Dynamics at the Horsetooth Volume 4, 2012.
Myla Kilchrist and Dave Packard
Department of Mathematics
Colorado State University
Report submitted to Prof. P. Shipman for Math 641, Spring 2012 mylak@rams.colostate.edu
drpackar@rams.colostate.edu
Abstract.
The Weierstrass-Enneper Representations are a great link between several branches of mathematics. They provide a way to study surfaces using both geometry and complex analysis. The Weierstrass-Enneper Representation for minimal surfaces says that any minimal surface may be represented by complex holomorphic functions.
With the use of differential forms, this idea may be generalized to constant mean curvature surfaces, which yields Hamiltonian systems. The ability to study a problem from several different angles can be a very useful tool in mathematics, which makes the
Weierstrass-Enneper Representations an exciting discovery.
Keywords: Weierstrass-Enneper Representations, minimal surfaces, constant mean curvature surfaces, holomorphic functions, differential forms, Hamiltonian systems
Mathematics is divided into many different categories and subjects. Sometimes it seems as if these subjects do not connect, especially when studied in high school or college classes that do not feel as if they overlap. It is always exciting, then, when a connection between two seemingly unrelated areas of math is made. In reality, the many different “subjects” of math are intertwined in beautiful ways. Sometimes these connections are obvious and at other times they take some thought and imagination to discover. Karl Weierstrass and Alfred Enneper discovered one of these connections. Their discovery is now known as the Weierstrass-Enneper Representations. They link together complex analysis, differential geometry, and Hamiltonian systems. Weierstrass and
Enneper figured out that minimal surfaces can be represented by holomorphic and meromorphic complex functions. This idea can then be generalized to constant mean curvature surfaces, and this generalization produces Hamiltonian systems. In order to dive into all of this, some background information about minimal surfaces, complex functions, and Hamiltonian systems is needed.
A surface M is called a minimal surface if the mean curvature, usually called H, is zero. The mean curvature is the average of the principal curvatures of that surface, so if we call the principal curvatures k
1 and k
2
, then
H
= k
1
+ k
2
.
2
(1)

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
The principal curvatures at a point p on a surface are the curvatures in the directions of maximal and minimal curvature at that point. They can be computed as the eigenvalues of the shape operator ( S p
), which is a linear transformation from the tangent space of the surface at that point to itself ( S p
∶
T p
M
→
T p
Let a surface to the surface is
M
⃗ =
⊆ x
R
⃗ u
M ) [1].
3 be parameterized by
× ⃗ v
∣ ⃗ u x v ∣ x
( u, v
) ∶
Ω
⊆
R
2 →
M . Then the unit normal vector
. The following equations can be used to compute the mean curvature and provide a way to define the shape operator as a matrix [1, 2]. Define E , F , G , l , m , and n as
E
= ⃗ u x u
,
F
= ⃗ u x v
,
G
= ⃗ v x v
, l
= ⃗ m
= ⃗
N , n
= ⃗
N .
(2)
Then the shape operator is
S
=
1
EG
−
F 2
(
Gl
−
F m Gm
−
F n
Em
−
F l En
−
F m
)
, (3) and the mean curvature is
H
=
En
+
Gl
−
2 F m
2
(
EG
−
F 2 )
=
1
2 tr
(
S
)
.
(4)
These equations work for any patch
⃗ ( u, v
)
, but sometimes it is useful to use a patch with special properties.
If
⃗ ( u, v
) ∶
Ω
→
M is a patch such that E
=
G and F
=
0, it is called an isothermal patch.
Geometrically this means that x
⃗ u and x
⃗ v are orthogonal, so angles are preserved, and the patch the same amount in the u and v directions. If
⃗ is an isothermal patch, then
⃗ stretches
H
=
En
+
El
2 E 2
= n
+ l
,
2 E
(5) so the mean curvature is very easy to compute. There is a theorem that states that isothermal coordinates exist on any minimal surface M
⊆
R
3
. In fact, any surface can be parameterized using an isothermal patch. The Weierstrass-Enneper Representation for minimal surfaces requires that the minimal surfaces be represented by isothermal patches [1].
Another type of patch that plays a role in the Weierstrass-Enneper Representations is a harmonic patch. A real function x
( u, v
) is harmonic if its second-order partial derivatives are continuous and then
△ x
△⃗
∶= ∂
= (
2
2 x
∂u 2
+
EH
∂
2 x
∂v 2
N
=
0. A theorem in differential geometry states that if
⃗ ( u, v
) is isothermal,
. The following corollary to this theorem is important in the proof of the
Dynamics at the Horsetooth 2 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Weierstrass-Enneper Representation for minimal surfaces [1].
Corollary: A surface M
∶ ⃗ ( u, v
) = ( x imal if and only if x
1
, x
2
, and x
3
1 ( u, v
)
, x
2 ( u, v are all harmonic.
)
, x
3 ( u, v
))
, with isothermal coordinates is min-
Proof [1]: (
Ô⇒
) If M is minimal, then H
=
0
Ô⇒ △⃗ = (
2 EH
) ⃗ =
0
Ô⇒ x
1
, x
2
, x
3
(
⇐Ô harmonic.
) If x
1
Now
, x
2
, x
3 are harmonic, then
△⃗ =
0
Ô⇒ (
2 EH
) ⃗
⃗ is the unit normal vector, so
So H
=
0
Ô⇒
M is minimal.
N
/=
0 and E
N
=
0.
= ⃗ u
⋅ ⃗ v x u
∣ 2
QED are
/=
0.
Intuitively this corollary makes sense because the curvature ( κ ) of a curve in a plane is essentially the rate that the tangent vector to the curve is changing. For a curve ( α ) parameterized by arclength, κ
= ∣ dT ds
∣ = ∣ d
2
α ds 2
∣
. Since
⃗ ( u, v
) is not parameterized by arc-length, the principle curvatures are not exactly the magnitude of the second derivatives related. So it makes sense to say that if x j uu
+ x j vv x uu
∣ and x vv
∣
, but they are certainly
=
0 for j
∈ {
1 , 2 , 3
}
, then k
1
+ k
2
=
0 and vice versa.
The Weierstrass-Enneper Representations do not only depend on differential geometry concepts, it is also important to understand some complex analysis. A complex function f
( z
) is holomorphic f
( z
0 + h
)− f
( z
0 ) at a point z
0 if lim h
→
0 h exists, so f is holomorphic in a region if it is differentiable at every point in that region. A complex function g
( z
) is meromorphic in a region if it is holomorphic everywhere in that region except at isolated singularities and all of these singularities are poles. A point u, v as
∂
∂z
∈ z
=
0
1
2 is a pole if
( ∂
∂u
− i
∂
∂v f
→ ∞
) and
∂
∂z as
= 1
2 z
(
→ z
0
. A complex number may be expressed as z
= u
+ iv where
R
, and its complex conjugate is z
= u
− iv . The derivatives of z and z may be expressed
∂
∂u
+ i
∂
∂v
)
. If f
∶
U
→
V is holomorphic and bijective, it is called a conformal map. Conformal maps preserve angles [3]. Isothermal patches also preserve angles, so this is the first connection between differential geometry and complex analysis that has been mentioned so far [4]. If a minimal surface can be represented by an isothermal patch, could it also be represented by a holomorphic function?
With all of this background information about minimal surfaces, isothermal patches, harmonic functions, and holomorphic and meromorphic functions, the Weierstrass-Enneper Representation for minimal surfaces may be constructed. First, it would be good to look at an example. Since it is called the Weierstrass-Enneper Representation, Enneper’s Surface makes a great example [4].
Enneper’s Surface
The most common parameterization for Enneper’s surface is
⃗ ( u, v
) = ( u
−
1
3 u
3 + uv
2
,
− v
− u
2 v
+
1
3 v
3
, u
2 − v
2 )
.
First show that this is an isothermal patch.
(6)
Dynamics at the Horsetooth 3 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard x
⃗ x
⃗ u v
= (
1
− u
2 + v
2
= (
2 uv,
−
1
− u
,
−
2 uv, 2 u
)
2 + v
2
,
−
2 v
)
E
G
F
= ⃗
= ⃗ u v
= ⃗ u
⋅ ⃗
⋅ ⃗ u v
⋅ ⃗ v
=
=
1
1
+
+
2
2 u u
2
2
+
2 v
2
+
2 v
=
2 uv
(
1
− u
2
2
+ v
2
+
2 u
2
+
2 u
2 v v
2
2
+ u
+ u
4
4
+ v
+ v
4
4
) −
2 uv
(−
1
− u
2 + v
2 ) −
4 uv
=
0
Since E
=
G and F
=
0,
⃗ ( u, v
) is isothermal.
Now let z
= u
+ iv and
⃗ = ⃗ u
− i x
⃗ v
.
Then
⃗ = (
1
− u
2 + v
2
= (
1
− ( u
+ iv
)
= (
1
− z
2
− i 2 uv,
−
2 uv
− i
(−
1
− u
2
, i
(
1
+ z
, i
(
1
+ ( u
+ iv
)
2 )
, 2 z
)
.
2
2 + v
2
)
, 2
( u
+ iv
))
)
, 2 u
+ i 2 v
)
Notice that φ
1 ( z
) =
1
− z
2
, φ
2 ( z
) = i
(
1
+ z
2 )
, and φ
3 ( z
) =
2 z are all holomorphic, so Enneper’s surface can be represented by holomorphic functions [4].
Is it possible to go backwards? Given φ , can Enneper’s surface be derived? Weierstrass figured out that, yes it is possible to obtain Enneper’s surface from φ .
We know
⃗ = ⃗ u be real-valued.
− ix v and φ
1 ( z
) =
1
− z
2
, φ
2 ( z
) = i
(
1
+ z
2 )
, and φ
3 ( z
) =
2 z , and we want
⃗ ( u, v
) to
Let x
1 =
Re
(∫ (
1
− z
=
Re
=
Re
( u
+ iv
−
= u
−
( z
1
3
− u
3
1
3 z
3
+ uv
3
2
2
)
) dz
)
1 ( u
+ iv
) 3 ) x
2 =
Re
(∫ i
(
1
+ z
=
Re
( i
( z
+ 1 z
=
Re
( i
( u
+ iv
+
= − v
− u
2
3 v
+ 1
3
2
3
) dz
) v
))
1
3
3
( u
+ iv
) 3 )) x
3 =
Re
(∫
2 zdz
)
=
Re
=
Re
(( u
+ iv
)
= u
2
(
− z
2 v
2
)
.
2 )
Then we get
⃗ ( u, v
) = ( u
− 1
3 u
3 + uv
2
,
− v
− u
2 v
+ 1
3 v
3
, u
2 − v
2 )
, which is Enneper’s surface [4]!
Enneper’s Surface
0
−0.5
−1
−1
1
0.5
x 10
4
−0.5
x 10
6
Dynamics at the Horsetooth
0
0.5
4
1
−1
0
−0.5
0.5
1 x 10
6
Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Enneper’s surface provides a great example of how a minimal surface may be represented by holomorphic functions, but now this idea needs to be generalized to all minimal surfaces.
From Isothermal Patches to Holomorphic Functions
Let
Let M be a minimal surface described by an isothermal patch
⃗ ( u, v
)
.
z
= u
+ iv , so then
∂
∂z
= 1
2
( ∂
∂u
− i
∂
∂v
)
, z
= u
− iv , and
Notice that z
+ z
=
2 u and z
− z
= i 2 v , so
∂
∂z
= 1
2
( ∂
∂u
− i
∂
∂v
)
.
u
= v
= z
+ z
2 z
− z
.
2 i
This means that
⃗ ( u, v
) may be written as
⃗ ( z, z
) = ( x
1 ( z, z
)
, x
2 ( z, z
)
, x
3 ( z, z
))
, and the derivative of the j th component is
∂x j
∂z
=
1
2
( x j u
− ix j v
)
.
Define
(
φ
)
⃗
=
2
∂
∂z
= ( x
⃗
1 z
= (
) 2 x
+ (
1 z
, x x
2 z
)
2 z
2
, x
3 z
)
+ ( x
3 z
) 2
.
(7)
(8)
(9)
(10)
(11)
Then
(
φ j so
(
φ
) 2 =
=
=
) 2
1
4
1
4
1
4
= (
(∑
(∣ ⃗ u x
3 j
=
1
∣ j z
2
)
((
2 x
= ( j u x
) v
2
2
∣ 2
1 ( x j u
− ( x
−
(
E
−
G
−
2 iF
)
.
2 i j v
− ix
) 2 x
⃗ u j v
)) 2
−
2 ix x v
)
= j u
1 x
4 j v
(( x
)) j u
) 2 − ( x j v
) 2 −
2 ix j u x j v
)
,
Since
⃗ is isothermal,
(
φ
) 2 = 1
4
(
E
−
E
) =
0 [1].
The following theorem relates minimal surfaces to holomorphic functions.
Theorem: Suppose M is a surface with patch
⃗
. Let mal). Then M is minimal if and only if each φ j
⃗
= ∂
⃗
∂z and suppose is holomorphic [1].
(
φ
) 2 =
0 (i.e.,
⃗ is isother-
The proof of this theorem requires the following lemma.
Lemma:
∂
∂z
( ∂
⃗
∂z
) = 1
4
△ ⃗
[1, 3].
Proof of lemma:
∂
∂z
( ∂
⃗
∂z
) =
=
=
∂
=
∂z
1
=
2
(
(
1
2
1
1
4
(
1
4
1
4
(
(
(
∂
⃗
∂u
∂
2
∂
2
∂u
∂
2
∂u x
2
∂u 2
+ x .
(
−
∂ i x
∂
⃗
∂v
))
− i
∂
⃗
− i
∂u
∂
⃗
∂
∂u
2
∂v 2
)
∂
⃗
∂v
∂v
+ i
) + i
∂
⃗
∂u ∂v
∂
∂v
∂
⃗
(
+
∂
⃗
∂u
∂
2
∂v x
2
− i
)
∂
⃗
∂v
)))
Dynamics at the Horsetooth 5 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
QED
Another useful tool in proving the above theorem is a theorem from complex analysis that says f is holomorphic if and only if
∂f
∂z
=
0 [3].
Proof of the above theorem: (
Ô⇒
) If M is minimal, then, by the corollary proven in the harmonic function section, x j is harmonic for j
∈ {
1 , 2 , 3
}
.
x j harmonic
Ô⇒ △⃗ =
0
Ô⇒
Ô⇒
1
4
∂
∂z
△ ⃗ =
0
( ∂
⃗
∂z
) =
0 by the above lemma
Ô⇒ ∂ ⃗
∂z
=
0.
By the theorem from complex analysis, since
∂
∂z
(
∂z
) =
0,
φ j is holomorphic.
∂
⃗
(
⇐Ô
) If φ j x j is holomorphic, then
∂ ⃗
∂z
=
0
Ô⇒ harmonic
Ô⇒
M is minimal.
∂
∂z
( ∂
⃗
∂z
Ô⇒ △⃗
Ô⇒ x j
) =
=
0
1
4
△ ⃗ = is harmonic.
0
QED
Now any minimal surface may be represented using
Given
⃗
⃗ with holomorphic components and
(
φ
) 2 =
0.
φ , how is an isothermal patch
⃗ for M constructed? The following corollary to the theorem proven above shows that the components of
⃗ may be integrated to obtain the components of x [1].
Corollary: x j ( z, z
) = c j
+
2 Re
(∫
φ j dz
)
[1].
Proof of corollary [1]: z
= u
+ iv
Ô⇒ dz
= du
+ idv
φ
φ j j dz dz
Ô⇒ x
=
= j
1
2
1
2
=
(
( x x
2 j u j u
−
+
Re ix ix
(∫ j v j v
Then we have dx j
φ
)(
)( j
= du du
+
−
∂x j
∂z
=
φ j
=
=
1
2 x j u idv idv dz dz
( x dz
) + c j j u
.
+
) =
) =
+ du
∂x
∂z j
φ du
+ x
=
2 Re
(
φ j
+ j v j
1
2
1
2 x dz
)
((
(( dz dz j v dv x x dv j u j u du du
) +
QED
+
+
1
2
( x x x j v j v j u dv dv
) − i
( x du
) +
+ i x
( j v x j u j u dv dv dv
)
−
− x x j v j v du
)) du
))
Now we know how to construct
⃗ if we have need each component, φ j to be holomorphic and
⃗
(
, but what is
φ
) 2 =
⃗ for a general minimal surface? We
0. A nice way to construct
⃗ is as follows [1].
Let f be a holomorphic function and g be a meromorphic function such that f g
2
Let is holomorphic.
φ
1
φ
2
φ
3
=
1 f
(
1
− g
2
2 i
=
2 f
= f g.
(
1
+ g
2
)
)
(12)
Dynamics at the Horsetooth 6 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Then φ
1
, φ
2
, and φ
3 are holomorphic and
(
φ
) 2 = 1
4 f
2 (
1
− g
2 ) 2 − 1
4 f
2 (
1
+ g
2 ) 2 + f
2 g
2 =
0 [1].
Now we have everything we need to understand the Weierstrass-Enneper Representation for minimal surfaces.
The Weierstrass-Enneper Representation for Minimal Surfaces
Theorem: The Weierstrass-Enneper Representation [1]: If f is holomorphic on a domain D, g is meromorphic on D, and f g x
( z, z
) = ( x
1 ( z, z
)
, x
2 ( z, z
)
, x
3
2 is holomorphic on D, then a minimal surface is defined by
( z, z
))
, where x
1 ( z, z
) =
Re
(∫ f
(
1
− g
2 ) dz
) x
2 ( z, z
) =
Re (∫ if
(
1
+ g
2 ) dz
) x
3 ( z, z
) =
Re
(∫
2 f gdz
)
.
(13)
In the Enneper’s surface example, we need f
=
1 and g
= z . Then
( f
(
1
− g
2 )
, if
(
1
+ g
2 )
, 2 f g
)
.
⃗
= (
1
− z
2
, i
(
1
+ z
2 )
, 2 z
) =
There is another way to write Weierstrass-Enneper using just one holomorphic function that is a composition of functions. If g is holomorphic with g means dτ dz
= dg dz
, so dτ
= dg . Define F
(
τ
) = f
/ dg dz
= f dz dg
−
1 also holomorphic, then set τ
= g which
. Then F
(
τ
) dτ
= f
( dz dg
)( dg
) = f dz . Substitute
τ for g and F
(
τ
) dτ for f dz in the Weierstrass-Enneper Representation to get the following version of Weierstrass-Enneper [1].
Theorem: Weierstrass-Enneper Representation II: For any holomorphic function F
(
τ
)
, a minimal surface is defined by x
( z, z
) = ( x
1 ( z, z
)
, x
2 ( z, z
)
, x
3 ( z, z
)) where x
1 ( z, z
) =
Re
(∫ (
1
−
τ
2 )
F
(
τ
) dz
) x
2 ( z, z
) =
Re
(∫ i
(
1
+
τ
2 )
F
(
τ
) dz
) x
3 ( z, z
) =
Re
(∫
2 τ F
(
τ
) dz
)
.
(14)
A good example of using this version of Weierstrass-Enneper is the helicoid.
The Helicoid
A helicoid may be obtained from and F
( e z
2 e 2 z
F
(
τ
) = i
2 τ 2 where τ
= e z
[1]. Notice that τ
= e , τ
) = i log
( z
) because Log
( z
) is the principal branch of the log and branches of the log are holomorphic, but log itself is not. Now compute
⃗ ( u, v
) in the following way.
z
−
1 =
Log
( z
)
, are all holomorphic on the domain of Log
( z
)
. I have used Log
( z
) instead of x
1 =
Re
(∫ (
1
−
τ
=
=
Re
Re
(
(
=
Re
(
−
− i
2
− i
2 i
2 τ
−
( e − i
2 z
τ
2
)
)
2 τ
+ e z
( e −( u
+ iv
) i
2 dτ
)
))
+ e u
+ iv ))
Dynamics at the Horsetooth 7 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
=
Re
(
=
Re
(
= 1
2 e
− i
2
− i
− u
2
( e e −
− u u ( cos
(− v
) + isin
(− v
)) + e cos
(− v
) + sin
(− v
) + 1
2 e u
1
2 e − u sin
( v
) sin
(− v
) − u i
2
( cos
( v
) + isin
( v
)))) e u cos
( v
) + 1
2 e u sin
( v
)) x
2 =
Re
(∫ i
(
1
+
τ
=
Re
(
=
Re
(
=
Re
(
=
Re
(
=
=
Re
1
2 e
(
− u
2
1
2
1
2
1
2
1
2 τ
1
−
( e − e − u
1 z
2
2
τ
)
− e z
) i
2 τ 2 cos
(− v
) + cos
(− v
) − 1
2 e u dτ
)
( e −(
( e − u u
+ iv
)
))
− e u
+ iv ))
( cos
(− v
) + isin
(− v
)) − e u i
2 e − u sin
(− v
) − cos
( v
)
1
2
( cos
( v
) + isin
( v
)))) e u cos
( v
) + i
2 e u sin
( v
)) x
3 =
Re
(∫
2 τ
( i
2 τ
=
Re
( iLog
∣
τ
∣)
2
=
Re
( iLog
∣ e z ∣)
=
Re
( iz
)
=
Re
( i
( u
+ iv
))
=
Re
( iu
− v
)
= − v
) dτ
)
So
⃗ ( u, v
) = ( the helicoid.
1
2
( e − u sin
(− v
) + e u sin
( v
))
,
1
2
( e − u cos
(− v
) − e u cos
( v
))
,
− v
) is an isothermal patch for
Helicoid
14
12
10
8
6
4
2
0
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
Now anyone who’s hobby is finding minimal surfaces can easily find them by integrating holomorphic functions.
What about other types of surfaces?
The Weierstrass-Enneper
Representation may be generalized to constant mean curvature surfaces, which yields some exciting results. Before doing so, a different derivation of the Weierstrass-Enneper Representations integrating the machinery of differential forms is necessary.
Dynamics at the Horsetooth 8 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Consider a surface Ω
⊂
R
3
, with orthonormal frame
{ e
1 each point on Ω. For 1-forms ω
1 and ω
2
, e
2
, e
3
} such that on the original patch there exist e
3
ω is the normal vector at
1 ′
, ω
2 ′ on Ω such that
(
ω
ω
1 ′
2 ′
) =
T
(
ω
ω
1
2
)
, where T is the shape operator, as defined in Eq. 3 [5]. If T is given by
T
= ( t
11 t
12 t
21 t
22
)
, then, by Eq. 4, Ω is a minimal surface if and only if t
11
+ t
22
Define the 1-form
τ
= ( e
1
− ie
2
)
ω
1 + ( e
2
=
0.
+ ie
1
)
ω
2
.
Then, dτ
= i
( t
11
+ t
22
) e
3
ω
1 ∧
ω
2
.
So, the following are equivalent:
1. Ω is a minimal surface.
2.
t
11
+ t
22
=
0.
3.
dτ
=
0
4.
τ is a closed form.
Define the function f such that
ω
1 + iω
2 = f dz and therefore ω
1 − iω
2 = f dz.
(15)
Taking the product of the two equations in Eq. 15 gives
∣ f
∣ 2 dz
∧ dz
=
ω
1 ∧
ω
1 − iω
1 ∧
ω
2 + iω
2 ∧
ω
1 +
ω
2 ∧
ω
2
∣ f
∣ 2 dz
∧ dz
= −
2 i ω
1 ∧
ω
2 dz
∧ dz
= −
2 i
∣ f
∣ 2
ω
1 ∧
ω
2 ≠
0
It follows that z can be used as a local coordinate on Ω, which makes Ω into a Riemann surface.
Finally, define F
( z, z
) = ( e
1
− ie
2
) f . It follows that
F dz
= ( e
1
− ie
2
) f dz
= ( e
1
− ie
2
)(
ω
1 + iω
2 ) =
τ.
Therefore, dτ
=
0 if and only if d
(
F dz
) = dF
∧ dz
=
0. This gives the condition that F is a function of z only, or
∂F
∂z
=
0. It follows that Ω is a minimal surface if and only if F is holomorphic. If Ω is indeed a minimal surface, then there exists a holomorphic function v
( z
) such that v
Using v , one can define the 1-form ξ
= dv , which gives rise to
′
( z
) =
F
( z
)
.
R
(
ξ
) =
R
( dv
) = e
1
ω
1 + e
2
ω
2 = dx, for a local position vector x on the surface. It follows that there exists some real-valued vector y such that v
( z
) = x
( z
) + iy
( z
)
.
Conversely, for any holomorphic
C
3
-valued function v , the surface R
( v
) is a minimal surface. This gives rise to Eq. 13 [5].
Dynamics at the Horsetooth 9 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Let’s try to generalize the new derivation for minimal surface to surfaces of constant mean curvature.
Let H
= c , for some constant c . Then t
11
+ t
22
=
2 H
=
2 c . Thus, dτ
=
2 ice
3
ω
1 ∧
ω
2
.
Since dτ
= d
(
F dz
)
, d
(
F dz
) = dF
∧ dz
=
2 ice
3
ω
1 ∧
ω
2 ≠
0 .
Therefore
∂F
∂z
≠
0, so F is not a holomorphic function.
OH NO!
There is no general Weierstrass
Enneper representation for constant-mean curvature surfaces. However, this does not mean that there are not other ways of representing constant mean curvature surfaces via holomorphic functions. Indeed, if a system has a Hamiltonian structure, than the corresponding functions can be found.
Example: Start with the linear system
ψ
1 z
= pψ
2
ψ
2 z
= − pψ
1
(16) where ψ
1 and ψ
2 are complex valued functions, but p
( z, z
) is real valued. This system leads to the surface given by
X
X
1
1
+ iX
− iX
X
2
2
3
=
2 i ∫ z
0 z
=
2 i ∫ z
0 z
(
ψ
1
2
(
ψ
2
2
= −
2 dz ′
∫ z
0 z
(
ψ
2
ψ
1 dz ′
− dz ′
ψ
−
ψ
2
1
+
2
2 dz
ψ
1 dz
′
)
ψ
2
′
) dz ′
)
,
(17) with mean curvature H
=
Eq. 16 becomes p
( z,z
)
∣
ψ
1 ∣
2
+∣
ψ
2 ∣
2
[6]. So, if H is constant, then p
( z, z
) =
H
(∣
ψ
1
∣ 2 + ∣
ψ
2
∣ 2 )
. Thus
ψ
1 z
=
H
(∣
ψ
1
∣ 2 + ∣
ψ
2
∣ 2 )
ψ
2
ψ
2 z
= −
H
(∣
ψ
1
∣ 2 + ∣
ψ
2
∣ 2 )
ψ
1
.
Splitting z into real and imaginary components, z
= t
+ ix , gives generalized momentum, Hamiltonian and Poisson bracket:
P
=
ψ
1 x
ψ
2
−
ψ
1
ψ
2 x dx
{
F
1
, F
H
= i
(
ψ
1 x
2
} = (
∂F
∂ψ
1
1
ψ
2
+
ψ
1
∂F
∂ψ
2
2
−
ψ
2
∂F
∂ψ x
1
2
) +
1
2
H
(∣
ψ
1
∣ 2
∂F
2
∂ψ
1
) − (
∂F
2
∂ψ
1
+ ∣
ψ
2
∣ 2 ) 2
∂F
∂ψ
1
2
−
∂F
∂ψ
2
2
∂F
1 )
.
∂ψ
1
This Possion bracket gives rise to the sympletic form
(18)
ξ
= dψ
1
∧ dψ
2
+ dψ
1
∧ dψ
2
.
This system is Hamiltonian because it can be represented in the form ψ
1 t
{
ψ
2
, H
}
[6]. This is shown in the following computations:
= {
ψ
1
, H
}
, ψ
2 t
=
{
ψ
1
, H
} =
∂H
∂ψ
2 by Eq .
18 .
Dynamics at the Horsetooth 10 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
= iψ
1 x
{
ψ
2
+
H
(∣
ψ
1
∣ 2
, H
} =
∂H
∂ψ
1
+ ∣
ψ
2
∣ 2 )
ψ
2 by Eq .
=
18
ψ
.
1 t
.
= iψ
2 x
+
H
(∣
ψ
1
∣ 2 + ∣
ψ
2
∣ 2 )
ψ
2
=
ψ
2 t
.
What symmetries are this Hamiltonian structure preserved under? Consider the case when p is a function of only time p
= p
( t
)
. Then, the system has solution
ψ
1
= r
( t
) exp
( iλx
)
ψ
2
= s
( t
) exp
( iλx
)
, where λ
∈
Then
R
− {
0
}
, r
( t
) = p
1
+ ip 2 & s
( t
) = q
1
+ iq
2 where p 1 , p
2
, q
1
, q
2 are real-valued functions.
H
0
=
H
2
( p
2
1
+ p
2
2
+ q
2
1
+ q
2
2
) 2 −
λ
( p
1 q
1
+ p
2 q
2
) and
{
F
1
, F
2
} = (
∂F
∂p
1
1
∂F
∂q
1
2 −
∂F
∂p
2
1
∂F
∂q
2
2
) − (
∂F
2
∂p
1
∂F
∂q
1
1 −
∂F
∂p
2
Then, the Hamiltonian and Poisson structure are preserved by an S
1
2
∂F
∂q
1
2 action
)
.
[ p
1
→ p p
2
→ p
1
1 cos φ
− p sin φ
+ p
2
2 sin φ cos φ
] [ q
1
→ q
1 q
2
→ q
1 cos φ
− q
2 sin φ
+ q
2 sin φ cos φ
]
.
As
H
0
=
−
λ
(( p
H
2
1
(( p
1 cos φ
− p cos φ
− p
2
2 sin φ
) sin φ
)( q
1
2 + ( p
1 sin cos φ
− q
2
φ
+ p
2 cos φ
) 2 sin φ
) + ( p
1
+ ( q
1 sin φ
+ p
2 cos φ
− q
2 cos φ
)( q
1 sin φ
) 2 + ( q
1 sin φ
+ q
2 sin φ
+ q cos φ
))
2 cos φ
) 2 ) 2
=
H
2
(( cos
2
φ
+ sin
2
φ
)( p
2
1
+ p
2
2
+ q
1
2 + q
2
2
)) 2 −
λ
( p
1 q
1
( cos
2
φ
+ sin
2
φ
) + p
2 q
2
( cos
2
φ
+ sin
2
φ
))
=
H
2
( p
2
1
+ p
2
2
+ q
2
1
+ q
2
2
) 2 −
λ
( p
1 q
1
+ p
2 q
2
)
.
A similar computation can be used to show that the Possion bracket is also conserved under the
S
1 action. It can be verified that this system is also invariant under the transformation
⎢⎢
⎢⎢
⎡⎢
X
X
1
2
→
X
→
X
X
3
1
1 cos τ
−
X sin τ
+
X
→
X
3
2
2 sin τ
+
4 M τ cos τ
⎥⎥
⎤⎥
, where τ is the helicoidial transformation discussed above and M is the induced metric on the space
[6].
What does this example show?
In the example, the constant mean curvature surface was parametrized in Eq. 17, which looks very similar to the Weierstrass-Enneper representation given in Eq. 13. Yet, these parameterizations are path integrals, so for either to be well defined, their exterior derivative must be zero. This means for Eq. 17,
∂
∂
∂z
∂
∂z
(
ψ
1
2
(
ψ
2
2
∂z
(
ψ
2
ψ
1
) = −
∂
∂z
) = −
∂
∂z
) =
∂
∂z
(
(
(
ψ
ψ
ψ
1
2
2
1
2
)
ψ
2
)
)
.
(19)
This gives a set of conserved quantities, which is the foundation of a Hamiltonian system. So, any surface given by Eq. 17, also satisfies Eq. 19 and is therefore a Hamiltonian system.
Dynamics at the Horsetooth 11 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
Connections between different areas of Mathematics are not presented or celebrated as much as they could be. The Weistrass-Enneper representations are one example of both how beautiful and powerful the connections between different areas in Mathematics can be, weaving together Differential Geometry, Complex Analysis and Hamiltonian Systems. All minimal surfaces can be given by the representation in Eq. 13, while all holomorphic functions can then generate a minimal surface by the same parametrization. This can be extended to the representation of Eq. 17 for a general surface, which will therefore have Hamiltonian structure satisfying Eq. 19. The duality also extends in the other direction; for a Hamiltonian system satisfying Eq. 19 will give yield to a surface via
Eq. 17. Classifying when these surfaces have constant mean curvature remains an area open for exploration in Mathematics, making the Wiestrass-Enneper representations promising for future results, despite having already provided beautiful and powerful connections in Mathematics.
(Picture courtesy of PIXAR)
Dynamics at the Horsetooth 12 Vol. 4, 2012

The Weierstrass-Enneper Representations Myla Kilchrist and Dave Packard
[1] Oprea, J. 2007.
Differential Geometry and Its Applications.
Mathematical Association of
America, Inc., USA.
[2] Pressley, A.
Elementary Differential Geometry.
2001. Springer, London.
[3] Stein, E., Shakarchi, R. 2003.
Princeton Lectures in Analysis II Complex Analysis.
Princeton
University Press, Princeton, New Jersey.
[4] Korevaar, N. March 26, 2002.
Making Minimal Surfaces with Complex Analysis.
University of
Utah.
[5] Clelland, J.
Lie Groups and the Method of the Moving Frame.
[6] Konopelchenko, B. G., Taimanov, I. A. May 26, 1995.
Constant mean curvature surfaces via integrable dynamical system.
Dynamics at the Horsetooth 13 Vol. 4, 2012