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.
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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 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.
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Quick Background on Surfaces
Recall that, if S ⊂ R3 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, Xv i
G(u, v) = hXv , Xv i
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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 × Xv kdudv = Area(R).
X −1 (R)
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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 × Xv kdudv = Area(R).
X −1 (R)
Note that
kXu × Xv k2 + hXu, Xv i2 = kXuk2kXv k2,
so the above integrand can be written as
p
kXu × Xv k = 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 → S 2
given by
Xu × Xv
N (p) =
(p).
kXu × Xv k
This map is called the Gauss map.
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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 → S 2
given by
Xu × Xv
N (p) =
(p).
kXu × Xv k
This map is called the Gauss map.
Remark:The differential dNp : TpS → TpS 2 ' TpS defines a self-adjoint
linear map.
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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 → S 2
given by
Xu × Xv
N (p) =
(p).
kXu × Xv k
This map is called the Gauss map.
Remark:The differential dNp : TpS → TpS 2 ' TpS defines a self-adjoint
linear map. Define
K = det dNp and H =
−1
trace dNp,
2
the Gaussian curvature and mean curvature, respectively, of S.
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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 → S 2
given by
Xu × Xv
(p).
N (p) =
kXu × Xv k
This map is called the Gauss map.
Remark:The differential dNp : TpS → TpS 2 ' TpS defines a self-adjoint
linear map. Define
−1
K = det dNp and H =
trace dNp,
2
the Gaussian curvature and mean curvature, respectively, of S.
Note: Since dNp is self-adjoint, it can be diagonalized:
dNp =
k1 0
0 k2
.
k1 and k2 are called the principal curvatures of S at p and the associated
eigenvectors are called the principal directions.
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Minimal Surfaces
Definition 1. A regular surface S ⊂ R3 is called a minimal surface if its
mean curvature is zero at each point.
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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,
dNp =
k 0
0 −k
.
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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,
dNp =
k 0
0 −k
.
If we give S 2 the opposite orientation (i.e. choose the inward normal
instead of the outward normal), then
dNp =
k 0
0 k
=k
1 0
0 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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Helicoid with genus
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Helicoid with genus
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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 these
surfaces.
The idea is that minimal surfaces should locally minimize surface area.
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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
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Normal Variations
Let X : U ⊂ R2 → R3 be a regular parametrized surface. Let D ⊂ U be
a bounded domain and let h : D̄ → R be differentiable. Then the normal
variation of X(D̄) determined by h is given by
ϕ : D̄ × (−, ) → R3
where
ϕ(u, v, t) = X(u, v) + th(u, v)N (u, v).
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Normal Variations
Let X : U ⊂ R2 → R3 be a regular parametrized surface. Let D ⊂ U be
a bounded domain and let h : D̄ → R be differentiable. Then the normal
variation of X(D̄) determined by h is given by
ϕ : D̄ × (−, ) → R3
where
ϕ(u, v, t) = X(u, v) + th(u, v)N (u, v).
For small , X t(u, v) = ϕ(u, v, t) is a regular parametrized surface with
Xut = Xu + thNu + thuN
Xvt = Xv + thNv + thv N.
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Normal Variations
Let X : U ⊂ R2 → R3 be a regular parametrized surface. Let D ⊂ U be
a bounded domain and let h : D̄ → R be differentiable. Then the normal
variation of X(D̄) determined by h is given by
ϕ : D̄ × (−, ) → R3
where
ϕ(u, v, t) = X(u, v) + th(u, v)N (u, v).
For small , X t(u, v) = ϕ(u, v, t) is a regular parametrized surface with
Xut = Xu + thNu + thuN
Xvt = Xv + thNv + thv N.
Then it’s straightforward to see that
E tGt − (F t)2 = EG − F 2 − 2th(Eg − 2F f + Ge) + R
= (EG − F 2)(1 − 4thH) + R
where limt→0 Rt = 0.
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Equivalence of curvature and variational definitions of
minimal
The area A(t) of X t(D̄) is given by
Z p
A(t) =
E tGt − (F t)2dudv
D̄
Z p
p
=
1 − 4thH + R̄ EG − F 2dudv,
D̄
where R̄ =
R
.
EG−F 2
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Equivalence of curvature and variational definitions of
minimal
The area A(t) of X t(D̄) is given by
Z p
A(t) =
E tGt − (F t)2dudv
D̄
Z p
p
=
1 − 4thH + R̄ EG − F 2dudv,
D̄
where R̄ =
R
.
EG−F 2
Hence, if is small, A is differentiable and
0
Z
A (0) = −
p
2hH EG − F 2dudv.
D̄
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Equivalence of curvature and variational definitions of
minimal
The area A(t) of X t(D̄) is given by
Z p
A(t) =
E tGt − (F t)2dudv
D̄
Z p
p
=
1 − 4thH + R̄ EG − F 2dudv,
D̄
where R̄ =
R
.
EG−F 2
Hence, if is small, A is differentiable and
Z
p
0
A (0) = −
2hH EG − F 2dudv.
D̄
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(D̄).
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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 , Xv i and hXu, Xv i = 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 coordinates
X : U → V ⊂ S is said to be isothermal if
hXu, Xui = hXv , Xv i and hXu, Xv i = 0.
i.e. G = E and F = 0.
Note: A parametrization X : U → V ⊂ S is isothermal if and only if it
is conformal.
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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 , Xv i and hXu, Xv i = 0.
i.e. G = E and F = 0.
Note: A parametrization X : U → V ⊂ S is isothermal if and only if it
is conformal.
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Facts about isothermal parametrizations
Proposition 3. Let X : U → V ⊂ S be an isothermal local
parametrization of a regular surface S. Then
∆X = Xuu + Xvv = 2λ2H,
where λ2(u, v) = hXu, Xui = hXv , Xv i and H = HN is the mean
curvature vector field.
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Facts about isothermal parametrizations
Proposition 3. Let X : U → V ⊂ S be an isothermal local
parametrization of a regular surface S. Then
∆X = Xuu + Xvv = 2λ2H,
where λ2(u, v) = hXu, Xui = hXv , Xv i 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.
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Facts about isothermal parametrizations
Proposition 3. Let X : U → V ⊂ S be an isothermal local
parametrization of a regular surface S. Then
∆X = Xuu + Xvv = 2λ2H,
where λ2(u, v) = hXu, Xui = hXv , Xv i 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.
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Facts about isothermal parametrizations
Proposition 3. Let X : U → V ⊂ S be an isothermal local
parametrization of a regular surface S. Then
∆X = Xuu + Xvv = 2λ2H,
where λ2(u, v) = hXu, Xui = hXv , Xv i 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 C 2 surface.
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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 6= 0. Therefore g is a
meromorphic function on S (now thought of as a Riemann surface).
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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 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 → S 2, which is meromorphic, extends to a
meromorphic 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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