minimal surfaces - colorado state universityclayton/research/talks/minimalsurfaces.pdfminimal...

Minimal Surfaces

Clay Shonkwiler

October 18, 2006

Soap Films

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

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Computer-generated soap film

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Double catenoid

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Costa-Hoffman-Meeks Surface

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Quick Background on Surfaces

Recall that, if S ⊂ R3 is a regular surface, the inner product 〈 , 〉p onTpS is a symmetric bilinear form. The associated quadratic form

Ip : TpS → R

given byIp(W ) = 〈W,W 〉p

is called the first fundamental form of S at p.

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Quick Background on Surfaces

Recall that, if S ⊂ R3 is a regular surface, the inner product 〈 , 〉p onTpS is a symmetric bilinear form. The associated quadratic form

Ip : TpS → R

given byIp(W ) = 〈W,W 〉p

is called the first fundamental form of S at p.

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

E(u, v) = 〈Xu, Xu〉F (u, v) = 〈Xu, Xv〉G(u, v) = 〈Xv, Xv〉

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Area

Proposition 1. Let R ⊂ S be a bounded region of a regular surfacecontained in the coordinate neighborhood of the parametrization X : U →S. Then ∫

X−1(R)‖Xu ×Xv‖dudv = Area(R).

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Area

Proposition 1. Let R ⊂ S be a bounded region of a regular surfacecontained in the coordinate neighborhood of the parametrization X : U →S. Then ∫

X−1(R)‖Xu ×Xv‖dudv = Area(R).

Note that

‖Xu ×Xv‖2 + 〈Xu, Xv〉2 = ‖Xu‖2‖Xv‖2,

so the above integrand can be written as

‖Xu ×Xv‖ =√

EG− F 2.

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The Gauss Map and curvature

At a point p ∈ S, define the normal vector to p by the map N : S → S2given by

N(p) =Xu ×Xv‖Xu ×Xv‖

(p).

This map is called the Gauss map.

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(p).


Remark:The differential dNp : TpS → TpS2 ' TpS defines a self-adjointlinear map.

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(p).


Remark:The differential dNp : TpS → TpS2 ' TpS defines a self-adjointlinear map. Define

K = det dNp and H =−12

trace dNp,

the Gaussian curvature and mean curvature, respectively, of S.

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(p).


Remark:The differential dNp : TpS → TpS2 ' TpS defines a self-adjointlinear map. Define

K = det dNp and H =−12

trace dNp,

the Gaussian curvature and mean curvature, respectively, of S.

Note: Since dNp is self-adjoint, it can be diagonalized:

dNp =(

k1 00 k2

).

k1 and k2 are called the principal curvatures of S at p and the associatedeigenvectors are called the principal directions.

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Minimal Surfaces

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

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Minimal Surfaces


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

dNp =(

k 00 −k

).

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Minimal Surfaces


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

dNp =(

k 00 −k

).

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

dNp =(

k 00 k

)= k

(1 00 1

),

so N is a conformal map.

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The Catenoid

Figure 1: The catenoid is a minimal surface

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The helicoid

Figure 2: The helicoid is a minimal surface as well

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The helicoid and the catenoid are locally isometric

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HelicoidCatenoid3.movMedia File (video/quicktime)

Helicoid with genus

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Helicoid with genus

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g1_mov.mpgMedia File (video/mpeg)

Catenoid Fence

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Riemann’s minimal surface

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Singly-periodic Scherk surface

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Doubly-periodic Scherk surface

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Fundamental domain for Scherk’s surface

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Sherk’s surface with handles

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Triply-periodic surface (Schwarz)

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Why are these surfaces “minimal”?

From this definition, it’s not at all clear what is “minimal” about thesesurfaces.

The idea is that minimal surfaces should locally minimize surface area.

22

Why are these surfaces “minimal”?

From this definition, it’s not at all clear what is “minimal” about thesesurfaces.

The idea is that minimal surfaces should locally minimize surface area.

Figure 3: The surface and a normal variation

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Normal Variations

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

ϕ : D̄ × (−�, �) → R3

whereϕ(u, v, t) = X(u, v) + th(u, v)N(u, v).

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Normal Variations


ϕ : D̄ × (−�, �) → R3


For small �, Xt(u, v) = ϕ(u, v, t) is a regular parametrized surface with

Xtu = Xu + thNu + thuN

Xtv = Xv + thNv + thvN.

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Normal Variations


ϕ : D̄ × (−�, �) → R3


For small �, Xt(u, v) = ϕ(u, v, t) is a regular parametrized surface with

Xtu = Xu + thNu + thuN

Xtv = Xv + thNv + thvN.

Then it’s straightforward to see that

EtGt − (F t)2 = EG− F 2 − 2th(Eg − 2Ff + Ge) + R

= (EG− F 2)(1− 4thH) + R

where limt→0Rt = 0.

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Equivalence of curvature and variational definitions ofminimal

The area A(t) of Xt(D̄) is given by

A(t) =∫

D̄

√EtGt − (F t)2dudv

=∫

D̄

√1− 4thH + R̄

√EG− F 2dudv,

where R̄ = REG−F 2.

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A(t) =∫

D̄


=∫

D̄

√1− 4thH + R̄

√EG− F 2dudv,


Hence, if � is small, A is differentiable and

A′(0) = −∫

D̄

2hH√

EG− F 2dudv.

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A(t) =∫

D̄


=∫

D̄

√1− 4thH + R̄

√EG− F 2dudv,


Hence, if � is small, A is differentiable and

A′(0) = −∫

D̄

2hH√

EG− F 2dudv.

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

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Isothermal coordinates

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

〈Xu, Xu〉 = 〈Xv, Xv〉 and 〈Xu, Xv〉 = 0.

i.e. G = E and F = 0.

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Isothermal coordinates

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


i.e. G = E and F = 0.

Note: A parametrization X : U → V ⊂ S is isothermal if and only if itis conformal.

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


i.e. G = E and F = 0.

Note: A parametrization X : U → V ⊂ S is isothermal if and only if itis conformal.

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Facts about isothermal parametrizations

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

∆X = Xuu + Xvv = 2λ2H,

where λ2(u, v) = 〈Xu, Xu〉 = 〈Xv, Xv〉 and H = HN is the meancurvature vector field.

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Corollary 4. Let X(u, v) = (x1(u, v), x2(u, v), x3(u, v)) be an isothermalparametrization of a regular surface S in R3. Then S is minimal (withinthe range of this parametrization) if and only if the three coordinatefunctions x1(u, v), x2(u, v) and x3(u, v) are harmonic.

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Corollary 5. The three component functions x1, x2 and x3 are locally thereal parts of holomorphic functions.

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Corollary 5. The three component functions x1, x2 and x3 are locally thereal 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.

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Connections with complex analysis

If σ : S2 → C ∪ {∞} is stereographic projection and N : S → S2 is theGauss map, then

g := σ ◦N : S → C ∪ {∞}is orientation-preserving and conformal whenever dN 6= 0. Therefore g is ameromorphic function on S (now thought of as a Riemann surface).

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Connections with complex analysis

If σ : S2 → C ∪ {∞} is stereographic projection and N : S → S2 is theGauss map, then

g := σ ◦N : S → C ∪ {∞}is orientation-preserving and conformal whenever dN 6= 0. Therefore g is ameromorphic function on S (now thought of as a Riemann surface).

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

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Enneper’s Surface

Figure 4: z ∈ C, g(z) := z

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Chen-Gackstatter Surface

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Chen-Gackstatter Surface with higher genus

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Symmetric 4-Noid

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Costa surface

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Meeks’ minimal Möbius strip

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Thanks!

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minimal surfaces - colorado state universityclayton/research/talks/minimalsurfaces.pdfminimal...

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