.4) we have

1

h(p)� H :

Then

F � Z 1=H

p

(1� Ht)2 dt = 1

3H

and by equation (5.3) we arrive at the sought equation (5.2).Equality gives us, by Willmore’s theorem (5.2), that S is in fact a round

sphere. �PROOF OF ALEXANDROV ’S THEOREM. By the divergence formulaZZS hX ;N i dA = ZZZ

D

div X dx dy dz

we have

A = H

ZZ hN ;X i dA = H

ZZZdiv X dx dy dz = 3HV :

Then by Lemma 5.3 the surface S must be a sphere sinceZZ1

HdA = 1

H

ZZdA = A

H= 3V : �

Recent developments

Wente was the first to show that there exists a compact CMC immmersionin R3 which is not the standard sphere. He provided this by explicitly con-structing an immersion of the torus T 2 in R3 having constant mean curvature.

THEOREM 5.4 (Wente’s counterexample). [5] There exists a countable num-ber of isometrically distinct conformal immersion of T 2 into R3 with constantmean curvature H 6= 0.

The problem whether there exists immersions of genus g � 2 was ad-dressed by N. Kapouleas in [7, 8]. He successfully showed that there existsCMC immersions of every genus g � 2 by fusing Delaunay surfaces andWente tori.

THEOREM 5.5. For any g � 2 there exists infinitely many smooth CMC-surfaces immersed in R3 having constant mean curvature H = 1 and genus g.

A recent development in the field is the DPW method [17, 18] namedafter its creators J. Dorfmeister, F. Pedit and H. Wu. This method gives adescription of all immersed CMC surfaces in R3 with or without umbilics. Itis often characterized as a Weierstrass type method as it allows the constructionof CMC-immersions from a meromorphic and a holomorphic function.

APPENDIX A

Harmonic maps

Harmonic mappings arise as critical points of a certain variational problem.Let f : (M ; g) ! (N ; h) be a smooth map between two Riemannian manifoldsof dimension m and n, respectively. We then define the energy density function,e(f), of f at p 2 M by

e(f) = 1

2traceg (f�h) = 1

2

mXi=1

h(df(ei); df(ei)) = 1

2

mXi=1

kdf(ei)k2h;

where f�h is the pull back of h via f and feigmi=1 is a local orthonormal frame

of TpM with respect to gp. Let p 2 M ,

(x1; x2; : : : ; xm) and (y1; y2; : : : ; yn)

be local coordinates around p and f(p) 2 N and for a = 1; 2; : : : ; n putfa = ya Æ f. Then at each point q in a neighborhood of p

e(f)(q) = 1

2

mXi;j=1

(g ij(q)

nXa;b=1

hab (f(q))�fa�xi

(q)�fb�xj

(q)

) ;where (g ij) is the inverse of the matrix (gij) with

gij = g(ei; ej)

We define the energy or action integral of f on M by

E(f) = ZM

e(f)dvolg ;where volg = det(gij)dx1 � � � dxn is the volume form on M .

A map F : (�e; e) � M ! N is said to be a smooth variation of f if itsatisfies (

F (0; p) = f(p) p 2 M ;F : (�e; e)�M ! N of class C1:

For every t 2 (�e; e) we define the smooth variation ft of f byft(p) = F (t; p); p 2 M :DEFINITION A.1. A map f : (M ; g) ! (N ; h) is said to be a harmonic

mapping if f is a critical point to E at C1(M ;N ) i.e. for any smooth variationft : M ! N with �e < t < e and f0 = f we have

d

dtE(ft)

��t=0

= 0

37

38 A. HARMONIC MAPS

Let f�1TN denote the pull-back of the tangent bundle TN of N via f,f�1TN = �(p; u) 2 M � TN j p 2 M ; u 2 Tf(p)M

:The variation vector field V along f defined by

V (p) = d

dtft(p)

��t=0

for all p 2 M ;is a C1 mapping from M into TN satisfying

V (p) 2 Tf(p)N for all p 2 M :We may then define a connection r� called the pull-back connection, on

the set of smooth sections of X 2 f�1TN by

(r�X V )(p) = Nrdf(X )V = d

dtN P�1fÆst

V (s(t))

��t=0

; p 2 M ;where t 7! s(t) 2 M is a C1 curve in M satisfying s(0) = p, s0(0) = Xp 2TpM , and st is a curve given by st(s) = s(s), 0 � s � t that is, the restrictionof s to the part between p and s(t). The map N PfÆst : Tf(p)N ! Tf(s(t))N isthe parallel transport along a C1 curve f Æ st with respect to the Levi-Civitaconnection Nr on N .

FACT A.2 (First variational formula). Let f : (M ; g) ! (N ; h) be a smoothmap. For a smooth variation ft of f put

V (p) = d

dtft(p)

��t=0

; p 2 M

thend

dtE(ft)

��t=0

= � ZM

h(V ; t(f))dvolM

Thus f is a harmonic mapping if and only ift(f)(p) = 0 for all p 2 M :This is known as the Euler-Lagrange equation for harmonic maps. The

section t(f) of the pull-back bundle f�1TN is called the tension field of f andis given by t(f)(p) = mX

i=1

�r�ek

df(ek)� df(rekek)� : (A.1)

The tension field can be given in local coordinates by the following: let(x1; x2; : : : ; xm) and (y1; y2; : : : ; yn) be local coordinates around p and f(p) inM and N , respectively. Then t(f) is given byt(f) = nXg=1

t(f)g ��yg ; where (A.2)t(f)g(p) = mXi;j=1

g ij

� �2fg�xi�xj� mX

k=1

G kij (p)

�fg�xk+ nXa;b=1

NGgab (f(p))�fa�xi

�fb�xj

�:

A. HARMONIC MAPS 39

Here G kij ,

NGgab are the Christoffel symbols on (Mm; g) and (N n; h), respec-tively.

Bibliography

[1] C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., 6 (1841),

309–320. With a note appended by M. Sturm.

[2] J. H. Jellet, Sur la Surface dont la Courbure Moyenne est Constant, J. Math. Pures Appl., 18 (1853), 163–167.

[3] H. Hopf, Differential Geometry in the Large (Seminar Lectures New York University 1946 and Stanford University1956), Lecture Notes in Mathematics 1000, Springer Verlag, 1983.

[4] A. D. Alexandrov, Uniqueness theorem for surfaces in the large, V. Vestnik, Leningrad Univ. 13, 19 (1958), 5–8,

Amer. Math. Soc. Trans. (Series 2) 21, 412–416.

[5] H. C. Wente, Counterexample to a conjecture of H. Hopf, Pac. Journal of Math., 245 (1986), 193–243.

[6] N. Kapouleas, Constant mean curvature surfaces in Euclidean three-space, Bull. AMS, 17 (1987), 318–320.

[7] N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differ. Geom., 33 (1991),

683–715.

[8] N. Kapouleas, Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math., 119 (1995),

443–518.

[9] M. Sphivak, A Comprehensive Introduction to Differential Geometry, vol. 4, Publish or Perish, Inc., 2 ed., 1979,

455–500.

[10] K. Weierstrass, Mathematische Werke von Karl Weierstrass, vol. 3, Mayer & Müller, 1903, 39–52.

[11] K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tôhoku Math. J., 32 (1980), 147–153.

[12] J. Eells, The Surfaces of Delaunay, The Math. Intelligences, 1 (1987), 53–57.

[13] E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Am. Math. Soc., 149 (1970), 569–573.

[14] K. Kenmotsu, Weierstrass Formula for Surfaces of Prescribed Mean Curvature, Math. Ann., 245 (1979), 89–99.

[15] R. Osserman, Curvature in the Eighties, Amer. Math. Monthly, 97 (1990), no. 8, 731–755.

[16] A. Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differ. Geom., 20

(1984), 215–220.

[17] J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces,Comm. Anal. Geom, 6 (1998), 633–668.

[18] J. Dorfmeister, I. McIntosh, F. Pedit and H. Wu, On the meromorphic potential for a harmonic surface in ak-symmetric space, Manuscripta Math., 92 (1997), 143–152.

41

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