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3 i Abstract The aim of this Master s dissertation is to give a survey of some basic results regarding surfacesëof constant mean ) 0 curvature (CMC) inê3. Such surfaces are often called soap bubbles since a soap film in equilibrium between two regions is characterized by having constant mean curvature. The surface area of these surfaces is critical under volume-preserving deformations. CMC surfaces may also be characterized by the fact that their Gauss map N :Ë S2 is harmonic i.e. it satisfies Ø(N whereø(n ) is the tension field of N, generalizing the classical Laplacian. This is a non-linear system of partial differential equations. It was proved in the 1990s that this system has global solutions on compact surfaces of any genus g 0. In this dissertation we study necessary and sufficient conditions for a surface to have CMC. We study the minimal case (characterized by mean curvature H 0) and the well-known Weierstrass representation for such surfaces. Also CMC surfaces with rotational symmetry are considered and a generalization of the Weierstrass representation to surfaces of non-zero constant mean curvature is presented. Finally we show that the only compact embedded CMC surfaces inê3 are spheres. It has been my intention throughout this work to give references to the stated results and credit to the work of others. The only part of this Master s dissertation which I claim is my own is the elementary proof of a special case of Ruh-Vilms theorem for surfaces inê3 given in Theorem 4.1.

4 ii

5 iii Acknowledgements I wish to thank my supervisor, Sigmundur Gudmundsson, for his time, knowledge and patience. In particular, I wish to express my gratitude for him inspiring me to study the wonderful subject of geometry. Carl Johan Lejdfors


7 Contents Short History 1 Chapter 1. Some basic surface theory Notation Isothermal coordinates The tension field 7 Chapter 2. Minimal surfaces Conformality of the Gauss map The Weierstrass representation formula 10 Chapter 3. CMC surfaces of revolution Kenmotsu s solution Delaunay s construction 17 Chapter 4. CMC surfaces Harmonicity of the Gauss map Kenmotsu s representation formula 24 Chapter 5. Compact CMC surfaces 33 Recent developments 35 Appendix A. Harmonic maps 37 Bibliography 41 v


9 Short History In 1841 Delaunay characterized in [1] a class of surfaces in Euclidean space which 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-trivial examples of surfaces having constant mean curvature, the sphere being the trivial case. In an appended note to Delaunay s paper M. Sturm characterized these surfaces variationally as the extremals of surfaces of rotation having fixed volume while maximizing lateral area. Using this characterization the following theorem was obtained: THEOREM (Delaunay s theorem). The complete immersed surfaces of revolution inê3 having constant mean curvature are exactly those obtained by rotating about their axis the roulettes of the conics. These surfaces where also recognized by Plateau using soap film experiments. In 1853 J. H. Jellet showed in [2] that ifëis a compact star-shaped surface inê3 having constant mean curvature then it is the standard sphere. A hundred years later, in 1956, H. Hopf conjectured that this, in fact, holds for all compact immersions: CONJECTURE (Hopf s conjecture). LetËbe an immersion of an oriented, compact hypersurface with constant mean curvature H 0inÊn. ThenËmust be the standard embedded (n 1)-sphere. Hopf proved the conjecture in [3] for the case of immersions of S 2 intoê3 having constant mean curvature and a few years later A. D. Alexandrov showed the conjecture to hold for any embedded hypersurface inên, see [4]. It was widely believed that this conjecture was true until 1982 when Wu-Yi Hsiang constructed a counterexample inê4. Two years later Wente constructed in [5] an immersion of the torus T 2 inê3 having constant mean curvature. Wente s construction has been thoroughly studied but has only been able to create surfaces having genus g 1. A different method for constructing surfaces inê3 having constant mean curvature of any genus g 3 was presented in 1987 by N. Kapouleas [6]. A proof of the fact that there exist CMC-immersions of compact surfaces of any genus was published in [8] in 1995 by the same author. 1


11 CHAPTER 1 Some basic surface theory In this chapter we introduce the notation to be used in this text. We also introduce some basic results concerning isothermal coordinates and the tension field of the Gauss map of a surface inê Notation DEFINITION 1.1. A non-empty subsetëonê3 is said to be a regular surface if for each point p¾ëthere exists an open neighborhood U inë around p and a bijective map (x y): U Ë Ê2 that its inverse X : (U ) U such i. is a homeomorphism, ii. is a differentiable map, iii. (X x X y )(q) 0for all q¾ (U ). The functions x y are called local coordinates around p. The map X is called a local parametrization ofëaround p. LetËbe a regular surface inê3 and p¾ëbe an arbitrary point. By a tangent vector toë, at the point p, we mean the tangent vector ¼(0) of a differentiable parametrized curve : ( ) Ëwith (0) p. y p¾ë The set of tangent vectors ofëat a point p¾ëis called the tangent space ofëat and is denoted by T pë. A local parametrization X determines a basis X x X of T pë, called the basis associated with X. On the tangent plane we have the usual induced metric from the ambient spaceê3 with the associated quadratic form I p : T pë Êcalled the first fundamental form ofëat p¾ë. Given a local parametrization X ofëand a parametrized curve (t) X (x(t) y(t)) for t¾( ) p (0) we have y¼«2 with the following form I p ( ¼(0)) ÅX x x¼ X y X x X x p (x¼) 2 2ÅX x X y«p x¼y¼ ÅX y X y«p (y¼) 2 (1.1) E(x¼) 2 2Fx¼y¼ G(y¼) 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 surfaceëinê3 at a point p¾ëthat the map y y¼ X x x¼ X 3

12 4 1. SOME BASIC SURFACE THEORY N :Ë S 2 defined by x X y N (p) X (1.2) X x X y (p) II p (v) ÅdN p (v) v«y¼ with p (0) X (x(0) y(0)), we get is well defined. This map is known as the Gauss map ofë. The quadratic form II p defined in T pëby is called the second fundamental form ofëat p. Given a local parametrization X onëat a point p¾ëand, as above, letting be a parametrized curve such that (t) X (x(t) y(t)) for t¾( ) dn p ( ¼) N¼(x(t) y(t)) N x x¼ N ( ¼) ( ¼) ¼«y Since N N 1 we must have that N x N y¾t y pëand hence N x a 11 X x a 21 x¼«x (1.3) N y a 12 X x a 22 X for some functions a ij. We find that II f p ÅdN p ÅN x x¼ N y y¼ X x x¼ X g e 2 v e(x¼) 2 2fx¼y¼ g(y¼) xx (1.4) where N x X x N X yy«yx«y«ån y X ÅN x«ån X xy«ån X x X ÅN y X y«ån X Using the terms from equations (1.1), (1.3) and (1.4) we arrive at eg a 11 ff 2 fg a 12 gf 2 EG F EG F fe ge a 21 ef a 22 ff EG F EG F known as the Weingarten equations. Continuing by using that X x X y is a basis for T pëand that N is orthogonal to both X x and X y we have that X x X y N is a basis forê3. We find that X xx 1 11 X x 2 11 X X y en yx 1 12 X x 2 12 X X xy 1 21 X x 2 21 X X y fn yy 1 22 X x 2 22 X y gn x 0 (1.5) The k ij are known as the Christoffel symbols and are invariant under isometries (i.e. can be computed from the first fundamental form alone). Using that (X xx ) y (X xy )

13 we find that y ISOTHERMAL COORDINATES 5 (X yy ) x (X xy ) N xy N yx ) g 2 11 e y f x e 1 12 f ( 2 12 f y g x e 1 22 f ( 2 22 E X x X x Ð2 ÅX K eg f 2 EG F 2 H eg 2fF ge 2(EG F 2 ) 1 12 ) g 2 12 (1.6) These equalities are known as the Mainardi-Codazzi equations. DEFINITION 1.2. LetËbe a surface inê3 and p¾ëan arbitrary point. Let dn p : T pë T pëbe the differential of the Gauss map. Then the determinant of dn p is called the Gaussian curvature K ofëat p. The negative half of the trace of dn p is called the mean curvature H ofëat p. In terms of the first and second fundamental forms K and H are given by (1.7) (1.8) 1.2. Isothermal coordinates In this section we introduce the notion of isothermal coordinates which is a useful tool in differential geometry. DEFINITION 1.3. LetËbe a surface y«g F 0 inê3. Then local coordinates (x y): U Ë Ê2 onëare said to be isothermal if there exists a strictly positive function, called the dilation,ð: U Ë Êsuch that y X We have the following result regarding existence of isothermal coordinates on an arbitrary surface inê2. THEOREM 1.4. LetËbe a differentiable surface inê3 and p¾ëbe a point onë. Then there exists an open neighborhood U of p and isothermal coordinates (x y): U Ë Ê2 2Ð2 around p. This was proved in the analytic case by Gauss. For a complete proof in the general case please see [9]. Having chosen isothermal coordinates the mean curvature simplifies 2fF H eg ge 2(EG F ) e g 2 The Christoffel symbols similarly 12 x simplify 22 1 y 2Ð2Ð (1.9) Ð2Ð

14 6 1. SOME BASIC SURFACE THEORY Using this we get the following form of the Mainardi-Codazzi equations (1.6): e y f x (e g) 2 g x f y (e g) (1.10) The Weingarten relations (eq. 1.3) reduce to a 11 e Ð2 a 21 a 12 fð2 a 22 gð2 (1.11) SupposeËis a surface inê3 and (x y): U Ë Ê2 are local isothermal coordinates onë. We may then consider the local coordinates (x y) as a complex-valued map z x iy : U Ë. The inverse X : z(u ) Ëcan then be considered as y map from an open subset z(u ) in intoëi.e. a local parametrization ofë. We then have X z 1 X x ix 2 U Ë The complex notation for surfaces inê3 has many advantages which we will be useful in chapters 2 and 3. Letting be the usual inner product inê3 and let ( ) be the complex bilinear extension of in 3 we have the following result. PROPOSITION 1.5. LetËbe a surface inê3 and let z x iy : be local isothermal coordinates onë. Then the inverse X : z(u ) Ëof z is conformal i.e. satisfies y«0 y«0 4 (X z X z ) X x 2 X y 2 2iÅX x X (1.12) 4 (X z X z) X x 2 X y 2 2iÅX x X Conversely, if z x iy are local coordinates onësatisfying equation U Ë (1.12) then they are isothermal. ) PROOF. The first statement follows by a direct computation. The reverse implication follows by considering real and imaginary parts of equation (1.12). PROPOSITION 1.6. LetËbe a surface inê3 and z x iy : be local isothermal coordinates onëwith dilationð. Then the inverse X : (U Ëof z satisfies where N :Ë S 2 4X zz X xx X yy 2Ð2 HN is the Gauss map ofë. PROOF. By a direct computation using the differentiated form of equation (1.12) we have 4X zz 2Ð2 [(X zz X z ) X z (X zz X z) X z] 4 (X zz N ) N ÅX xx X yy N«N 4(e g)n 2Ð2 HN

15 1.3. THE TENSION FIELD 7 This immediately gives our sought result The tension field In this section we give an explicit formula for the tension field (see appendix A) for maps from a surface inê3 into S 2 in terms of local isothermal coordinates. PROPOSITION 1.7. LetËbe a surface inê3 and :Ë S2 be a map into the unit sphere S 2 inê3. If (x y): U Ë Ê2 are local isothermal coordinates onëthen the tension fieldø( ) of is locally given by T Ø( ) 1Ð2 i.e. as the tangential part of the classical Laplacian x y 2 ek) inê2. PROOF. By the definition of the tension field of a smooth map :Ë S2 we have Ø( ) 2 k 1 d (ek) Ö ek d (Öek whereö is the pull-back connection on the pull-back bundle 1Ð2 1 TS overë 2 via. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be isothermal coordinates around p. We then have k Ð2 Öe k e k e k e k ( 2 2e 1Ð2 )e k g(e k e k ) grad Ö1Ðx 1Ðx Ö1Ðy 1Ðy 1Ð2 k Ð2 e k e k 1Ð2 ( 2 2e 1Ð2 )e k grad for k 1 2. Then using the definition of the gradient grad we obtain Öe 1 e 1 Öe 2 e 2 e 1 e 1 e 2 e 2 Ð2 e 1 ( 1Ð2 )e 1 e 2 ( 1Ð2 )e 2 grad ) 1Ðx 1Ð x 1Ðy 1Ð 1Ðx 1Ð x 1Ðy 1Ð y This implies that d (Öe 1 e 1 Öe 2 e 2 (1.13) y

16 ) 1Ðx 1Ð ) 1Ðy 1Ð y T 1Ð y 1Ð x T 1Ð x 1Ð y 1Ð SOME BASIC SURFACE THEORY The other term is given by Ö e 1 d (e 1 Ö e 2 d (e 2 1Ð T It follows by equations (1.13), (1.14) and (1.15) that 2 T 1Ð2 T x y Ø( ) Ö e 1 d (e 1 ) Ö e x 1Ð2 2 T 2 d (e 2 ) d (Öe 1 e 1 ) d (Öe 2 e 2 ) (1.14) (1.15) Harmonic maps generalize the concept of harmonic functions well known from complex analysis. A harmonic map is one for which the tension field vanishes everywhere and, as stated in Appendix A, arises as a critical point of a certain variational problem. THEOREM 1.8. LetËbe a surface inê3 and :Ë S2 be a map into the unit sphere Å xx y«s 2 inê3. If is conformal then it is harmonic. PROOF. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be xx x Å yx y«å yx x«å yy x«å yy y«å xy x«å xy y«local isothermal coordinates around p. Then the conformality of means that By differentiating we then obtain and thereforeå xx yy x«å yx yy«å xx yy y«thatø( ) 1Ð2 T 0 Å yx xx«å xy xx«0 Å yx yy«0 These relations imply Å x y«0 and x x Å y y«

17 CHAPTER 2 Minimal surfaces In this chapter we introduce some results concerning minimal surfaces. We also prove the famous Weierstrass representation for minimal surfaces. DEFINITION 2.1. A surfaceëinê3 is said to be minimal if its mean curvature H satisfies H Conformality of the Gauss map PROPOSITION 2.2. LetËbe a minimal surface inê3. Then the Gauss map N :Ë S ofëis conformal. PROOF. Let p¾ëbe an arbitrary point onëand (x y) be local isothermal coordinates around p. Then it follows by H e g y 2Ð2 0 y and equation (1.11) that N x 1Ð2 ex x fx N y 1Ð2 fx x 0 ex This implies that ÅN y«e2 f 2 x N y«0 and N y N Ð2 Hence N is conformal. A partial reverse implication of the previous theorem is obtained via the following. PROPOSITION 2.3. LetËbe a real analytic surface inê3 and N :Ë S2 be a Gauss map ofë. If N is conformal thenëis either minimal y or part of a sphere. PROOF. For local isothermal coordinates (x y) onëwe have ex x fx y 1Ð2 fx x gx 1Ð4 ef X e g 2fH y«x X x fgåx y X fð2 0 ÅN x N y«1ð2 x N x ÅN Let p¾ëbe a point. Suppose H(p) 0 then there exists an open neighborhood V Ëaround p such that f V 0. For every point in q¾v we have, 9

18 10 2. MINIMAL SURFACES by equation (1.11), that N x and N y are parallel with X x and X y, respectively. By x conformality we y have that N x N y and, since (x y) are isothermal, that X x X y. Hence q is umbilical i.e. the principal curvatures coincide. Let k k 1 k 2, where k 1 and k 2 are the principal curvatures. Differentiating N kx x and N kx y gives (kx x ) y (N x ) y (N y ) x (kx y ) x and since X x and X y N are linearly independent we must have k x k y 0 so k is constant. If k 0 then H 0 which contradicts 2 the assumption. Hence k 0 and kx a where a is a constant vector. Then X is a local parametrization for a sphere having radius 1 k centered at a k since 1 X a 2 1 N 2 1 k k k Thus by real analyticityëis either minimal or part of a sphere The Weierstrass representation formula The Weierstrass representation formula was first presented by Karl Weierstrass in [10]. It states that given two holomorphic functions defined on some simply connected subset of there exists an associated minimal surface. This surface is unique up to motions. THEOREM 2.4. LetËbe a surface inê3 and x iy : U Ë be local isothermal coordinates onë. Suppose U is an open simply connected subset ofë. If X : z(u ) Ëis the inverse of z thenëis minimal if and only if the derivative X z U Ë : z(u ) 3 is holomorphic. PROOF. This is a direct consequence of Proposition 1.6 and the fact that a map f : U is holomorphic if and only if f z 0. Integration gives us the following corollary. COROLLARY 2.5. LetËbe a surface inê3 and x iy : be local isothermal coordinates onë. Suppose U is an open simply connected subset ofë. Then the inverse X : z(u ) Ëof z is given by X (z) 2 Re z X z z 0 (z)dz C (2.1) where C is some constant vector inê3. PROOF. We have X z dz 1 (X x ix y )(dx idy) dx dx 2 1 X x dx X y dy i(x x dy X y 2 X zd z 1 X x dx X y dy i(x x dy X y 2

19 2.2. THE WEIERSTRASS REPRESENTATION FORMULA 11 Integrating dx X z dz X zd z 2 Re X z dz 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 connected subset of. Suppose f : V is holomorphic on V, g : V is meromorphic on V and the product fg 2 is holomorphic on V. Then X : V Ê3 defined by X (z) Re z X z z 0 (z)dz (2.2) where X z (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 that equation (2.2) define isothermal coordinates. This, however, follows by direct computation using Proposition 1.5. Examples of minimal surfaces are the surfaces of Sherk (Fig. 2.1) and Catalan (Fig. 2.2). FIGURE 2.1. Sherk s minimal surface. (f g) ( 2 1 z 4 z) An interesting observation is the fact that the Gauss map of minimal surfaces generated using Theorem 2.6 can be identified with the complex valued function g. PROPOSITION 2.7. LetËbe a minimal surface inê3 given by the Weierstrass representation X (z) Re z f (z)(1 g(z) ) 2g(z))dz 2 i(1 g(z) 2 Then the Gauss map N :Ë S projection, with g. z 0 2 ofëmay be identified, via stereographic

20 12 2. MINIMAL SURFACES G 1 FIGURE 2.2. Catalan s minimal surface. (f g) 3 z 1 i z z PROOF. Let 1Æg then Re g 2Im g g G (2 2 1) 1 g2 Now the real and imaginary parts of f (1 g 2 ) if (1 g ) 2fg 2 represent two orthogonal tangent vectors onëinê3. Then fg g i 2 1 g 2 (1 g g) 1 G fg 2 g g g g g and since G 1it is clear that G is a Gauss map forë. We can conclude that the above examples (Figures 2.1, 2.2) have bijective Gauss map i.e. for every point p¾s2 there exists only one point onëhaving that point as a normal. Amongst the Enneper surfaces, defined by (f g) (1 zn ) for every n¾æ, only the case of n 1 satisfies this property (see Figures 2.3, 2.4 and 2.5). FIGURE 2.3. First order Enneper surface. (f g) (1 z)

21 2.2. THE WEIERSTRASS REPRESENTATION FORMULA 13 FIGURE 2.4. Second order Enneper surface. (f g) (1 z 2 ) FIGURE 2.5. Third order Enneper surface. (f g) (1 z 3 )


23 CHAPTER 3 CMC surfaces of revolution In this chapter we study complete CMC surfaces with rotational symmetry. We present Kenmotsu s modern solution given in [11] to the problem of finding all such surfaces. Furthermore we describe the classical construction of the same due to Delaunay, see [1]. DEFINITION 3.1. A surfaceëinê3 is said to have constant mean curvature (CMC) if and only if there exists a c¾êsuch that H c Kenmotsu s solution Let : with (s) I Ê Ê2 x(s) y(s) be a parametrization of some regular planar C 2 curve. Assume that is an arclength parametrization and that 0 is contained in the open interval I. LetËbe the surface of revolution inê3 defined by (s ) (x(s) y(s) cos y(s) sin ) s¾i 0 2Ô Then the first and second p (x¼¼y¼ d 2 x¼ fundamental x¼¼yy¼ x¼yy¼¼ 0 2 x¼yd 2 forms are given by I p ds 2 y 2 II x¼y¼¼)ds x¼x¼¼ y¼y¼¼ 0 s¾i s¾i Assuming y(s) 0for s¾i we have, by definition of H, that (3.1) Multiplying by x¼and y¼, respectively, and simplifying using the fact that (x¼) 2 (y¼) 2 1 and we obtain 2Hyx¼ (yy¼)¼ 1 0 and 2Hyy¼ (yx¼)¼ 0 Z¼ 1 0 s¾i Setting Z(s) y(s)y¼(s) iy(s)x¼(s) and combining these equations the following first order complex linear differential equation is obtained 2iHZ (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 c 1 ic 2 15

24 y c CMC SURFACES OF REVOLUTION for some C c 1 ic 2¾. 2 This gives us y(s) Z(s) Õ(s c 1 ) 2 c2 Z 2 (3.3) x¼(s) Im Ô(s c 1 ) 2 c2 2 By integrating x we obtain 1 x c 2 arcsinh s c s c 1 sinh x c 2 hence c 2 c 2 Substituting into equation (3.3) we obtain y Õ(s c 1 ) 2 c 2 sinh 2 2 x c 2 c 2 c 2 2 c 2 2 cosh x c It is clear that this is a parametrization of a catenary. If H 0 then Z(s) 1 1 e 2iHs C e 2iHs 2iH 1 (1 2iHC) e 2iHs e 2iHs (3.4) 2iH Be i(2hs ) 1 2iH where Be i 1 2iHC for y some B ¾Êand C¾ is an arbitrary constant. Using the fact that y(s) 0 we have by translation of the arclength and by restricting our attention to H 0 y(s) Z 1 2HÔ1 B2 2B sin 2Hs Z 1 B sin 2Hs x¼(s) Im Ô1 B dt 2 2B sin 2Hs Hence the solution to equation (3.4) is the one-parameter family of surfaces of revolution having constant mean curvature H given by 1 Bsin 2Ht Ô1 B2 2B sin 2Ht (s; H B) s 0 1 2HÔ1 B 2 2B sin 2Hs (3.5) for any B¾Êand H 0. Studying for varying B (see Fig. 3.1) we find that (s; H 0) is a generating curve for a right circular cylinder and (s; H 1) is a generating curve for a sequence of continuous half-circles centered on the x-axis. For 0 B 1 the function x(s) increases monotonously whereas in the case of B 1 it does not.

25 3.2. DELAUNAY S CONSTRUCTION 17 THEOREM 3.2 (Delaunay s theorem). Any complete surface of revolution with constant mean curvature is either a sphere, a catenoid or a surface whose generating curve is given by (s; H B) for some B¾Ê. PROOF. Let H¾Êbe given and let H(s) be a generating curve parametrized by arclength for a complete surface of revolution having constant mean curvature H. By uniqueness of solution of (3.2) we have H(s) (s; H B) for some B¾Ê. FIGURE 3.1. Solutions for H 0 5 and B Delaunay s construction The surfaces of Delaunay are constructed by rolling a conic along a straight line in the plane and taking the trace of the focus F. This is called a roulette of the conic. This trace then describes a planar curve which is rotated about the axis along which it was rolled. This gives a surface of revolution having constant mean curvature. The construction presented here is based on the article [12] by J. Eells is a parabola. Let be a parabola given by : t (t at 2 ) for some a which we take to be strictly positive. Let F be the focus and A be the vertex of (see Figure 3.2). Let K be a point on and denote by P the intersection of the tangent line of at K with the horizontal axis. By solving the line equation for the tangent at K we find that for K (t) (t at 2 ) then P (t 2 0). This implies that PK OP. And since FOP PKF we F K A P O FIGURE 3.2. is parabola

26 18 3. CMC SURFACES OF REVOLUTION FIGURE 3.3. Catenary also. By definition of the trigonometric functions have OPF KPF Ô2 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 (3.6) ds cos Õ dx where denotes the angle made by the tangent of F with the x-axis. We then arrive at c y dx dx y ds y dx dy ds 2 ds 2 2 Õ1 dy dx 2 or, equivalently, dy c 2 (3.7) c dx Öy 2 The solution to this differential equation is given by y c e x c e x c c cosh x 2 c which is a catenary (Fig. 3.3). The corresponding surface of revolution is the catenoid (Fig. 3.4). The Gauss map of the locus of F into S 1 is given by x x where cos x dx ds c y showing that the Gauss map is injective onto an open semicircle.

27 3.2. DELAUNAY S CONSTRUCTION 19 FIGURE 3.4. Catenoid is an ellipse. Let F and F¼be foci of and O its center. Take a point K on and let P and P¼be the points on the tangent at K closest to F and F¼, respectively (Fig. 3.5). As above, letting PK be the x-axis and PF (P¼F¼) the ordinate y (y¼). Let T and T¼denote the intersection with the x-axis y¼ of the tangent of the locus of F and F¼, respectively. We have FKP F¼KP¼. Also the tangent of the locus of F (F¼) is orthogonal to FK (FL¼) and. This gives us hence KFT KF¼T¼ Ô2 y FK sin FKP cos FTP dx ds F¼K sin F¼KP¼ cos F¼T¼P¼ dx ds 2 From the characterization of the ellipse, FK F¼K 2a for some a 0, and the pedal equation, PF P¼F¼ b for some b 0, we find y y¼ 2a dx and yy¼ b 2 so ds T P K P F O F FIGURE 3.5. is ellipse

28 20 3. CMC SURFACES OF REVOLUTION 2 0 FIGURE 3.6. Undulary, H 0 5, B y 2 2ay dx ds b Taking a b we get the following cases (when the angle is obtuse and acute, respectively) dt dx y 2 2ay (3.8) ds b A solution to this problem is given in Section B sin 2Ht x(s) s 0Ô1 B2 2B sin 2Ht (3.9) y(s) 1 2HÔ1 B2 2B sin 2Hs b where H 1 2. The locus of either foci is called 4H 2 the undulary (Fig. 3.6). The corresponding surfaces is called the unduloid (Fig. 3.7). The Gauss map of the undulary is given by where x x cos x dx 2ay ds y2 b2 2a and B Õ1 FIGURE 3.7. Unduloid, H 0 5, B 0 5.

29 PF P¼F¼ is a hyperbola. We proceed as in the case of the ellipse but instead use the characterization FK F¼K 2a 0 of the hyperbola and the pedal equation b equation 3.2. DELAUNAY S CONSTRUCTION (Fig. 3.8). We arrive at the differential dx y 2 2ay b ds K P F O F P FIGURE 3.8. Hyperbola 2 This differential equation can be solved in the same manner as for the ellipse with the exception that B in equation (3.9) is given by B Ö1 b2 4H 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 map of the nodary is given by where x x cos x y2 b 2ay 2 This map has no extreme points and is clearly surjective. FIGURE 3.9. Nodary, H 0 5, B 1 5.

30 22 3. CMC SURFACES OF REVOLUTION FIGURE Nodoid, H 0 5, B 1 5.

31 CHAPTER 4 CMC surfaces M The main aim of this chapter is to give a new elementary proof of a special case of the Ruh-Vilms theorem. We also present the Kenmotsu representation formula for CMC surfaces with H Harmonicity of the Gauss map Let (M g) be an orientable m-dimensional Riemannian manifold, i : ) möh Êm p be an isometric immersion and N : M G p o(êm p ) be the associated Gauss map, mapping x¾mto the oriented normal space of i(m) at i(x). Then Ruh-Vilms theorem presented in [13] states that the tension fieldø(n ) of N satisfies Ø(N whereöh 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 curvature vector field H is parallel. For surfaces inê3 this is equivalent to the ) surface having constant mean curvature. THEOREM 4.1. LetËbe an oriented surface inê3. ThenËhas constant mean curvature if and only if its Gauss map N :Ë S2 is harmonic. PROOF. We prove that the following equation Ø(N 2 grad H holds. Our sought result then follows trivially. Let p¾ëbe an arbitrary point and (x y): U Ë Ê2 be isothermal coordinates around p. Then we have y (N xx ) T N xx X x X x ÅN xx X y«x y Ð2 1Ð2 ( e x N x X xx )X x ( f x ÅN y x X yx«)x 1Ð2 ( e x N x X xx )X x ( e y (e g) 2 22 xx ÅN x X yx«)x This follows by using yx«e x x N x X x N xx X x N x X f x xån x X y«ån xx X y«ån x X 23

32 24 4. CMC SURFACES y and the Mainardi-Codazzi equations (1.10). Similarly we have (N yy ) T 1Ð2 ( g x (e g) 1 11 ÅN y X xy«)x y ( g y ÅN y X yy«)x By using the Christoffel relations (1.9) we 11 y«find N x X xx Åa 11 X x a 21 X y 1 12 X x 2 11 X Ð2 a a 21 2 e 1 11 f 2 ÅN x X 21 yx«ð2 a a 21 2 f 1 11 e 2 ÅN y X yx«ð2 a a 22 2 g 1 11 f 2 ÅN y X 22 x«22 yy«ð2 a a 22 2 f 1 11 g 2 Adding and using the above equations we get 22 N X x ÅN xx N yy X 1Ð2 (e x g x ) (e g) 1 N 11 x X xx xy«y«11 ÅN y X 1Ð2 (e x g x ) 2(e g) 1 22 Å N X ) y«ån xx N T yy X 1Ð2 (e y g y ) 2(e g) 2 Hence we have 1Ð2 Ø(N ( N 1Ð2 e x g x e g ) x X Ð2 x 1Ð2 e y g y e g y X Ð2 y 1Ð2 e x g x 1Ð2 ) X x 1Ð2 e y g y 1Ð2 H ) X y 1Ðx (e g) 1Ð2 (e g) 1Ðx ( 1Ð2 ) 1ÐX x 1Ðy (e g) 1Ð2 (e g) 1Ðy ( 1Ð2 ) 1ÐX y 2 grad This proves our theorem Kenmotsu s representation formula Ð4Ð2 Ð4Ð2 Ð2 (e g)x ( Ð2 (e g)y ( In this section we show a corresponding result to Weierstrass representation formula for surfaces having non-zero constant mean curvature.

33 Let S 2 be the unit sphere inê3. Cover S 2 by open sets U i, i 1 2 where U 1 S2 n and U 2 S2 s and n and s are the north and south pole, respectively. Let be the stereographic projection with respect 1 to the north pole n: (x) x 1 ix 2 2 for x (x 1 x 2 x 3 )¾U (4.1) 1 3 For a surfaceëinê3 having Gauss map N :Ë S2 consider the following composition :ËN S which we also call the Gauss map ofë. This map is considered as a complex mapping from a 1-dimensional complex manifoldëinê3 into the Riemann sphere. Using this notation we have the following theorem presented in [14] due to K. Kenmotsu KENMOTSU S REPRESENTATION FORMULA 25 THEOREM 4.2 (Kenmotsu s representation formula). Let V be an open simply connected subset of and H be an arbitrary non-zero real constant. Suppose : V is a harmonic function into the Riemann sphere. If z 0 then X : V Ê3 defined by X (z) Re z X z z 0 (z)dz (4.2) with ( 1) X z (z) ) 2 (z) (z) 1 (z) 2 i(1 (z) 2 H(1 (z) (z)) 2 z for z¾v, is a regular surface having as a Gauss map and mean curvature H. Ø( ) 4Ð z z First we derive an explicit formula of the tension field of the Gauss map :Ë of an arbitrary surfaceë. PROPOSITION 4.3. LetËbe a surface inê3 and :Ë be a Gauss map onë. If z x iy are local isothermal coordinates with dilationðthen (4.3) z z PROOF. Let z x iy be local isothermal coordinates and denote (z) u iv. Then 4 (1 u2 v2 ) (du2 dv 2 ) 2 which gives us Christoffel symbols 12 on the Riemann sphere u u 2 v 2v u 2 v h g Ð2 (dx 2 dy 2 ) and

34 y CMC SURFACES 11 u x 2 u Ø(u) 1Ð2 u 1 22 v x 2 v 1 By the explicit formula for the tension field (eq. A.2) we get 12 u xv y y x u yv x 2 y 2 v x v y Ø(u) iø(v) where is the classical Laplacian. Adding this and the similar formula forø(v) we arrive at Ð2Ø( ) Ð2 (u iv) u iv 2 u x 2 2 2iu xv u 1 2 v z z u y 2 2iu yv 8 z 8 z z z This proves our sought formula. H z Ý z :Ë Ê Next we show that a surface having prescribed mean curvature H satisfies the following equation. LEMMA 4.4. LetËbe a surface inê3 having mean curvature H and let :Ë be a Gauss map ofë. Then z whereý 1Ð2 (e g) 2 if. PROOF. We show that the following equation holds H 1 2(X 1 ix 2 ) 3 2 x 2 By direct computation using the Weingarten equations (1.11) together with equation (4.1) we have z z N 1 in N 2Ð2 (1 N 3 ) Xx N 1 3 Xx 1 N 1 X 3 i Xx 2 N 3 Xx N 2 2 Xx e 3 z (4.4) (1 N 3 ) X 2 y X 2 x ixy ix 2 x (N 1 1 in 2 ) Xy ix 3 x f 3