On the Weierstrass-Enneper Representation of Minimal Surfaces Albin Ingelström Bachelor s thesis 2017:K15 Faculty of Science Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM
Abstract The Weierstrass-Enneper representation of minimal surfaces gives an important link between differential geometry and complex analysis. In short, it tells us that any regular minimal surface can be represented by two meromorphic functions of one complex variable. The main focus of this paper is to prove this fact and to give examples of this representation for well known minimal surfaces. We will assume some basic knowledge of complex analysis while some basic results from differential geometry will be stated in Chapter 1. This to remind the reader of these and to establish the notation that will be used in the rest of the thesis. In Chapter 2 the Weierstrass-Enneper representation and its preliminaries will be discussed and in Chapter 3 we will look at some classical examples. Throughout this work it has been my firm intention to give reference to the stated results and credit to the work of others. All theorems, propositions, lemmas and examples left unmarked are assumed to be too well known for a reference to be given.
Acknowledgments This text would not have been if not for the people around me who have encouraged me towards this goal. First I would like to thank my thesis supervisor, Sigmundur Gudmundsson, for his guidance during the writing process and his teaching prior and parallel to this project. My parents have always given me their unlimited support, and for that I am forever grateful. Finally, I would like to thank my partner Malin and my dear friend Kenneth, for making my time here in Lund as good as it could ever be. Albin Ingelström
Contents 1 Minimal Surfaces 1 1.1 General Surface Theory......................... 1 1.2 A Variational Problem.......................... 2 2 The Weierstrass-Enneper Representation 7 2.1 Preliminary Theory............................ 7 2.2 Defining the Weierstrass-Enneper Representation........... 10 3 Examples and Applications 13 3.1 The Catenoid and the Helicoid...................... 13 3.2 Reflection Principles........................... 14 3.3 The Enneper Surface........................... 16 3.4 Scherk s Doubly Periodic Surface.................... 17 4 Summing Up 21 Bibliography 23
Chapter 1 Minimal Surfaces To be able to discuss the properties of minimal surfaces and their relationship with meromorphic functions we first need to establish some terminology. The notation is similar to that used in [4], and complete proofs may be found there. 1.1 General Surface Theory The partial derivatives of X with respect to u and v will be denoted by X u and X v, respectively, and a unit normal vector field to the surface by N. In this text we will be concerned with regular parametrisations. Definition 1.1. A differentiable map X : U R 2 R 3 is said to be a regular parametrised surface if for every q U. X u (q) X v (q) 0, The coefficients of the first and second fundamental forms will be denoted by E, F, G, e, f and g, respectively. The following formulas serve as definitions of these coefficients and for a further discussion of the first and second fundamental forms the reader is referred to [2] and [4]. Definition 1.2. Let X : U R 2 R 3 be a regular parametrised surface. Then its first fundamental form is defined as where E du 2 + 2F dudv + G dv 2, E =< X u, X u >, F =< X u, X v > and G =< X v, X v >. Definition 1.3. Let X : U R 2 R 3 be a regular parametrised surface with the unit normal vector field N given by N = X u X v X u X v. Then its second fundamental form is defined as e du 2 + 2f dudv + g dv 2, 1
where e =< X uu, N >, f =< X uv, N > and g =< X vv, N >. It is well-known that holomorphic functions are conformal i.e. angle-preserving. For all results presented here, we assume that the minimal surface is parametrised by isothermal coordinates. Definition 1.4. A regular parametrisation X : U R 2 R 3 of a surface is said to be conformal if it preserves angles. This means that the surface is parametrised by isothermal coordinates i.e. E =< X u, X u >=< X v, X v >= G and F =< X u, X v >= 0. It is a classical result that, locally, every surface can be parametrised by isothermal coordinates. A complete proof of this can be found in [5]. We now define the important notion of the area of a surface. Definition 1.5. Let X : U R 2 R 3 be a regular parametrisation of a surface. Then the area A of the subset X(D) of X(U) is defined by A(X(D)) = EG F 2 da. D The mean curvature H of a surface is equal to the mean of its principal curvatures. The first and second fundamental forms give us a useful formula for calculating H without determining the principal curvatures. This standard formula is stated here without a proof. Theorem 1.6. Let X : U R 2 R 3 be a regular parametrised surface. Then its mean curvature H satisfies the following equation H = eg 2fF + ge, 2(EG F 2 ) where E, F, G, e, f and g are the coefficients of the first and second fundamental forms, respectively. 1.2 A Variational Problem Definition 1.7. A regular surface in R 3 is said to be minimal if its mean curvature is constantly equal to zero. To explain why a surface with vanishing mean curvature is called a minimal surface we need the following notion of a normal variation. Definition 1.8. A normal variation of a regular parametrised surface X, determined by the function h : D R, is a map ϕ : D ( ɛ, ɛ) R 3 such that where ɛ R and ɛ > 0. ϕ(u, v, t) = X(u, v) + t h(u, v) N(u, v), 2
We now employ the notion of a normal variation to show that a minimal surface is a critical point of the area functional. The following argument may be found in [1]. Let X : U R 2 R 3 be a parametrisation of a regular surface, let D be an open subset of U and let h : D R be a continuous function which is differentiable on D. Then for any fixed small t R the map X t (u, v) = ϕ(u, v, t) = X(u, v) + t h(u, v) N(u, v) also parametrises a surface. Calculating the partial derivatives gives us X t u = X u + thn u + th u N, X t v = X v + thn v + th v N. Then calculating the first fundamental form for each member of this family of surfaces X t yields Similarly, we get We then obtain E t =< X t u, X t u > =< X u, X u > +2th < X u, N u > +2th u < X u, N > + t 2 h 2 < N u, N u > +2t 2 hh u < N u, N > +t 2 h 2 u < N, N > = E 2the + t 2 (h 2 < N u, N u > +h 2 u). F t =F 2thf + t 2 (h 2 < N u, N v > +h u h v ), G t =G 2thg + t 2 (h 2 < N v, N v > +h 2 v). E t G t (F t ) 2 = EG F 2 2th(Eg 2F f + Ge) + R, where every term of R is a multiple of t 2 so that R/t 0 as t 0. By using the general formula we then obtain H = Eg 2F f + Ge 2(EG F 2 ) E t G t (F t ) 2 =(EG F 2 )(1 4thH) + R =(EG F 2 R )(1 4thH + EG F ). 2 Calculating the area for each member X t of this family of surfaces and differentiating with respect to the family parameter t gives us A(t) = Et G t (F t ) 2 da D R = (EG F 2 )(1 4thH + EG F ) da. 2 This shows that D A (0) = D 2hH EG F 2 da. These results provide us with the necessary tools to prove the following: 3
Proposition 1.9. A regular parametrised surface X : U R 2 R 3 is minimal if and only if, for every differentiable function h : D R and every open subset D of U, the area functional A(t) = Et G t (F t ) 2 da has a critical point at t=0. Proof. If X is minimal then H = 0 which implies that A (0) = 2hH EG F 2 da = 0. D D For the converse, let us assume that X is not minimal. Then there exists a point q D such that H(q) 0. Choose h such that h(q) = H(q) and h = 0 outside a small neighbourhood of q. Then, for the area corresponding to this h, we have that A (0) < 0 which contradicts our assumption. Hence X must be minimal. For a conformal regular parametrisation X of a surface, there exists a simple relationship between the harmonicity of its components and the minimality of X. This will be used later on when looking at the relationship between holomorphic functions and minimal surfaces. Theorem 1.10. Let X : U R 2 R 3 be a conformal regular parametrisation of a surface. Then H 0 if and only if the map X is harmonic i.e. X uu + X vv = 0. Proof. By the general formula H = Eg 2F f + Ge 2(EG F 2 ) and the fact that X is conformal, we get that H 0 if and only if e + g = 0. By definition, this means that But by the product rule, we have or equivalently, Similarly, we have < X v, N v >= < X u, N u >. < X v, N v > + < X vv, N >=< X v, N > v = 0, < X v, N v >= < X vv, N >. < X u, N u >= < X uu, N >. This immediately implies that H 0 if and only if or equivalently, < X vv, N > + < X uu, N >= 0, < X uu + X vv, N >= 0. 4
Then, if X uu + X vv = 0 it is clear that H 0. Next, we assume that < X uu + X vv, N >= 0 and will show that X uu + X vv = 0 holds. By using the fact that {X v, X u, N} is a local orthogonal frame for R 3 we only need to show that X uu + X vv is orthogonal to X u and X v i.e. < X uu + X vv, X u >= 0 and < X uu + X vv, X v >= 0. Calculating the first scalar product yields < X uu + X vv, X u >= 1 2 < X u, X u > u + < X u, X v > v < X uv, X v > = 1 2 (< X u, X u > < X v, X v >) u + < X u, X v > v. But since X is conformal this vanishes identically. By the symmetry of u and v, we also have This proves that X is harmonic. < X uu + X vv, X v >= 0. 5
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Chapter 2 The Weierstrass-Enneper Representation In this chapter we will consider the construction of minimal surfaces using complex analysis. Our goal is to define the so called Weierstrass-Enneper representation. To do this we need some theory concerning the function ϕ : U R 2 C 3 defined by ϕ(z) = X u ix v = f(z) 2 ((1 g(z)2 ), i(1 + g(z) 2 ), 2g(z)). It will be shown that if f : C C is a holomorphic function and g : C C is a meromorphic function satisfying certain special conditions, then the map X : C R 3, whose partial derivatives are found in the formula above, will define a minimal surface. To prove this we need the following two lemmas. 2.1 Preliminary Theory Lemma 2.1. Let X : U R 2 R 3 be a conformal parametrisation of a regular surface and let ϕ : U R 2 C 3 be the map given by ϕ = X u ix v. Then X is minimal if and only if each component of ϕ is holomorphic. Proof. The result of Theorem 1.10 tells us that H 0 if and only if which is equivalent to the following X uu + X vv = 0 (X u ) u = ( X v ) v. This is one of the Cauchy-Riemann equations. The other Cauchy-Riemann equation follows directly from the regularity of the surface, since X uv = X vu is just another way of writing (X u ) v = ( X v ) u. 7
Lemma 2.2. Let X : U R 2 R 3 be a conformal parametrisation of a regular minimal surface. Then the holomorphic map ϕ : U C C 3, defined by satisfies the following two conditions ϕ(u + iv) = (ϕ 1, ϕ 2, ϕ 3 ) = X u ix v, ϕ 2 1 + ϕ 2 2 + ϕ 2 3 0 and ϕ is never zero. Moreover, any holomorphic map ϕ satisfying these conditions will give rise to a conformal parametrisation of a minimal surface X by the identity ϕ(u + iv) = X u ix v. Proof. Assuming that X = (X 1, X 2, X 3 ) is a conformal parametrisation of a minimal surface, we see that ϕ 2 1 + ϕ 2 2 + ϕ 2 3 = 3 ((Xu) k 2 (Xv k ) 2 2i Xu k Xv k ) k=1 = X u 2 X v 2 2i < X u, X v >. This vanishes since X is conformal. Further, ϕ vanishes only if both X u and X v are zero, but this can never occur since X is a regular parametrisation. For the converse, we assume that the map ϕ satisfies the given conditions and will prove that X is conformal and minimal. Now define X(u, v) = Re ϕ(ξ)dξ, where γ is a curve in a simply connected domain U C from the starting point z 0 = u 0 +iv 0 to an arbitrary z = u+iv. Note that X is the real part of a holomorphic map. Then calculating the partial derivatives and using the fact that f u = df dz and for a holomorphic function f, we have X u = ( ) ( d u γ Re ϕ(ξ)dξ = Re dz γ X v = ( ) ( d v Re ϕ(ξ)dξ = Re γ dz i γ f v = i df dz, γ ) ϕ(ξ)dξ = Re ϕ(z), ) ϕ(ξ)dξ = Im ϕ(z). This implies that the real part of ϕ is X u and the imaginary part of ϕ is X v, hence ϕ(u + iv) = X u (u, v) ix v (u, v). Since every component of ϕ is holomorphic, we see from Lemma 2.1 that X is a parametrisation of a minimal surface. To show that X is conformal we use the fact that ϕ 2 1 + ϕ 2 2 + ϕ 2 3 = X u 2 X v 2 2i < X u, X v >= 0. 8
This directly implies that F =< X u, X v >= 0 and E =< X u, X u >=< X v, X v >= G. The second condition, namely that ϕ is nowhere zero, implies that X u (w) and X v (w) do not simultaneously vanish for any w U. Together with the identity < X u, X u >=< X v, X v > this implies that neither X u nor X v is zero anywhere. Hence X is regular. Theorem 2.3. Let f : U C C be a holomorphic function and g : U C C a meromorphic function such that fg 2 is holomorphic. Further, assume that if ξ U is a pole of order n of g then ξ is a zero of order 2n of f, and these are the only zeroes of f. Then the map ϕ(z) = f(z) 2 ((1 g(z)2 ), i(1 + g(z) 2 ), 2g(z)) ( ) satisfies the conditions for ϕ of Lemma 2.2. Further, for every such ϕ there exist a holomorphic map f and a meromorphic map g such that ( ) holds. Proof. Given a map ϕ such that ( ) holds, we have that ϕ 2 1 + ϕ 2 2 + ϕ 2 3 = 1 4 f(z)2 (1 g(z) 2 ) 2 1 4 f(z)2 (1 + g(z) 2 ) 2 + f(z) 2 g(z) 2 = f(z) 2 g(z) 2 + f(z) 2 g(z) 2 = 0. If ϕ were to equal zero we would have 0 = ϕ(z) = f(z) 2 ((1 g(z)2 ), i(1 + g(z) 2 ), 2g(z)). The restrictions of the zeroes and poles of f and g, respectively, imply that f(z)g(z) 2 0. So for a fixed z the first and second coordinates can not both equal zero. Hence, ϕ is nowhere zero. Next, we assume that ϕ is a holomorphic map satisfying ϕ 2 1+ϕ 2 2+ϕ 2 3 0, ϕ is never zero and let f(z) = ϕ 1 (z) iϕ 2 (z) and g(z) = ϕ 3 (z) ϕ 1 (z) iϕ 2 (z). Then f is a holomorphic function and g is the quotient of holomorphic functions. If the denominator in g is identically zero, we instead let g(z) = ϕ 3 (z) ϕ 1 (z) + iϕ 2 (z), and proceed with the proof in a similar way. Now the denominator of g is not identically zero, hence g is meromorphic. Further, the relation ϕ 2 1 + ϕ 2 2 + ϕ 2 3 0 9
implies that which, in terms of f and g, becomes (ϕ 1 + iϕ 2 )(ϕ 1 iϕ 2 ) = ϕ 2 3 (ϕ 1 + iϕ 2 ) = fg 2. This last equation, along with the definitions of f and g, show that ϕ(z) = f(z) 2 ((1 g(z)2 ), i(1 + g(z) 2 ), 2g(z)). This proves the statement of the theorem. This relationship between the functions f, g and the minimality of X by the equation ϕ(z) = X u ix v = f(z) 2 ((1 g(z)2 ), i(1 + g(z) 2 ), 2g(z)). give us the necessary tools for defining the Weierstrass-Enneper representation of minimal surfaces. 2.2 Defining the Weierstrass-Enneper Representation Definition 2.4. Let U be a simply connected open subset of the complex plane C = R 2 and γ be a curve contained in U from a fixed point z 0 to the arbitrary z = u + iv. Let f be a holomorphic function, not constantly zero, and g be a meromorphic function such that fg 2 is holomorphic. Further, assume that at every pole of g of order n f has a zero of order 2n and that f has no other zeroes. Then ( f(ξ) ) X(u, v) = x 0 + Re 2 (1 g(ξ)2, i(1 + g(ξ) 2 ), 2g(ξ))dξ, γ where x 0 is a point in R 3, is a regular parametrisation of a minimal surface. Furthermore, for every nonplanar minimal surface there exist such a representation. This is called the Weierstrass-Enneper representation of minimal surfaces. The Weierstrass-Enneper representation allows us to find regular and conformal parametrisations of minimal surfaces by simply integrating holomorphic functions. Another very interesting fact is that the meromorphic function g, occurring in the representation formula, has a close relationship with the normal vector field N of the surface. In fact, as shown in the following theorem, the normal vector field N is equal to the inverse of the stereographic projection of g. Theorem 2.5. Let Ĉ be the extended complex plane and σ 1 : Ĉ S2 R 3 be the inverse of the stereographic projection from the north pole given by σ 1 1 (z) = 1 + z (z + z, i(z z), 2 z 2 1). Then one of the unit normal vector fields N is the inverse of the stereographic projection of the function g in the Weierstrass-Enneper representation of a minimal surface i.e. N = σ 1 1 (g) = 1 + g (g + g, i(g g), 2 g 2 1). 10
Proof. It follows directly from the proof of Lemma 2.2 that X u = Re ϕ and X v = Im ϕ. We now use this fact and calculate the cross product of these partial derivatives X u X v = Re ϕ ( Im ϕ) = (Re ϕ 2 Im ϕ 3 Re ϕ 3 Im ϕ 2, Re ϕ 3 Im ϕ 1 Re ϕ 1 Im ϕ 3, Re ϕ 1 Im ϕ 2 Re ϕ 2 Im ϕ 1 ) = (Im ϕ 2 ϕ 3, Im ϕ 3 ϕ 1, Im ϕ 1 ϕ 2 ). In f and g the first coordinate becomes Im ϕ 2 ϕ 3 = Im ( if 2 (1 + g2 )fg) = Im ( i f 2 2 (g + g 2 g)) = f 2 2 Re (g + g 2 g) = f 2 2 (1 + g 2 ) Re g. A similar calculation for the other coordinates gives us the following X u X v = f 2 4 (1 + g 2 )(2 Re g, 2 Im g, g 2 1) = f 2 4 (1 + g 2 ) 2 1 1 + g 2 (g + g, i(g g), g 2 1) = f 2 4 (1 + g 2 ) 2 σ 1 (g). Here we see that N is a scalar multiple of σ 1 (g), but since N is of unit length this scalar has to be 1. Then This completes the proof. N = X u X v X u X v 1 = 1 + g (g + g, i(g g), 2 g 2 1). 11
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Chapter 3 Examples and Applications We will now look at how the Weierstrass-Enneper representation can be used to find minimal surfaces by simply integrating complex functions. We will look at four classical minimal surfaces that were the first ones to be discovered beside the plane, see [1] and [2]. We will introduce their Weierstrass-Enneper representations and prove that this representation does in fact yield a well-known parametrisation of the corresponding surfaces. 3.1 The Catenoid and the Helicoid A historically important example of a minimal surface is the catenoid. This surface can be represented by the globally defined holomorphic functions f, g : C C with f(z) = e z and g(z) = e z. Substituting these into the Weierstrass-Enneper representation formula and integrating from z 0 = (0, 0) to the general point z = u + iv we obtain ( z f(ξ) ) X(u, v) = x 0 + Re 0 2 (1 g(ξ)2, i(1 + g(ξ) 2 ), 2g(ξ))dξ ( z e ξ ) = x 0 + Re 0 2 (1 e2ξ, i(1 + e 2ξ ), 2e ξ )dξ ( z 1 ) = x 0 + Re 0 2 (e ξ e ξ, i(e ξ + e ξ ), 1)dξ ( 1 = Re 2 ( e z e z ), 1 ) 2i ( e z + e z ), z ( ) = Re cosh z, i sinh z, z = ( cosh u cos v, cosh u sin v, u). Apart from the plane, the catenoid is the only minimal surface of revolution, see [1], and locally there exists an isometry between the catenoid and the helicoid. Moreover, the catenoid can be continuously deformed to a part of the helicoid through a family of minimal surfaces. This can be explained using the interesting notion of an associate family. 13
Definition 3.1. Let X : U R 2 R 3 be a parametrised minimal surface, represented by the functions f and g. Then the family of surfaces defined by X θ = Re (e iθ f(ξ) ) γ 2 (1 g(ξ)2, i(1 + g(ξ) 2 ), 2g(ξ))dξ is called the associate family of X. Of particular interest is the surface parametrised by X π/2 which is called the conjugate of X. Notice that the surfaces of the same associate family can be represented by the same function g, but would then differ by the function f with a rotation about the origin. Then f θ = e iθ f is still holomorphic and not constantly zero, hence X θ is a minimal surface for every θ. The helicoid is the conjugate of the catenoid and can therefore be represented by the functions f(z) = ie z and g(z) = e z. 3.2 Reflection Principles With the notion of an associate family defined, we would like to introduce the reader to two reflection principles that apply to minimal surfaces. The proofs rely on Schwarz s reflection principle for harmonic functions and may be found in [6]. We will state the theorems and discuss an interesting fact about the different types of reflections and how they relate to a surface and its conjugate. Theorem 3.2. Let U be a subset of R 2 which is symmetric with respect to the real axis and define I = {(x, y) U y = 0}, U + = {(x, y) U y > 0}, U = {(x, y) U y < 0}. Let X : U + I R 3 be a conformal and regular parametrisation of a minimal surface that is twice differentiable on U + and continuous on U + I. Further, assume that X maps I to a straight line L in R 3. Then the minimal surface X can be extended to be defined on the whole of U by a reflection about the line L. This means that X(u, v) = (X(u, v)) for (u, v) U where A denotes the reflection image of A about the line L. Theorem 3.3. Let U, U +, U, I and X be defined as in Theorem 3.2 but assume that X maps I to a curve contained in a plane P in R 3 such that P is perpendicular to X along X(I). Then the minimal surface X can be extended to be defined on the whole of U by a reflection about the plane P. This means that X(u, v) = (X(u, v)) for (u, v) U, where A denotes the mirror image of A about the plane P. 14
Figure 3.1: The deformation of the catenoid into a part of the helicoid. The first picture is of the catenoid, the only minimal surface of revolution apart from the plane. The last image is of the helicoid which is the only ruled minimal surface apart from the plane, see [1]. 15
The symmetries of minimal surfaces are summed up well in the following statement of H. Karcher, see [8]: If a planar geodesic resp. a straight line lies on a complete minimal surface, then reflection in the plane of the planar geodesic resp. 180 -rotation around the straight line is a congruence of the minimal surface. There is also an intimate relationship between planar geodesics on a surface and straight lines on its conjugate. Looking at the deformation of the catenoid into the helicoid, we can see that a meridian in the catenoid, which is a planar geodesic, turns into a straight line on the helicoid. This is in fact true in general. A planar geodesic of X will be deformed into a straight line contained in X π/2 and vice versa, see [3]. 3.3 The Enneper Surface Another interesting example of a minimal surface is Enneper s surface. Its Weierstrass- Enneper representation is obtained by choosing (f(w), g(w)) = (1, w). This means that the conformal parametrisation X becomes ( 1 z ) X(z) = Re ((1 ξ 2 ), i(1 + ξ 2 ), 2ξ)dξ 2 0 = 1 ( ) 2 Re z z3 iz3, iz + 3 3, z2 = 1 u3 (u 2 3 + uv2, v + v2 3 u2 v, u 2 v 2 ) for z = u + iv. The Enneper surface is both complete and minimal but contains self intersections. Enneper s surface is conjugate to itself i.e. Enneper s surface and its conjugate are the same geometric objects. For a proof of this fact we refer the reader to [6]. 16
Figure 3.2: A plot of Enneper s surface, a self-intersecting minimal surface. 3.4 Scherk s Doubly Periodic Surface The famous doubly periodic Scherk surface, also called Scherk s first surface, provides us with another example of a historically important minimal surface. We will discuss the Weierstrass-Enneper representation of the surface and the existence of vertical lines in the surface. A more detailed discussion of the surface can be found in [6]. Scherk s doubly periodic surface is defined by the equation e z = cos y cos x. We will now prove that its Weierstrass-Enneper representation is given by the functions f : C \ {±1, ±i} C and g : C C where ( 2 ) (f(w), g(w)) = 1 w, w. 4 We first note that f(1 g 2 2 ) = 1 + w = i 2 w + i i w i, i f(1 + g 2 ) = 2i 1 w = i 2 w + 1 i w 1, 2fg = 4w 1 w = 2w 4 w 2 + 1 2w w 2 1. 17
Figure 3.3: A plot of a fundamental piece of Scherk s first surface. The surface continues in a checkerboard fashion as it is periodic in both the x and y directions. 18
Integrating this gives us which simplifies to ( X(w) = Re i log w + i w i, i log w + 1 w 1, log w2 + 1 ), w 2 1 ( X(w) = arg w + i w i, arg w + 1 w 1, log w 2 + 1 ) w 2 1 when using the principal value of the logarithm function. Now, by using the following identities, w + i w i = w 2 1 w i + i w + w 2 w i, 2 w + 1 w 1 = w 2 1 w 1 + i w w 2 w 1, 2 we find expressions for cos x and cos y. We have that Similarly, This implies that cos x = cos( arg w + i w i ) = cos(arg w + i w i ) ( w i = cos(arg w + i w + i ) ) w i ( w i = Re w + i w + i ) w i w i ( w + i ) = w + i Re w i w i = w + i w 2 i w i 2 = w 2 1 w 2 + 1. cos y = cos( arg w + 1 w 1 ) w 1 = w + 1 z 2 i w 1 2 = w 2 1 w 2 1. cos y cos x = w2 + 1 w 2 1 = ez. Hence, the maps f and g as above define Scherk s doubly periodic surface. 19
Further, to show that Scherk s doubly periodic surface contains straight lines parallel to the z-axis, we consider the coordinate functions (x(w), y(w), z(w)) under the restriction w = e iθ for some θ (0, π ). Then 2 ( e x(e iθ iθ + i ) ) = arg e iθ i = arg (i eiθ + e iθ ) e iθ i 2 = π 2. Similarly, we have ( e y(e iθ iθ + 1 ) ) = arg e iθ 1 = arg (i e iθ e iθ ) e iθ 1 2 = 3π 2. Hence, the x and y coordinates are constant while z(e iθ ) = log w2 + 1 w 2 1 tends to ± as θ tends to 0 and π, respectively. This means that X maps the 2 arc {e iθ C 0 < θ < π} onto the straight line {( π, 3π, s) 2 2 2 R3 s R}. Hence, Scherk s doubly periodic surface contains straight lines, which is what we wanted to prove. Since the surface is doubly periodic it contains an infinite amount of lines of symmetry like this one. 20
Chapter 4 Summing Up We have discussed the theory needed to state the Weierstrass-Enneper representation of minimal surfaces and we have defined the representation itself. We have shown that there exists a simple relationship between the function g found in the representation and the normal vector field N of the minimal surface. We have then looked at four examples of minimal surfaces and defined the associate family of a minimal surface. We have also mentioned two reflection principles and how they relate to a minimal surface and its conjugate. There are numerous more examples of interesting minimal surfaces with known Weierstrass-Enneper representations and we have also seen only a fraction of how the representation can be used to analyse a minimal surface. The representation has been used to prove more deep results about minimal surfaces. For example, to illustrate the usefulness of the representation, the following two theorems may be proven using the relationship between the function g and the vector field N. The theorems make no mention of the Weierstrass-Enneper representation although the proofs rely on it. Theorem 4.1. [6] The normal vector field N of a minimal surface X : C R 3 omits at most two points of the unit sphere unless X(C) is contained in a plane. Theorem 4.2. [7] If X is a complete minimal surface in R 3, then either X is a plane or the normals to X are everywhere dense in the sphere. 21
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Bibliography [1] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall (1976). [2] A. Pressley, Elementary Differential Geometry, Second Edition, Springer Undergraduate Mathematics Series, Springer (2010). [3] M. Weber, Classical Minimal Surfaces in Euclidean Space by Examples (2001). http://www.indiana.edu/ minimal/research/claynotes.pdf [4] S. Gudmundsson, An Introduction to Gaussian Geometry, Lecture Notes in Mathematics, University of Lund (2017). www.matematik.lu.se/matematiklu/personal/sigma/gauss.pdf [5] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish (1979). [6] U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab Minimal Surfaces I, Boundary Value Problems, Springer (1992). [7] R. Osserman, A Survey of Minimal Surfaces, Dover (2002). [8] H. Karcher, Construction of Minimal Surfaces, Tokyo (1989). http://www.math.uni-bonn.de/people/karcher/karchertokyo.pdf 23
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Bachelor s Theses in Mathematical Sciences 2017:K15 ISSN 1654-6229 LUNFMA-4062-2017 Mathematics Centre for Mathematical Sciences Lund University Box 118, SE-221 00 Lund, Sweden http://www.maths.lth.se/