Minimal Surfaces

Clay Shonkwiler

October 18, 2006 Soap Films

A soap film seeks to minimize its surface energy, which is proportional to area. Hence, a soap film achieves a minimum area among all surfaces with the same boundary.

1 2 Computer-generated soap film

3 Double catenoid

4 Costa-Hoffman-Meeks Surface

5 Quick Background on Surfaces

3 Recall that, if S ⊂ R is a regular surface, the inner product h , ip on TpS is a symmetric bilinear form. The associated quadratic form

Ip : TpS → R given by Ip(W ) = hW, W ip is called the first fundamental form of S at p.

If X : U ⊂ R2 → V ⊂ S is a local parametrization of S with coordinates (u, v) on U, then the vectors Xu and Xv form a basis for TpS with p ∈ V . The first fundamental form I is completely determined by the three functions:

E(u, v) = hXu,Xui

F (u, v) = hXu,Xvi

G(u, v) = hXv,Xvi

6 Area

Proposition 1. Let R ⊂ S be a bounded region of a regular surface contained in the coordinate neighborhood of the parametrization X : U → S. Then Z kXu × Xvkdudv = Area(R). X−1(R)

Note that

2 2 2 2 kXu × Xvk + hXu,Xvi = kXuk kXvk , so the above integrand can be written as

p 2 kXu × Xvk = EG − F .

7 The Gauss Map and curvature

At a point p ∈ S, define the normal vector to p by the map N : S → S2 given by X × X N(p) = u v (p). kXu × Xvk This map is called the Gauss map.

2 Remark:The differential dNp : TpS → TpS ' TpS defines a self-adjoint linear map. Define

−1 K = det dN and H = trace dN , p 2 p the Gaussian curvature and mean curvature, respectively, of S.

Note: Since dNp is self-adjoint, it can be diagonalized:   k1 0 dNp = . 0 k2 k1 and k2 are called the principal curvatures of S at p and the associated eigenvectors are called the principal directions.

8 Minimal Surfaces

Definition 1. A regular surface S ⊂ R3 is called a minimal surface if its mean curvature is zero at each point.

If S is minimal, then, when dNp is diagonalized,

 k 0  dN = . p 0 −k

If we give S2 the opposite orientation (i.e. choose the inward normal instead of the outward normal), then

 k 0   1 0  dN = = k , p 0 k 0 1 so N is a conformal map.

9 The Catenoid

Figure 1: The catenoid is a minimal surface

10 The helicoid

Figure 2: The helicoid is a minimal surface as well

11 Helicoid with genus

12 Catenoid Fence

13 Riemann’s minimal surface

14 Singly-periodic Scherk surface

15 Doubly-periodic Scherk surface

16 Fundamental domain for Scherk’s surface

17 Sherk’s surface with handles

18 Triply-periodic surface (Schwarz)

19 Why are these surfaces “minimal”?

From this definition, it’s not at all clear what is “minimal” about these surfaces. The idea is that minimal surfaces should locally minimize surface area.

Figure 3: The surface and a normal variation

20 Normal Variations

Let X : U ⊂ R2 → R3 be a regular parametrized surface. Let D ⊂ U be a bounded domain and let h : D¯ → R be differentiable. Then the normal variation of X(D¯) determined by h is given by

3 ϕ : D¯ × (−, ) → R where ϕ(u, v, t) = X(u, v) + th(u, v)N(u, v). For small , Xt(u, v) = ϕ(u, v, t) is a regular parametrized surface with

t Xu = Xu + thNu + thuN t Xv = Xv + thNv + thvN.

Then it’s straightforward to see that EtGt − (F t)2 = EG − F 2 − 2th(Eg − 2F f + Ge) + R = (EG − F 2)(1 − 4thH) + R

R where limt→0 t = 0.

21 Equivalence of curvature and variational definitions of minimal

The area A(t) of Xt(D¯) is given by Z A(t) = pEtGt − (F t)2dudv D¯ Z p p = 1 − 4thH + R¯ EG − F 2dudv, D¯

¯ R where R = EG−F 2 . Hence, if  is small, A is differentiable and

Z p A0(0) = − 2hH EG − F 2dudv. D¯

Therefore: Proposition 2. Let X : U → R3 be a regular parametrized surface and let D ⊂ U be a bounded domain. Then X is minimal (i.e. H ≡ 0) if and only if A0(0) = 0 for all such D and all normal variations of X(D¯).

22 Isothermal coordinates Definition 2. Given a regular surface S ⊂ R3, a system of local coordinates X : U → V ⊂ S is said to be isothermal if

hXu,Xui = hXv,Xvi and hXu,Xvi = 0.

i.e. G = E and F = 0. Note: A parametrization X : U → V ⊂ S is isothermal if and only if it is conformal.

23 Facts about isothermal parametrizations

Proposition 3. Let X : U → V ⊂ S be an isothermal local parametrization of a regular surface S. Then

2 ∆X = Xuu + Xvv = 2λ H,

2 where λ (u, v) = hXu,Xui = hXv,Xvi and H = HN is the mean curvature vector field. Corollary 4. Let X(u, v) = (x1(u, v), x2(u, v), x3(u, v)) be an isothermal parametrization of a regular surface S in R3. Then S is minimal (within the range of this parametrization) if and only if the three coordinate functions x1(u, v), x2(u, v) and x3(u, v) are harmonic. Corollary 5. The three component functions x1, x2 and x3 are locally the real parts of holomorphic functions.

Theorem 6. Isothermal coordinates exist on any minimal surface S ⊂ R3. Remark: In fact, isothermal coordinates exist for any C2 surface.

24 Connections with complex analysis

If σ : S2 → C ∪ {∞} is stereographic projection and N : S → S2 is the Gauss map, then g := σ ◦ N : S → C ∪ {∞} is orientation-preserving and conformal whenever dN 6= 0. Therefore g is a meromorphic function on S (now thought of as a Riemann surface).

In fact, we have: Theorem 7 (Osserman). If S is a complete, immersed minimal surface of finite total curvature, then S can be conformally compactified to a Riemann surface Σk by closing finitely many punctures. Moreover, the Gauss map N : S → S2, which is meromorphic, extends to a meromorphic function on Σk.

25 Enneper’s Surface

Figure 4: z ∈ C, g(z) := z

26 Chen-Gackstatter Surface

27 Chen-Gackstatter Surface with higher genus

28 Symmetric 4-Noid

29 Costa surface

30 Meeks’ minimal M¨obiusstrip

31 Thanks!

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