minimal surfaces as holomorphic functionsgzitelli/pdf/undergrad/minimalsurfaces.pdfminimal surfaces...

MINIMAL SURFACES AS HOLOMORPHIC FUNCTIONS

GREGORY ZITELLI

Abstract. In the context of differential geometry, minimal surfaces are defined assurfaces with vanishing mean curvature, and appear in problems related to findingsurfaces of minimal area. An interesting consequence of their definition is that it isalways possible to construct a coordinate patch for a minimal surface whose componentsare harmonic. Using complex analysis, we can connect these harmonic functions with thecomponents of holomorphic complex functions. Furthermore, the representation formulaof Weierstrass allows us to draw a one-to-one correspondence between holomorphicfunctions and the local coordinate patches of minimal surfaces in the absence of umbilicpoints.

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Minimal Surfaces as Holomorphic Functions Gregory Zitelli

Contents

1. Introduction 31.1. Principal Curvatures 31.2. Mean Curvature 52. Minimal Surfaces as Harmonic Maps 62.1. Isothermal Coordinates 62.2. Harmonic Components of Minimal Surfaces 83. The Weierstrass-Enneper Representation 93.1. Holomorphic Functions 103.2. Construction of the Representation Functions 103.3. Refinement of the Representation Functions 123.4. The Gauss Map and Stereographic Projection 134. Single Function Representations 154.1. The Inverse Gauss Map 154.2. Adjoint Surfaces 17References 18

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1. Introduction

Intuitively, a minimal surface is a surface with certain properties that generalize thenotion of tightness and minimal area. Manifesting physically in the shape of soap films, aminimal surface’s curvature is balanced, which agrees with physical concepts like surfacetension. While not all minimal surfaces necessarily minimize area, they do often appearas the solutions to problems of that nature. Provided that we are given a sufficientlynice boundary and consider all surfaces stretched across it, if a surface with minimalarea exists we will see that it must be a minimal surface. The converse, that a minimalsurface exists for any nice curve, is known as Plateau’s problem. Though it has beenproven to be true, it is beyond the scope of this discussion.

Minimal surfaces are defined as surface with vanishing mean curvature, and so inorder to analyze minimal surfaces further, we must briefly establish some elements ofdifferential geometry.

1.1. Principal Curvatures.

Let M be a Ck regular surface in R3, with k ≥ 2, and let ~x : U → M be a localcoordinate patch with U ⊆ R2 open, we can consider some point ~x(u1, u2) = ~p ∈ Mand the tangent space T~pM . We defined the unit normal ~n and the first and secondfundamental forms (gij) and (Lij) with naming conventions as in [1], with subscriptsdenoting the partial derivative for the coordinate patch ~x, while other subscripts likethose for the fundamental forms may not.

~n =~x1 × ~x2|~x1 × ~x2|

gij =⟨~xi, ~xj

⟩i, j = 1, 2

Lij =⟨~n, ~xij

⟩i, j = 1, 2

Consider Figure 1, which shows a curve ~γ traversing part of a surface. The solid blacklines show the tangent and normal vectors corresponding to the curve. The dotted linerepresents the surface normal ~n to the surface, as we just defined. If ~γ is a unit speedcurve, then the tangent vector can be given by ~T = ~γ′, and we can define the curvatureκ at the point to be the number κ = |~T ′|.

Now in the context of surfaces, it is useful to consider how much a curve on a surface iscurving in the natural direction of the surface. This is done by observing how much thevector κ ~N points in the direction of the normal ~n. In other words, we wish to computethe following.

κn =⟨κ ~N,~n

⟩

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Figure 1. Part of a pseudosphere with a curve traced over it. The solidarrows show the tangent and normal vectors emanating from the point. Thedotted line shows the normal vector to the surface. The normal curvatureκn to the surface is the amount that the vector κ ~N points in the directionof the unit normal. In our picture, this quantity would be negative, sincethe normal vector to the curve and the unit normal to the surface point inopposite directions.

The Weingarten map L : T~pM → T~pM is defined as the negative of the directionalderivative of the unit normal ~n in the direction of vectors in T~pM . The coefficients of theWeingarten map can be given by the following equation, where the matrix (gij) is theinverse of the first fundamental form (gij).

Li j =2∑

k=1

Lkjgki i, j = 1, 2

The Weingarten map possesses a number of important properties that we shall givewithout proof. The map is self-adjoint, and in fact we have conveniently that for any~X, ~Y ∈ T~pM the second fundamental form can be written as follows. Here, the super-scripts denote the components of the vectors.

II(~X, ~Y

)=

2∑i,j=1

LijXiY j =

⟨L(~X), ~Y⟩

=⟨~X, L

(~Y)⟩

It becomes natural to wonder which directions maximize and minimize the normalcurvature, as expressed by the second fundamental form. By considering the set of unit

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vectors in T~pM , which is itself a compact set, Lagrange multipliers can be used to de-termine that there exist eigenvectors and corresponding eigenvalues of L that do exactlythat. We denote the eigenvalues, called principal curvatures, of M at point ~p by κ1 (theminimum) and κ2 (the maximum).

An important result of the Weingarten map begin self-adjoint is that the principalcurvatures for any surface always occur in orthogonal directions. This is given by thefollowing theorem, inspired by [2].

Theorem 1.1. Given ~p ∈ M , let ~X ∈ T~pM be an eigenvector for L with eigenvalue λ1.

Let ~Y ∈ T~pM be some nonzero vector orthogonal to ~X. Then ~Y is also an eigenvectorfor L.

Proof. The result follows from the fact that L is self-adjoint, and that T~pM has dimension2. Using the second fundamental form, we have that

II(~X, ~Y

)=⟨~X, L

(~Y)⟩

=⟨L(~X), ~Y⟩

=⟨λ1 ~X, ~Y

⟩= 0

Thus, L(~Y ) is orthogonal to ~X. Since ~Y is also orthogonal to ~X, and L is non-

degenerate, we have that L(~Y ) and ~Y are parallel. Therefore, there exists some λ2 such

that L(~Y ) = λ2~Y . �

Corollary 1.2. The minimum and maximum values κ1, κ2 achieved by the second fun-damental form over the set of unit vectors in T~pM occur in orthogonal directions.

Definition 1.3. A point on a surface is said to be an umbilical point if κ1 = κ2.

1.2. Mean Curvature.

We now define the mean curvature H of the surface M as the average of κ1 and κ2.Because of their relationship with the Weingarten map, H can be expressed as follows.

H =κ1 + κ2

2=

1

2Tr(L)

Definition 1.4 (Minimal Surface). A surface M as described above is a minimal surfaceif H ≡ 0 for all points on the surface.

Since we saw that the directions where κ1 and κ2 are achieved are always orthogonal,a minimal surface is one in which every point is curving equal and opposite amounts intwo orthogonal directions. Therefore, each point on a minimal surface is a saddle point,although not all surfaces made up entirely of saddle points will necessarily be minimal.

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Figure 2. A portion of a catenoid, an example of a minimal surface. Thelines shown run orthogonal to one another, and trace out curves with equaland opposite curvature.

2. Minimal Surfaces as Harmonic Maps

Because minimal surfaces are characterized by the vanishing of the mean curvature, wemay wonder where the mean curvature may pop up in our calculations. Under certain,unusually pleasant coordinate patches, we will show that the Laplacian of the componentsof the coordinate patch involves the mean curvature, and when the mean curvaturevanishes for minimal surfaces correlates it results in the vanishing of the Laplacian. Sucha coordinate patch is known as an isothermal coordinate patch, and the involvement ofthe Laplacian allows us to show that the components of an isothermal coordinate patchfor a minimal surface are harmonic. Using complex analysis, we can then connect thecomponents of minimal surface coordinate patches to the real (or imaginary) parts ofholomorphic functions.

2.1. Isothermal Coordinates.

Definition 2.1. A coordinate patch ~x : U → M is said to be isothermal if the firstfundamental form satisfies g11 = g22, and g12 = g21 = 0 for all points on the surface M .That is to say, the first fundamental form can be expressed as follows:

I(~X, ~Y

)=

2∑i,j=1

gijXiY j = c

⟨~X, ~Y

⟩where c(u, v) = g11 = g22 varies for different points ~p ∈M .

While it is true that there exist isothermal coordinates for all surfaces, the proof isfar too lengthy for our purposes. Instead, it is fairly straightforward to show that such

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coordinates not only exist locally for any minimal surface, but also that they can becomputed rather easily when the surface is expressed as a graph.

Theorem 2.2 (Osserman [2]). For any minimal surface M , and point ~p ∈ M , thereexists some local coordinate patch ~x : U →M such that ~p is contained in the image of ~x,and that ~x is isothermal.

Proof. As in [3], choose a Cartesian coordinate system such that M is locally given bythe graph of some function f(x, y) = z. We can then construct the local coordinate patch~x(x, y) = (x, y, f(x, y)). The first fundamental form is computed as[

1 + f 21 f1f2

f1f2 1 + f 22

]Now letting W =

√det(gij) =

√1 + f 2

1 + f 22 , we can then construct the coefficients of

the Weingarten map for this coordinate patch:

(Li j)

=1

W 3

[f11 (1 + f 2

2 )− f12f1f2 f12 (1 + f 21 )− f11f1f2

f12 (1 + f 22 )− f22f1f2 f22 (1 + f 2

1 )− f12f1f2

]From the fact that M is a minimal surface, we then have that

f11(1 + f 2

2

)− 2f12f1f2 + f22

(1 + f 2

1

)= 0

Now observe that(f1f2W

)1

−(

1 + f 21

W

)2

=f2W

(f11(1 + f 2

2

)− 2f12f1f2 + f22

(1 + f 2

1

) )= 0(

f1f2W

)2

−(

1 + f 22

W

)1

=f1W

(f11(1 + f 2

2

)− 2f12f1f2 + f22

(1 + f 2

1

) )= 0

Therefore, there exist functions P (x, y) and Q(x, y) such that

P1 =(1 + f 2

1

)/W Q1 = (f1f2) /W

P2 = (f1f2) /W Q2 =(1 + f 2

2

)/W

Using the change of coordinates u = x + P (x, y) and v = y + Q(x, y), we have thatthe Jacobian of the transformation is (as in Osserman)

J =∂ (u, v)

∂ (x, y)= 2 +

2 + f 21 + f 2

2

W> 0

Therefore, there exists a local inverse ψ : V → U such that the first fundamental formof the coordinate patch ~x ◦ ψ : V → M is described as g11 = g22 = W/J and g12 = 0.Therefore, the coordinate patch ~x ◦ ψ is isothermal. �

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2.2. Harmonic Components of Minimal Surfaces.

Now consider the computation of the mean curvature for a surface expressed in isother-mal coordinates. Through the use of the Weingarten map, we have the following.

H =1

2Tr (L) =

L11 + L2

2

2=

∑k Lk1g

k1 +∑

k Lk2gk2

2=L11 + L22

2√g

As promised, we will now see that the components of a coordinate patch written inisothermal coordinates of a minimal surface are all harmonic. The result occurs when weconsider the ”Laplacian” of the isothermal coordinate patch ~x, given by ~x11 + ~x22. Evenif the coordinate patch is not minimal, the fact that is it isothermal eventually gives usthat ~x11+~x22 = 2

√gH~n, which only vanishes if the surface is minimal. As we mentioned,

isothermal coordinates exist for any surface. However, since we have only proven theirexistence for minimal surfaces, we will restrict our consideration to them exclusively.

Theorem 2.3. Let xi be the components of an isothermal coordinate patch ~x for someminimal surface M . Then each xi is harmonic. That is to say, xi11 + xi22 = 0.

Proof. While it is not easy to show that each of the components is harmonic individually,we can show that ~x11 + ~x22 = ~0, which will prove it for all three simultaneously. Usingthe fact that the coordinate patch ~x is isothermal, we have the following.

〈~x1, ~x1〉 = g11 = g22 = 〈~x2, ~x2〉〈~x1, ~x2〉 = g12 = 0

〈~x1, ~x2〉1 = 〈~x11, ~x2〉+ 〈~x1, ~x12〉 = 0 〈~x2, ~x11〉 = −〈~x1, ~x12〉

〈~x1, ~x2〉2 = 〈~x12, ~x2〉+ 〈~x1, ~x22〉 = 0 〈~x2, ~x12〉 = −〈~x1, ~x22〉

Furthermore, 〈~x1, ~x1〉1 = 〈~x1, ~x11〉 + 〈~x11, ~x1〉 = 2 〈~x1, ~x11〉, and likewise 〈~x2, ~x2〉1 =2 〈~x2, ~x12〉. By the equality of 〈~x1, ~x1〉 and 〈~x2, ~x2〉, we then have that

〈~x1, ~x11〉 = 〈~x2, ~x12〉 = −〈~x1, ~x22〉

It follows that 〈~x1, ~x11 + ~x22〉 = 0. Similarly, 〈~x2, ~x11 + ~x22〉 = 0, and so ~x11 + ~x22 isparallel to ~n. Therefore,

~x11 + ~x22 = 〈~x11 + ~x22, ~n〉~n = (L11 + L22)~n = 2√gH~n

However, since M is minimal, we have that ~x11 + ~x22 = ~0, and so each component of ~xis harmonic. �

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We can now describe a minimal surface locally as a triplet of harmonic functions of twovariables. Not all triplets of harmonic functions will necessarily create a minimal surface,however. We’ve only shown that these components are harmonic when they are part ofan isothermal coordinate patch, and so a triplet of harmonic functions must satisfy theappropriate conditions on its first fundamental form. As we will see, these restrictioncan be very elegantly described once these harmonic functions are described in terms ofholomorphic functions.

Figure 3. A portion of an Enneper surface, a minimal surface. The sur-face is described with harmonic components. This one is given as follows:

~r(u, v) =(x− x3/3 + xy2, y3/3− x2y − y, x2 − y2

)

3. The Weierstrass-Enneper Representation

In the context of the complex analysis, holomorphic functions of complex variables areintimately related to harmonic functions. Each holomorphic function f(z) = u(x, y) +iv(x, y) of a single complex variable z = x + iy can be represented by its real andimaginary components, which can in turn be described as real valued functions of tworeal variables x and y. As we will see, the fact that f is holomorphic places certainrestrictions on these components, including that they be harmonic.

The converse of this result, that all harmonic functions can be used to construct holo-morphic functions, is also true. This can be proven rather straightforwardly by performingthe actual construction. While these facts can be found in [4] and [5], the constructionwe will use to create holomorphic functions mimics that which is typically found in theproof that they exist.

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3.1. Holomorphic Functions.

We begin with some preliminary results from complex analysis. Our goal is to de-fine what a holomorphic function is, and to construct holomorphic functions which willrepresent our local isothermal coordinate patch.

Definition 3.1. Let f : D → C = R2 be a function of a complex variable, with D ⊆ Copen. Then f is said to be holomorphic on D if the derivative f ′(z0) = limz→z0

f(z)−f(z0)z−z0

exists for all points z0 ∈ D. If such a derivative exists for a function defined on all of Dexcept for a set of isolated points, we say that function is meromorphic.

Theorem 3.2. Suppose that f : D → C = R2 is a C1 function, with D as before. Writef(x + iy) = u(x, y) + iv(x, y), where u, v : D → R are real valued functions of twovariables. Then the necessary and sufficient condition for f to holomorphic on D is thatthe Cauchy-Riemann equations are met for all points (x0, y0) ∈ D:

∂u

∂x=∂v

∂y

∂u

∂y= −∂v

∂x

Furthermore, the complex derivative f ′ can be expressed in terms of the partial deriva-tives with respect to the real variables:

f ′(z) =∂u

∂x+ i

∂v

∂x=∂v

∂y− i∂u

∂y

An example of these results can be seen if we consider f : C→ C given by f(z) = z2.We can express the function as f(x+ iy) = (x+ iy)2 = (x2− y2) + i(2xy), and thereforeour two real valued functions are u(x, y) = x2 − y2 and v(x, y) = 2xy. These functionsdo indeed satisfy the Cauchy-Riemann equations.

∂u

∂x= 2x =

∂v

∂y

∂u

∂y= −2y = −∂v

∂x

Additionally, the derivative f ′(z) may be written as 2x+ 2iy = 2z.

3.2. Construction of the Representation Functions.

Now we have the tools needed to construct holomorphic functions using the harmoniccomponents or our coordinate patch. To avoid the confusion of using x and y as variables,we will write u1 + iu2 = (u1, u2) ∈ D for arbitrary points in the domain.

Now given a minimal surface M , let ~x : D → R3 be an isothermal coordinate patch.We write ~x = (x1, x2, x3). Then define the functions φj : D → C by

φj = µj + iνj =∂xj

∂u1− i∂x

j

∂u2j = 1, 2, 3

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We write φ = (φ1, φ2, φ3) for the total triplet, φ2 = φ21 + φ2

2 + φ23 for its square, and

|φ|2 = |φ1|2 + |φ2|2 + |φ3|2 = φ1φ1 + φ2φ2 + φ3φ3 for the square of the norm of φ.

The construction of these functions is chosen specifically so that they satisfy theCauchy-Riemann equations. This can be computed immediately, relying on the fact thateach component xj is harmonic and mixed partials are equal.

∂µj∂u1

=∂2xj

∂u21= −∂

2xj

∂u22=∂νj∂u2

∂µj∂u2

=∂2xj

∂u2∂u1=

∂2xj

∂u1∂u2= −∂νj

∂u2

From the previous theorem, this shows that each function φj is holomorphic on D. Oneresult of being holomorphic is that our functions have, at least locally, a well definedantiderivative given by the integral of the function. We define the local antiderivativesof our φj functions as follows.

Φj(ω) =

∫ ω

ω0

φj(z)dz + αj

for some ω0 ∈ D, and with αj chosen such that xj(ω0) = Re(αj). Then Φj is the uniquecomplex antiderivative of φj such that Φj(ω0) = αj. For a comprehensive theory ofcomplex integration, see [5].

Theorem 3.3. The coordinate functions xj defined on the open interval D ⊆ R2 areidentically equal to Re (Φj). That is to say, for any (u1, u2) ∈ D,

xj(u1, u2) = Re(Φj(u1 + iu2)

)Proof. By properties of the complex integral, we have that Φ′j = φj. From the Cauchy-Riemann equations, it follows that

∂xj

∂u1= Re (φj) = Re

(∂Φj

∂u1

)=∂ Re(Φj)

∂u1

∂xj

∂u2= Re (iφj) = Re

(∂Φj

∂u2

)=∂ Re(Φj)

∂u2

Furthermore, xj(ω0) = Re(αj) = Re(Φj(ω0)) by definition. Since the two functionsare both the real-valued anti-derivatives of the function φj, they must be equal by itsuniqueness. �

Our results thus far can be summarized in the following theorem, which we will expandas our discussion continues. In particular, we will begin to examine the restrictions placedon the functions φj that we have constructed.

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Theorem 3.4 (Preliminary Representation Theorem). Given a minimal surface M , andsome point ~p ∈ M , there exists a triplet of complex analytic functions φ = (φ1, φ2, φ3)such that the triplet

Re

(∫ ω

ω0

φ(z)dz + α

)is an isothermal coordinate patch for M over some open neighborhood D ⊆ R2 for someω0 ∈ D and α ∈ C3.

The converse of this statement will now become our main concern, and therefore ex-pand our ability to represent minimal surfaces as holomorphic functions. Given a tripletof complex holomorphic functions, will the real component of such an integral produceisothermal coordinates of some minimal surface? If the triplet is chosen randomly, cer-tainly not. What, then, should be expected of such a triplet?

3.3. Refinement of the Representation Functions.

The easiest way to explore this question is to determine what properties are presentwhen we construct such functions from a minimal surface with an isothermal coordinatepatch. Consider the previously constructed φj functions, and the isothermal coordinate

patch ~x. Since the surface is non-degenerate, the two partials of ~x cannot be ~0, and so|φ|2 6= 0. From the isothermal conditions, we also have the following equation. Recallthat xij refers to derivative with respect to uj of the ith component of the coordinatepatch ~x.

φ21 + φ2

2 + φ23 =

(x11 − ix12

)2+(x21 − ix22

)2+(x31 − ix32

)2=(x11)2

+(x21)2

+(x31)2

−(x12)2 − (x22)2 − (x32)2− 2i

(x11x

12

)− 2i

(x21x

22

)− 2i

(x31x

32

)= 〈~x1, ~x1〉 − 〈~x2, ~x2〉 − 2i 〈~x1, ~x2〉 = 0

The immediate consequence of this observation is that we can now reduce such atriplet of functions by solving for one of them in terms of the other two. The most ob-vious way that this can be done is by transforming the equation into φ2

3 = −φ21 − φ2

2.If we solve for φ3, we would have a representation of the minimal surface as the triplet(φ1, φ2, i

√φ21 + φ2

2

), which only depends on two holomorphic functions. The disadvan-

tage of this representation is that it relies on the complex square root, which is multi-valued. This would make things considerably more difficult, and so we concern ourselveswith finding another way to eliminate the need for all three functions. We can accom-plish this by observing that the above equation can be factored, such that we have thefollowing.

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(φ1 − iφ2)(φ1 + iφ2) + φ23 = 0

φ1 + iφ2

φ3

= − φ3

φ1 − iφ2

From this we could construct other holomorphic functions, our goal being to representall three of these functions by only two, simpler functions. As an example, consider thefunctions P = φ1 − iφ2 and Q = φ3. Then we can reconstruct our functions using theequations above.

(φ1, φ2, φ3) =

(−1

2

Q2 − P 2

P,i

2

Q2 + P 2

P,Q

)While this makeshift representation accomplishes our goal of reducing our problems to

two holomorphic functions, it is in no way intuitive. In order to find a more appropriateformulation, we consider the complex formulation of the Gauss map.

3.4. The Gauss Map and Stereographic Projection.

Given a coordinate patch ~x : U → M to a surface (not necessarily minimal), we canconstruct the Gauss map G : U → S2 from the domain of the coordinate patch ontothe unit sphere, by assigning to each points ~p ∈ U the unit normal ~n correspondingto the point ~x(~p). In calculating the unit normal, we must take the cross product ofthe two partial coordinate patches ~x1, ~x2. However, by our construction we have thatxj1 = Re(φj) = µj and xj2 = Im(φj) = −νj.

~x1 × ~x2 =(x21x

32 − x31x22, x31x12 − x11x32, x11x22 − x21x12

)=(µ2(−ν3)− µ3(−ν2), µ3(−ν1)− µ1(−ν3), µ1(−ν2)− µ2(−ν1)

)=(µ3ν2 − µ2ν3, µ1ν3 − µ3ν1, µ2ν1 − µ1ν2

)=(

Im(φ2φ3), Im(φ3φ1), Im(φ1φ2))

By normalizing this expression, we have the following unit normal:

~n =2(

Im(φ2φ3), Im(φ3φ1), Im(φ1φ2))

|φ|2

Because the Gauss map sends elements in the domain of the coordinate patch ontothe unit sphere, we have the opportunity to take advantage of stereographic projectionbetween the sphere and the complex plane. See [3] for more details on stereographicprojection. The projection of a point on the sphere S2 with coordinates (x, y, z) is givenby the complex point Γ = (x + iy)/(1 − z). By considering the coordinates of ~n as our

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point on S2, we can compute the corresponding point on the complex plane. Followingfrom the fact that Im(z) = (z − z)/2i for a complex number z, we have the following:

Γ =2 Im(φ2φ3)/|φ|2 + 2i Im(φ3φ1)/|φ|2

1− 2 Im(φ1φ2)/|φ|2=

2 Im(φ2φ3) + 2i Im(φ3φ1)

|φ|2 − 2 Im(φ1φ2)

=

(φ2φ3 − φ2φ3 + iφ3φ1 − iφ3φ1

)/i

|φ|2 − 2 Im(φ1φ2)=φ3(φ1 + iφ2)− φ3(φ1 + iφ2)

|φ|2 − 2 Im(φ1φ2)

As in [3], we can use the relationship between the representation functions to simplifythe numerator.

Γ =φ3(φ1 + iφ2)− φ3(φ1 + iφ2)

|φ|2 − 2 Im(φ1φ2)

=φ3(φ1 + iφ2)− φ2

3 (−φ3/(φ1 − iφ2))

|φ|2 − 2 Im(φ1φ2)

=

(φ3

φ1 − iφ2

) (φ1 + iφ2

)(φ1 − iφ2) + φ3φ3

|φ|2 − 2 Im(φ1φ2)

=

(φ3

φ1 − iφ2

)φ1φ1 + φ2φ2 + φ3φ3 + iφ1φ2 − iφ1φ2

φ1φ1 + φ2φ2 + φ3φ3 + iφ1φ2 − iφ1φ2

Therefore, the expression reduces simply to φ3/(φ1− iφ2). Consequently, given a mini-mal surface and representation functions (φ1, φ2, φ3), the function φ3/(φ1− iφ2) : D → Cis a holomorphic map that takes points in the original parameter domain D, looks at theirimage on the unit sphere S2 through the Gauss map, and uses stereographic projectionto project the points back down to the complex plane.

Theorem 3.5 (Weierstrass-Enneper Representation). Given a minimal surface M , andsome point ~p ∈M , we can construct a holomorphic function F and meromorphic functionG, where FG2 is holomorphic, such that the triple of functions

Re

(∫ ω

ω0

1

2F(1−G2

)dz + α1

)Re

(∫ ω

ω0

i

2F(1 +G2

)dz + α2

)Re

(∫ ω

ω0

FG dz + α3

)is an isothermal coordinate patch for M over some open neighborhood D ⊆ R2 for someω0 ∈ D and α ∈ C3. Such a construction is obtained by defining

F = φ1 − iφ2 G =φ3

φ1 − iφ2

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Furthermore, any pair of functions (F,G) satisfying these conditions can be used toconstruct an isothermal coordinate patch for a minimal surface from the integrals above.

The function G is taken to represent the Gauss map. The other function, F , appears tobe somewhat arbitrarily chosen, however it allows us to reconstruct our original represen-tation by the relation (φ1, φ2, φ3) =

(12F (1−G2), i

2F (1 +G2), FG

). Most texts demand

that F is not identically zero in order to ensure that |φ|2 6= 0, however since F appearsin the denominator of G, the restriction that G is meromorphic guarantees this for us.

4. Single Function Representations

The advantage of reducing minimal surfaces down to two complex functions, one holo-morphic and one meromorphic, is that we can smoothly manipulate such surfaces bymanipulating their representations. We proceed as in [6] to show that we can further re-duce the representation to a single holomorphic function, as long as there are no umbilicpoints close by. This, in fact, shows that there exists a local one-to-one correspondencebetween minimal surfaces and holomorphic functions. This is known as the representationformula of Weierstrass.

4.1. The Inverse Gauss Map.

Consider a minimal surface M with coordinate patch ~x and appropriate functions(F,G), where G represents the Gauss map. Since G is given to be meromorphic, thecondition that G′(ω) 6= 0 is enough to ensure that there exists a local inverse Ψ, alsomeromorphic, in some neighborhood of ω.

Theorem 4.1. Given the function G representing the Gauss map, G′(ω) = 0 if and onlyif ~x(ω) is an umbilic point of the minimal surface M .

We provide this theorem without proof, as it would involve the introduction of Gauss-ian curvature, and numerous calculations. We differ the interested reader to [6]. Now con-sider the new coordinate patch ~x◦Ψ. The work to confirm that such a reparametrizationis still isothermal is trivial. Computing the representation functions φ for this coordinatepatch yields the following:

φ(ω) = φ(Ψ(ω)

)Ψ′(ω)

As in [6], the Weierstrass-Enneper representation for such a reparametrization yields

F (ω) = F(Ψ(ω)

)Ψ′(ω) and G(ω) = ω. Then we let Ξ(ω) = 1

2F (ω) = 1

2F(Ψ(ω)

)Ψ′(ω),

with a factor of 1/2 added for convenience. This leads us to the following theorem.

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Figure 4. A minimal surface given by the function Ξ(ω) = 1/ω − 1/ω3,the adjoint of Catalan’s surface. Though the meromorphic function has apole at the origin, the coordinate patch is defined over part of an annulusthat avoids it.

Theorem 4.2 (Representation Formula of Weierstrass). Given a minimal surface M ,and some point ~p ∈M , there exists a holomorphic function Ξ such that

Re

(∫ ω

ω0

Ξ(z)(1− z2

)dz + α1

)

Re

(∫ ω

ω0

i Ξ(z)(1 + z2

)dz + α2

)

Re

(∫ ω

ω0

2z Ξ(z) dz + α3

)is an isothermal coordinate patch for M over some open neighborhood D ⊆ R2 for someω0 ∈ D and α ∈ C3. Furthermore, any holomorphic function Ξ can be used to constructan isothermal coordinate patch for a minimal surface from the integrals above.

Beginning with two arbitrary functions (F,G), we have essentially chosen G perma-nently to be the identity function. Such a choice does not make it terribly easy for usto construct the Ξ function from a given minimal surface, however it does allow us tocreate minimal surfaces given only one holomorphic function.

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Figure 5. The deformation of a catenoid into a helicoid using the functionΞθ(ω) = eiθ ·1/ω2. Images show θ = 0, π

10, π

5, 3π

10, 2π

5, π

2. Each intermediate

surface is minimal, due to the construction.

4.2. Adjoint Surfaces.

Recall that the original representation for a minimal surface essentially reduced toshowing that its isothermal coordinate patch was the real component of a holomor-phic function. More specifically, we constructed a triplet of holomorphic functions Φ =(Φ1,Φ2,Φ3), and showed that ~xj = Re(φj). We may now ask, what does the imaginarypart of Φ represent? Clearly, it too could be used to construct a minimal surface, us-ing the fact that Im(Φ) = Re(−iΦ). Such a surface is called the adjoint of the originalsurface. Our new representation of minimal surfaces as a single holomorphic function Ξmakes it incredibly easy to compute adjoint surfaces, as they can be obtained simply bymultiplying Ξ by a factor of ±i.

The traditional example of this concept is the helicoid and the catenoid, the latter ofwhich we have already seen. The catenoid can be given by the function Ξ(ω) = 1/ω2,

while the helicoid can similarly be described by Ξ(ω) = i/ω2. Initially looking at thetwo surfaces, it is not clear how they are connected to one another. Our representationbecomes even more advantage, however, when one considers that the multiplication ofholomorphic functions is itself holomorphic. We can define the holomorphic functionΞθ = eiθ · Ξ for arbitrary 0 ≤ θ < 2π. Such a function is the rotation of the function Ξ

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through the angle θ in the complex plane, and in fact we have that Ξ0 = Ξ and Ξπ/2 = Ξ.Additionally, Ξθ also represents a one parameter family of minimal surfaces, smoothlytransforming the catenoid into the helicoid. Such a family is called the associated familyof minimal surfaces with the catenoid (or helicoid, as they are part of the same families).

Figure 6. The deformation of Catalan’s surface, given by the functionΞθ(ω) = eiθ ·

(1ω− 1

ω3

). The final surface on the lower right side is periodic,

although our coordinate patch does not extend to further periods.

References

[1] Richard S. Millman and George D. Parker. Elements of Differential Geometry. Englewood Cliffs, NJ:Prentice-Hall, 1977.

[2] Robert Osserman. A Survey of Minimal Surfaces. New York: Van Nostrand Reinhold, 1969.[3] A. T. Fomenko and A. A. Tuzhilin. Elements of the Geometry and Topology of Minimal Surfaces in

Three-dimensional Space. Providence, RI: American Mathematical Society, 1991.[4] Tristan Needham. Visual Complex Analysis. Oxford University Press: London, 2006.[5] Kunihiko Kodaira. Complex Analysis. Cambridge: Cambridge Univ., 2008.[6] U. Dierkes, S. Hildebrandt, A. Kuster, and O. Wohlrab. Minimal Surfaces I. Springer-Verlag: Berlin,

1992.

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