Minimal Surfaces. Clay Shonkwiler

Transcription

1 Minimal Surfaces Clay Shonkwiler October 18, 2006

2 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

3 2

4 Computer-generated soap film 3

5 Double catenoid 4

6 Costa-Hoffman-Meeks Surface 5

7 Quick Background on Surfaces Recall that, if S R 3 is a regular surface, the inner product, p on T p S is a symmetric bilinear form. The associated quadratic form I p : T p S R given by I p (W ) = W, W p is called the first fundamental form of S at p. 6

8 Quick Background on Surfaces Recall that, if S R 3 is a regular surface, the inner product, p on T p S is a symmetric bilinear form. The associated quadratic form I p : T p S R given by I p (W ) = W, W p is called the first fundamental form of S at p. If X : U R 2 V S is a local parametrization of S with coordinates (u, v) on U, then the vectors X u and X v form a basis for T p S with p V. The first fundamental form I is completely determined by the three functions: E(u, v) = X u, X u F (u, v) = X u, X v G(u, v) = X v, X v 6

9 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 X u X v dudv = Area(R). X 1 (R) 7

10 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 X u X v dudv = Area(R). Note that X 1 (R) X u X v 2 + X u, X v 2 = X u 2 X v 2, so the above integrand can be written as X u X v = EG F 2. 7

11 The Gauss Map and curvature At a point p S, define the normal vector to p by the map N : S S 2 given by N(p) = X u X v X u X v (p). This map is called the Gauss map. 8

12 The Gauss Map and curvature At a point p S, define the normal vector to p by the map N : S S 2 given by N(p) = X u X v X u X v (p). This map is called the Gauss map. Remark:The differential dn p : T p S T p S 2 T p S defines a self-adjoint linear map. 8

13 The Gauss Map and curvature At a point p S, define the normal vector to p by the map N : S S 2 given by N(p) = X u X v X u X v (p). This map is called the Gauss map. Remark:The differential dn p : T p S T p S 2 T p S defines a self-adjoint linear map. Define K = det dn p and H = 1 2 trace dn p, the Gaussian curvature and mean curvature, respectively, of S. 8

14 The Gauss Map and curvature At a point p S, define the normal vector to p by the map N : S S 2 given by N(p) = X u X v X u X v (p). This map is called the Gauss map. Remark:The differential dn p : T p S T p S 2 T p S defines a self-adjoint linear map. Define K = det dn p and H = 1 2 trace dn p, the Gaussian curvature and mean curvature, respectively, of S. Note: Since dn p is self-adjoint, it can be diagonalized: dn p = ( ) k k 2 k 1 and k 2 are called the principal curvatures of S at p and the associated eigenvectors are called the principal directions. 8

15 Minimal Surfaces Definition 1. A regular surface S R 3 is called a minimal surface if its mean curvature is zero at each point. 9

16 Minimal Surfaces Definition 1. A regular surface S R 3 is called a minimal surface if its mean curvature is zero at each point. If S is minimal, then, when dn p is diagonalized, dn p = ( k 0 0 k ). 9

17 Minimal Surfaces Definition 1. A regular surface S R 3 is called a minimal surface if its mean curvature is zero at each point. If S is minimal, then, when dn p is diagonalized, dn p = ( k 0 0 k ). If we give S 2 the opposite orientation (i.e. instead of the outward normal), then dn p = so N is a conformal map. ( k 0 0 k ) = k ( choose the inward normal ), 9

18 The Catenoid Figure 1: The catenoid is a minimal surface 10

19 The helicoid Figure 2: The helicoid is a minimal surface as well 11

20 Helicoid with genus 12

21 Catenoid Fence 13

22 Riemann s minimal surface 14

23 Singly-periodic Scherk surface 15

24 Doubly-periodic Scherk surface 16

25 Fundamental domain for Scherk s surface 17

26 Sherk s surface with handles 18

27 Triply-periodic surface (Schwarz) 19

28 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. 20

29 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

30 Normal Variations Let X : U R 2 R 3 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 ϕ : D ( ɛ, ɛ) R 3 where ϕ(u, v, t) = X(u, v) + th(u, v)n(u, v). 21

31 Normal Variations Let X : U R 2 R 3 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 ϕ : D ( ɛ, ɛ) R 3 where ϕ(u, v, t) = X(u, v) + th(u, v)n(u, v). For small ɛ, X t (u, v) = ϕ(u, v, t) is a regular parametrized surface with X t u = X u + thn u + th u N X t v = X v + thn v + th v N. 21

32 Normal Variations Let X : U R 2 R 3 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 ϕ : D ( ɛ, ɛ) R 3 where ϕ(u, v, t) = X(u, v) + th(u, v)n(u, v). For small ɛ, X t (u, v) = ϕ(u, v, t) is a regular parametrized surface with Then it s straightforward to see that X t u = X u + thn u + th u N X t v = X v + thn v + th v N. E t G t (F t ) 2 = EG F 2 2th(Eg 2F f + Ge) + R = (EG F 2 )(1 4thH) + R where lim t 0 R t = 0. 21

33 Equivalence of curvature and variational definitions of minimal The area A(t) of X t ( D) is given by A(t) = = D D Et G t (F t ) 2 dudv 1 4thH + R EG F 2 dudv, where R = R EG F 2. 22

34 Equivalence of curvature and variational definitions of minimal The area A(t) of X t ( D) is given by where R = A(t) = R EG F 2. = D D Et G t (F t ) 2 dudv 1 4thH + R EG F 2 dudv, Hence, if ɛ is small, A is differentiable and A (0) = D 2hH EG F 2 dudv. 22

35 Equivalence of curvature and variational definitions of minimal The area A(t) of X t ( D) is given by where R = A(t) = R EG F 2. = D D Et G t (F t ) 2 dudv 1 4thH + R EG F 2 dudv, Hence, if ɛ is small, A is differentiable and A (0) = 2hH EG F 2 dudv. D Therefore: Proposition 2. Let X : U R 3 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 A (0) = 0 for all such D and all normal variations of X( D). 22

36 Isothermal coordinates Definition 2. Given a regular surface S R 3, a system of local coordinates X : U V S is said to be isothermal if X u, X u = X v, X v and X u, X v = 0. i.e. G = E and F = 0. 23

37 Isothermal coordinates Definition 2. Given a regular surface S R 3, a system of local coordinates X : U V S is said to be isothermal if X u, X u = X v, X v and X u, X v = 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

38 Isothermal coordinates Definition 2. Given a regular surface S R 3, a system of local coordinates X : U V S is said to be isothermal if X u, X u = X v, X v and X u, X v = 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

39 Facts about isothermal parametrizations Proposition 3. Let X : U V S be an isothermal local parametrization of a regular surface S. Then X = X uu + X vv = 2λ 2 H, where λ 2 (u, v) = X u, X u = X v, X v and H = HN is the mean curvature vector field. 24

40 Facts about isothermal parametrizations Proposition 3. Let X : U V S be an isothermal local parametrization of a regular surface S. Then X = X uu + X vv = 2λ 2 H, where λ 2 (u, v) = X u, X u = X v, X v and H = HN is the mean curvature vector field. Corollary 4. Let X(u, v) = (x 1 (u, v), x 2 (u, v), x 3 (u, v)) be an isothermal parametrization of a regular surface S in R 3. Then S is minimal (within the range of this parametrization) if and only if the three coordinate functions x 1 (u, v), x 2 (u, v) and x 3 (u, v) are harmonic. 24

41 Facts about isothermal parametrizations Proposition 3. Let X : U V S be an isothermal local parametrization of a regular surface S. Then X = X uu + X vv = 2λ 2 H, where λ 2 (u, v) = X u, X u = X v, X v and H = HN is the mean curvature vector field. Corollary 4. Let X(u, v) = (x 1 (u, v), x 2 (u, v), x 3 (u, v)) be an isothermal parametrization of a regular surface S in R 3. Then S is minimal (within the range of this parametrization) if and only if the three coordinate functions x 1 (u, v), x 2 (u, v) and x 3 (u, v) are harmonic. Corollary 5. The three component functions x 1, x 2 and x 3 are locally the real parts of holomorphic functions. 24

42 Facts about isothermal parametrizations Proposition 3. Let X : U V S be an isothermal local parametrization of a regular surface S. Then X = X uu + X vv = 2λ 2 H, where λ 2 (u, v) = X u, X u = X v, X v and H = HN is the mean curvature vector field. Corollary 4. Let X(u, v) = (x 1 (u, v), x 2 (u, v), x 3 (u, v)) be an isothermal parametrization of a regular surface S in R 3. Then S is minimal (within the range of this parametrization) if and only if the three coordinate functions x 1 (u, v), x 2 (u, v) and x 3 (u, v) are harmonic. Corollary 5. The three component functions x 1, x 2 and x 3 are locally the real parts of holomorphic functions. Theorem 6. Isothermal coordinates exist on any minimal surface S R 3. Remark: In fact, isothermal coordinates exist for any C 2 surface. 24

43 Connections with complex analysis If σ : S 2 C { } is stereographic projection and N : S S 2 is the Gauss map, then g := σ N : S C { } is orientation-preserving and conformal whenever dn 0. Therefore g is a meromorphic function on S (now thought of as a Riemann surface). 25

44 Connections with complex analysis If σ : S 2 C { } is stereographic projection and N : S S 2 is the Gauss map, then g := σ N : S C { } is orientation-preserving and conformal whenever dn 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 Σ k by closing finitely many punctures. Moreover, the Gauss map N : S S 2, which is meromorphic, extends to a meromorphic function on Σ k. 25

45 Enneper s Surface Figure 4: z C, g(z) := z 26

46 Chen-Gackstatter Surface 27

47 Chen-Gackstatter Surface with higher genus 28

48 Symmetric 4-Noid 29

49 Costa surface 30

50 Meeks minimal Möbius strip 31

51 Thanks! 32