A Comprehensive Overview of Minimal-Surface Theory

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A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN’S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP

A Thesis Presented to the Faculty of the Department of Mathematics University of Houston

In Partial Fulﬁllment of the Requirements for the Degree Master of Science

By Gregory Patrick Ozbolt August 2014 A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN’S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP

Gregory Patrick Ozbolt

APPROVED:

Dr. Min Ru, Chairman

Dr. Shanyu Ji

Dr. Qianmei (May) Feng

Dean, College of Natural Sciences and Mathematics

ii Acknowledgement

I owe a debt of gratitude to Dr. Min Ru for his guidance and encouragement throughout this endeavor. During my years as an undergraduate student, it was Dr. Ru who helped instill in me a deep appreciation for the power and elegance of differential geometry, a branch of mathematics that soon became one of my favorites. It seemed only natural to base my master’s thesis on a topic in differential geometry, so after consulting with Dr. Ru, I decided to base it on minimal-surface theory. By devoting myself to this topic and completing this project, I now have a clear understanding of this rich theory and an even greater appreciation for differential geometry, and I have Dr. Ru to thank for this. His expertise on the subject has been invaluable to me.

iii A COMPREHENSIVE OVERVIEW OF MINIMAL-SURFACE THEORY, WITH A THOROUGH MATHEMATICAL TREATMENT OF BERNSTEIN’S THEOREM, THE WEIERSTRASS-ENNEPER REPRESENTATIONS, AND PROPERTIES OF THE GAUSS MAP

An Abstract of a Thesis Presented to the Faculty of the Department of Mathematics University of Houston

In Partial Fulﬁllment of the Requirements for the Degree Master of Science

By Gregory Patrick Ozbolt August 2014

iv Abstract

Notable results in minimal-surface theory include Bernstein’s Theorem, the Weierstrass- Enneper representations, and properties of the Gauss map of a minimal surface. For a solution f(x1, x2) of the minimal-surface equation on the whole x1, x2-plane, Bern- stein’s Theorem guarantees the existence of a nonsingular linear transformation given by x1 = u1, x2 = au1 + bu2, b > 0, such that (u1, u2) are (global) isothermal parameters for the surface defined by xk = fk(x1, x2), k = 3, ..., n. We prove Bernstein’s Theorem, and we state and prove three important corollaries of Bernstein’s Theorem. The Weierstrass-Enneper representations indicate that a minimal surface is defined by the parametrization ~x(z) = (x1(z), x2(z), x3(z)), where the coordinate functions x1(z), x2(z), and x3(z) are expressed in terms of a holomorphic function f and a meromorphic function g in the first case, and in terms of τ = u + iv and a holomorphic function F (τ) in the second case. We construct the Weierstrass-Enneper representations. Finally, we state and prove a number of results regarding the Gauss map of a minimal surface in E3. For instance, we prove that if M is a complete regular minimal surface in E3, then either M is a plane or else the image of M under the Gauss map is everywhere dense in the unit sphere; we prove that the Gauss map of a complete non-flat regular minimal surface in E3 can omit at most four points of the sphere. Pertinent results in differential geometry and minimal-surface theory have been included throughout this thesis in an effort to make it fairly self-contained. We conclude that the theory of minimal surfaces has broadened into a rich field of study with exciting results that are often unrelated to notions of area from which the theory originally arose.

v Contents

1 Introduction...... 1 2 Deﬁning a Minimal Surface ...... 3 3 Classical Examples of Minimal Surfaces...... 13 4 The Minimization of Area...... 18 5 The Minimal-Surface Equation...... 23 6 Isothermal Parameters ...... 28 7 Bernstein’s Theorem ...... 33

8 The Weierstrass-Enneper Representations of a Minimal Surface in E3 ...... 43 9 The Gaussian Curvature of a Minimal Surface in E3 ...... 51 10 The Gauss Map of a Minimal Surface in E3 ...... 55 11 Conclusion ...... 78 References ...... 80

vi 1 Introduction

Differential geometry has long been regarded as a tremendously important branch of mathematics. By utilizing techniques of differential and integral calculus as well as linear and multilinear algebra to study problems in geometry, the mathematical field of differential geometry has proven to be a powerful tool in subjects ranging from topology to Einstein’s General Theory of Relativity. A significant discovery that was made in the field of differential geometry was the concept of a minimal surface. The mathematical definition of a minimal surface is a surface whose mean curvature is everywhere zero. The meaning of a surface’s mean curvature will be developed in the next section. A minimal surface is said to be a surface that minimizes area. In the preface to the book, Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics 2001 Summer School [3], David Hoffman writes that a minimal surface “has the defining property that every sufficiently small piece of it (small enough, say, to be a graph over some plane) is the surface of least area among all surfaces with the same boundary.” Classical examples of minimal surfaces include the plane, which is a trivial example; the catenoid, which is the surface formed by rotating a cate- nary curve about its directrix; the helicoid, which is the surface formed by drawing horizontal rays from the axis of a helix curve to points on the helix curve; and the Enneper surface, which is an intricate, self-intersecting surface that was introduced by Alfred Enneper in connection with minimal-surface theory. Before delving into the mathematics associated with minimal surfaces, we shall briefly review the history of minimal surfaces as explained in Hoffman [3]. The problem of determining the surfaces of rotation that minimize area was introduced by the Swiss mathematician Leonhard Euler in 1744. In that same year, he solved the problem and concluded that the catenoid is the only such surface. In 1755, the

1 nineteen-year-old Italian-born mathematician Joseph-Louis Lagrange became inter- ested in the problem of surfaces that minimize area and began to write letters to Euler. In these letters, Lagrange discussed the problem of finding a graph over a region in the plane, with prescribed boundary values, that was a critical point for the area function. In one of his letters to Euler, Lagrange included a second-order, nonlinear, elliptic equation that he had formulated which solved the problem. This equation would become known as the Euler-Lagrange equation. Although Lagrange included this newly-discovered equation for the solution in his letter to Euler, he did not include any new solutions. It is interesting to note that their correspondence led to the formulation of the calculus of variations, a term coined by Euler in 1766. The helicoid was first discovered by Jean Baptiste Meusnier in 1776. He demonstrated that it was also a solution to the Euler-Lagrange equation. Beyond this discovery, he gave an extremely important geometric interpretation of the Euler-Lagrange equation in terms of the vanishing of the average of the principal curvatures of the surface. This quantity became known as the “mean curvature,” thanks to the sugges- tion of the French mathematician Sophie Germain. The concept of a minimal surface being a surface whose mean curvature is everywhere zero was born. Since their discovery, minimal surfaces have been the subject of immense interest among mathematicians because they are the surfaces that usually minimize area, a trait that can be very useful in many different fields. Over the past fifteen years, the study of minimal surfaces has become the focus of much mathematical and scientific research, notably in the areas of molecular engineering and materials science due to their anticipated applications in nanotechnology.

2 2 Deﬁning a Minimal Surface

In order to study the mathematical developments made in the field of minimal surfaces, we will need to review several fundamental concepts in differential geometry that will lead us to the definition of a minimal surface. The concepts presented in this section and the next have been drawn from Shifrin [7].

We deﬁne a surface in R3 to be a connected subset M ⊂ R3 such that each point in M has a neighborhood that is regularly parametrized. Now, we shall deﬁne

~x : U → M ⊂ R3,

(u, v) 7→ ~x(u, v) to be a C3 one-to-one function from some open set U ⊂ R2 to the aforementioned 3 ~ connected subset M ⊂ R . We will require that ~xu × ~xv 6= 0, where ~xu and ~xv denote ∂~x/∂u and ∂~x/∂v, respectively. We call ~x a regular parametrization of the surface

3 ~ M ⊂ R , regular because ~xu × ~xv 6= 0. We can think of a surface M as containing infinitely many curves within it. Two very important types of curves on M are the u-curves and the v-curves.A u-curve is obtained by fixing v = v0 and varying u, such that u 7→ ~x(u, v0), and a v-curve is obtained by fixing u = u0 and varying v, such that v 7→ ~x(u0, v).

If we let P = ~x(u0, v0) be a point on M, then ~xu(u0, v0) is tangent to the u- curve and ~xv(u0, v0) is tangent to the v-curve. Assuming M is a regular parametrized surface and ~x is a regular parametrization ~x : U → M ⊂ R3, then we shall deﬁne the tangent plane, or tangent space, of M at P , denoted by TP M, to be the subspace spanned by ~xu and ~xv evaluated at P = ~x(u0, v0). We note that ~xu × ~xv is a normal

3 vector to TP M. We shall deﬁne the unit normal ~n of a parametrized surface by

~x × ~x ~n = u v . |~xu × ~xv|

p Note that we are using | · | to represent the Euclidean norm k · k = h· , ·i on R3, where h· , ·i denotes the standard inner product on R3, which is also called the dot product “ · ” on R3. A very useful concept in diﬀerential geometry is that of the ﬁrst fundamental

form. The ﬁrst fundamental form is the inner product on the tangent space TP M of a surface M ⊂ R3. It is induced canonically from the dot product on R3. The ﬁrst fundamental form is useful because it allows the curvature and metric properties of a surface, such as length and area, to be calculated in a way that is consistent with the

surrounding space of the surface. We deﬁne the ﬁrst fundamental form IP such that

~ ~ ~ ~ ~ ~ ~ ~ IP (U, V ) = hU, V i = U · V, for U, V ∈ TP M,

where IP : TP M → TP M is a bilinear map. Since ~xu and ~xv span TP M and are

linearly independent, then the set {~xu, ~xv} is a basis for TP M. We therefore deﬁne:

E = IP (~xu, ~xu) = ~xu · ~xu,

F = IP (~xu, ~xv) = ~xu · ~xv = ~xv · ~xu = IP (~xv, ~xu),

G = IP (~xv, ~xv) = ~xv · ~xv.

IP can take the form of a symmetric matrix,

    a a EF  11 12    IP =  =   , a21 a22 FG

4 ~ ~ so that a11 = E, a12 = a21 = F , and a22 = G. If U and V are vectors contained in ~ ~ TP M, then U = a~xu + b~xv and V = c~xu + d~xv for scalars a, b, c, d ∈ R. Thus, we have

~ ~ ~ ~ IP (U, V ) = U · V = (a~xu + b~xv) · (c~xu + d~xv)

= E(ac) + F (ad + bc) + G(bd),

~ ~ ~ ~ 2 2 IP (U, U) = U · U = (a~xu + b~xv) · (a~xu + b~xv) = E(a ) + 2F (ab) + G(b ).

Note that E,F, and G are called the coeﬃcients of the ﬁrst fundamental form.

If we let θ be the angle between ~xu and ~xv at the point P ∈ M, then

~x · ~x F cos θ = u v = √ . |~xu||~xv| EG

Therefore, the u-curve and v-curve are orthogonal at P if and only if F = 0 at P . Now, suppose that ~α(t) = ~x(u(t), v(t)) is a curve on a parametrized surface ~x : U → M such that ~α(t0) = ~x(u0, v0) = P . Then by the chain rule,

0 0 0 ~α (t0) = u (t0)~xu(u0, v0) + v (t0)~xv(u0, v0).

We let ds/dt be the element of arc length of the curve ~α on ~x, where ds/dt = |~α 0|. Then we have

ds2 = |~α 0|2 = ~α 0 · ~α 0 = I (~α 0, ~α 0) dt ~α

0 2 0 0 0 2 = (u ) (~xu · ~xu) + 2u v (~xu · ~xv) + (v ) (~xv · ~xv)

= E(u0)2 + 2F u0v0 + G(v0)2.

5 Thus, the length of ~α from ~α(a) to ~α(b), a ≤ t ≤ b, is given by

Z b 0 0 1/2 s(t) = [I~α(t)(~α (t), ~α (t))] dt a

Z b = [E(u, v)(u0)2 + 2F (u, v)u0v0 + G(u, v)(v0)2 ]1/2 dt, a

where u = u(t), v = v(t), u0 = u0(t), and v0 = v0(t). Next, let us consider the identity given by

2 2 2 2 |~xu| |~xv| = (~xu · ~xv) + |~xu × ~xv|

for vectors ~xu, ~xv ∈ TP M. This can be rewritten as

2 2 2 2 2 |~xu × ~xv| = |~xu| |~xv| − (~xu · ~xv) = EG − F .

Hence, √ 2 |~xu × ~xv| = EG − F

It is a fact that the element of area on a surface ~x is given by

dA = |~xu × ~xv| du dv.

Therefore, to calculate the surface area of the parametrized surface ~x : U → M, we evaluate the double integral

ZZ ZZ √ 2 |~xu × ~xv| du dv = EG − F du dv. U U

6 Earlier, we deﬁned the unit normal of a parametrized surface by

~x × ~x ~n = u v . |~xu × ~xv|

The German mathematician Carl Friedrich Gauss realized that instead of differenti- ating the tangent vectors ~xu and ~xv to define the concept of curvatures for a surface M, it would be much more convenient to differentiate the unit normal ~n, which can be viewed as a vector-valued function of two variables. Instead of bothering with two vectors, namely ~xu and ~xv, there would be a unique choice ~n. He understood that the curvatures of the surface could be described by the rate of change of ~n. If we let S2 represent the unit 2-sphere, then the Gauss map of a regular parametrized surface M is defined as the function ~n : M → S2 that assigns to each point P ∈ M the unit normal ~n(P ). The mapping ~n encapsulates most of the geometric information regarding the surface M. If we take any scalar-valued or vector-valued function f on M and any tangent ~ vector V ∈ TP M, then we can choose a parameterized curve ~α :(−, ) → M on the surface with ~α(0) = P and ~α 0(0) = V~ and compute the directional derivative

0 DV~ f(P ) by computing (f ◦ ~α) (0). Now, if we choose a point P ∈ M, then we can form a plane that is spanned by ~ ~ the vectors ~n(P ) and V ∈ TP M, where V is a unit vector. This plane will slice the surface M creating a curve on M. This curve is the result of what is referred to as a normal slice of M at P . If we look at various normal slices of M at P , we can gain an understanding of the shape of M at P . Suppose that ~α is the arc-length-parametrized

0 ~ curve that is formed from the normal slice. Then ~α(0) = P and ~α (0) = V ∈ TP M, where V~ is a unit vector since ~α is an arc-length-parametrized curve. Now, the curve must lie in the plane spanned by ~n(P ) and V~ . Therefore, the principal normal N~ of

7 the curve at ~α(0) = P must be equal to ± ~n(P ), with N~ (0) = +~n(P ) if ~α is curving towards ~n at P , and N~ (0) = −~n(P ) if ~α is curving away from ~n at P . Letting T~ denote the unit tangent vector of the curve, we deduce that (~n◦~α(s))·T~(s) = 0 for all s in the close vicinity of zero since N~ ·T~ ≡ 0. Note that κN~ (0)·~n(P ) = ±κ(P ), where κ(P ) denotes the curvature of the normal slice at P . Using the Frenet-Serret formula κN~ = T~ 0, the deﬁnition of the directional derivative, and the fact that T~(0) = ~α 0(0), we have

±κ(P ) = T~ 0(0) · ~n(P ) = −T~(0) · ~n 0(P ) = −T~(0) · (~n ◦ ~α)0(0)

0 ~ ~ = −~α (0) · DV~ ~n(P ) = −V · DV~ ~n(P ) = −DV~ ~n(P ) · V.

This tells us that the curvature of the normal slice can be determined by ﬁrst knowing which direction ~n(P ) points with respect to N~ (0), and then by computing ~ −DV~ ~n(P ) · V . ~ It is a fact that for any V ∈ TP M, the directional derivative DV~ ~n(P ) is an element

of TP M [7]. Using this important fact, we deﬁne the shape operator SP at P to be

a symmetric linear map SP : TP M → TP M deﬁned by

~ SP (V ) = −DV~ ~n(P ).

~ ~ ~ ~ ~ ~ Since SP is symmetric, then SP (U) · V = U · SP (V ) for any U, V ∈ TP M. This brings us to the second fundamental form. The second fundamental form is a quadratic form on the tangent plane of a smooth surface in three-dimensional

Euclidean space. We deﬁne the second fundamental form IIP to be a bilinear map IIP : TP M × TP M → R deﬁned by

~ ~ ~ ~ IIP (U, V ) = SP (U) · V.

8 ~ ~ Thus, if V ∈ TP M and |V | = 1, then

~ ~ ~ ~ ~ ±κ = −DV~ ~n(P ) · V = SP (V ) · V = IIP (V, V ), where κ is the curvature of the normal slice in the direction of the unit vector ~ V ∈ TP M as described above. Hence,

~ ~ ~ IIP (V, V ) = ±κ = κN · ~n,

~ ~ ~ 0 ~ so that IIP (V, V ) gives the component of the curvature vector T = κN of the arc- length-parametrized curve ~α, which is normal to the surface M at P . We shall deﬁne the normal curvature κn of ~α at P by

~ ~ ~ κn = κN · ~n = IIP (V, V ).

It is interesting to note that the normal curvature depends only on the direction of ~α at P and otherwise not on the curve.

If we once again consider the set {~xu, ~xv} as a basis for TP M, then we shall deﬁne:

e = IIP (~xu, ~xu) = −D~xu~n · ~xu = −~nu · ~xu = ~xuu · ~n,

f = IIP (~xu, ~xv) = −D~xu~n · ~xv = −~nu · ~xv = ~xvu · ~n = ~xuv · ~n

= −~nv · ~xu = −D~xv ~n · ~xu = IIP (~xv, ~xu),

g = IIP (~xv, ~xv) = −D~xv ~n · ~xv = −~nv · ~xv = ~xvv · ~n.

9 IIP can also take the form of a symmetric matrix,

    b (~n) b (~n) e f  11 12    IIP =  =   , b21(~n) b22(~n) f g

so that b11(~n) = e, b12(~n) = b21(~n) = f, and b22(~n) = g. As we did with the ﬁrst ~ ~ ~ fundamental form, we can write the vectors U, V ∈ TP M as U = a~xu + b~xv and ~ V = c~xu + d~xv for scalars a, b, c, d ∈ R. Then we have the following:

~ ~ IIP (U, V ) = IIP (a~xu + b~xv, c~xu + d~xv) = −Da~xu+b~xv ~n · (c~xu + d~xv)

= −Da~xu+b~xv ~n · c~xu − Da~xu+b~xv ~n · d~xv

= −aD~xu~n · c~xu − bD~xv ~n · c~xu − aD~xu~n · d~xv − bD~xv ~n · d~xv

= −acD~xu~n · ~xu − bcD~xv ~n · ~xu − adD~xu~n · ~xv − bdD~xv ~n · ~xv

= acIIP (~xu, ~xu) + (bc + ad)IIP (~xu, ~xv) + bdIIP (~xv, ~xv)

= eac + f(bc + ad) + gbd.

Note that e, f, and g are called the coeﬃcients of the second fundamental form.

With the set {~xu, ~xv} being a basis for TP M, it can be shown that the matrix representation of the shape operator SP with respect to {~xu, ~xv} is given by

 −1     EF e f a c −1      e e  IP IIP =    =   , FG f g eb de

eG − fF fE − eF fG − gF gE − fF where a = , eb = , c = , and de= , e EG − F 2 EG − F 2 e EG − F 2 EG − F 2 so that SP (~xu) = ea~xu + eb~xv and SP (~xv) = ec~xu + d~xe v [7]. If {~xu, ~xv} happens to be an orthonormal basis for TP M, then E = ~xu · ~xu = 1,F = ~xu · ~xv = 0, and

10 G = ~xv · ~xv = 1. Therefore, the matrix representation of SP with respect to the orthonormal basis {~xu, ~xv} is given by the following:

 −1     1 0 e f e f −1       IP IIP =     =   = IIP . 0 1 f g f g

This makes sense because SP is a symmetric linear map and thus its matrix representation with respect to an orthonormal basis must be symmetric. In the process of deﬁning a minimal surface, we need to recall several facts from linear algebra as discussed in the appendix of [7]. If we let A be an n × n matrix

(or T : Rn → Rn be a linear map), then a nonzero vector ~v is called an eigenvector of A (or of T ) if A~v = λ~v (or if T (~v) = λ~v) for some scalar λ, called the associated eigenvalue. The spectral theorem states that if A is a symmetric 2 × 2 matrix (or

2 2 if T : R → R is a symmetric linear map), then it has two real eigenvalues λ1 and

λ2 and, provided λ1 6= λ2, the corresponding eigenvectors ~v1 and ~v2 are orthogonal.

Since the shape operator SP is a symmetric linear map, then SP has two real eigenvalues by the spectral theorem. These two eigenvalues of SP are called the principal curvatures of a surface M at the point P and will be denoted by k1(P ) and k2(P ). The eigenvectors that correspond to k1(P ) and k2(P ) are called principal directions. Now, the principal directions are orthogonal by the spectral theorem.

Therefore, we can choose an orthonormal basis for TP M that consists of principal directions. Euler utilized these facts and came up with what is now referred to as

Euler’s curvature formula. The idea is that if ~e1 and ~e2 are unit vectors in the principle directions at P with corresponding principal curvatures k1 = k1(P ) and ~ k2 = k2(P ), and if there is a unit vector V ∈ TP M which makes an angle θ ∈ [0, 2π)

11 ~ with ~e1 such that V = cos θ~e1 + sin θ~e2, then

~ ~ 2 2 IIP (V, V ) = k1 cos θ + k2 sin θ.

Using Euler’s curvature formula, we can show that the principal curvatures are the maximum and minimum signed curvatures of the various normal slices at a point P ~ ~ on the surface M. Suppose V ∈ TP M is a unit vector such that V = cos θ~e1 + sin θ~e2 for some θ ∈ [0, 2π). Let k1 = k1(P ) and k2 = k2(P ), and without loss of generality, assume k1 ≥ k2. Then

~ ~ 2 2 2 2 κn(P ) = IIP (V, V ) = k1 cos θ + k2 sin θ = k1(1 − sin θ) + k2 sin θ

2 = k1 + (k2 − k1) sin θ ≤ k1.

Also,

~ ~ 2 2 2 2 κn(P ) = IIP (V, V ) = k1 cos θ + k2 sin θ = k1 cos θ + k2(1 − cos θ)

2 = k2 + (k1 − k2) cos θ ≥ k2.

We are now ready to deﬁne two very important curvatures in the study of surfaces. We deﬁne the Gaussian curvature K of a surface M at a point P to be the product of the principal curvatures at P . That is,

K = det SP = k1(P )k2(P ).

A surface M is considered flat if K ≡ 0. We note that the Gaussian curvature can also be expressed in terms of the coefficients of the first and second fundamental forms, so that eg − f 2 K = . EG − F 2

12 Gauss discovered that the Gaussian curvature is determined by only the ﬁrst

fundamental form IP . That is, K can be computed from just E,F , and G and their ﬁrst and second partial derivatives. He found that

1 ∂ E ∂ G K = − √ √ v + √ u . 2 EG ∂v EG ∂u EG

This is referred to as Gauss’s Theorema Egregium or Gauss’s Remarkable Theorem. It is one of the crowning results of local diﬀerential geometry. Next, we deﬁne the mean curvature H of a surface M at a point P to be the average of the principal curvatures at P . That is,

1 1 H = tr S = (k (P ) + k (P )). 2 P 2 1 2

We note that the mean curvature can also be expressed in terms of the coeﬃcients of the ﬁrst and second fundamental forms, so that

eG − 2fF + gE H = . 2(EG − F 2)

This brings us to the deﬁnition of a minimal surface. A minimal surface is a surface M whose mean curvature H is identically equal to zero (H ≡ 0).

3 Classical Examples of Minimal Surfaces

There are two general classes of surfaces that are relevant to this section. One of these general classes is the class of ruled surfaces. If we let I ⊂ R be an interval,

13 then we can deﬁne a parametrized surface by

~x(u, v) = ~α(u) + vβ~(u),

where u ∈ I, v ∈ R, and where ~α : I → R3 is a regular parametrized curve and β~ : I → R3 is an arbitrary smooth function with β~(u) 6= ~0 for all u ∈ I. Here, ~x is the parametrization of a ruled surface with ~α as the directrix and β~ representing the rulings. The other relevant class of surfaces is the class of surfaces of revolution. As previously mentioned in the introduction, it was the problem of determining the surfaces of rotation (or revolution) which minimize area that was ﬁrst posed and

solved by Euler in 1744. Now, if we let I ⊂ R be an interval and f, g : I → R be C2 functions with f > 0, then we can obtain the surface of revolution by rotating a regular parametrized plane curve given by

~γ(u) = (0, f(u), g(u)),

with u ∈ I, about the z-axis. The parametrization of the resulting surface is

~x(u, v) = (f(u) cos v, f(u) sin v, g(u)), where u ∈ I, 0 ≤ v < 2π. We call the u-curves the proﬁle curves or meridians, and we call the v-curves the parallels. The proﬁle curves are in the shape of ~γ and the parallels are circles. We note that ~x is a regular parametrization [7]. We mentioned in the introduction that the plane, which is both a ruled surface and a surface of revolution, is a trivial example of a minimal surface. This follows from the fact that there is no curvature on the plane. We stated that another ruled

14 surface, the helicoid, is also a minimal surface. To see this, we let

~x(u, v) = (u cos v, u sin v, bv),

where u ≥ 0, v ∈ R, be the parametrization of the helicoid [7]. Then

~xu = (cos v, sin v, 0),

~xv = (−u sin v, u cos v, b).

Thus, 2 2 E = ~xu · ~xu = cos v + sin v = 1,

F = ~xu · ~xv = −u cos v sin v + u cos v sin v = 0,

2 2 2 2 2 2 2 G = ~xv · ~xv = u sin v + u cos v + b = u + b .

Next,

2 2 ~xu × ~xv = (b sin v, −b cos v, u cos v + u sin v) = (b sin v, −b cos v, u).

Then

~x × ~x (b sin v, −b cos v, u) (b sin v, −b cos v, u) ~n = u v = √ = √ . 2 |~xu × ~xv| b2 sin v + b2 cos2 v + u2 b2 + u2

Also,

~xuu = (0, 0, 0),

~xuv = (− sin v, cos v, 0),

~xvv = (−u cos v, −u sin v, 0).

15 Therefore,

e = ~xuu · ~n = 0,

−b sin2 v − b cos2 v b f = ~xuv · ~n = √ = − √ , b2 + u2 b2 + u2

−ub cos v sin v + ub cos v sin v g = ~xvv · ~n = √ = 0. b2 + u2

Finally, we have

eG − 2fF + gE 0 − 0 − 0 H = = = 0. 2(EG − F 2) 2(u2 + b2 − 0)

We conclude that the helicoid is a minimal surface. Here, we see that a surface is minimal if and only if the numerator of its mean curvature H, given by eG−2fF +gE, is equal to zero. We note that the helicoid and the plane are the only ruled minimal surfaces [6]. Another surface that we earlier claimed is a minimal surface is the catenoid, which is a surface of revolution. To see that it is indeed minimal, we let

~x(u, v) = a(cosh u cos v, cosh u sin v, u), where a, u ∈ R, v ∈ [0, 2π), be the parametrization of the catenoid [7]. Then

~xu = a(sinh u cos v, sinh u sin v, 1),

~xv = a(− cosh u sin v, cosh u cos v, 0).

16 Thus,

2 2 2 2 2 2 2 2 2 E = ~xu · ~xu = a sinh u cos v + a sinh u sin v + a = a (sinh u + 1)

= a2 cosh2 u,

2 2 F = ~xu · ~xv = −a cosh u sinh u cos v sin v + a cosh u sinh u cos v sin v = 0,

2 2 2 2 2 2 2 G = ~xv · ~xv = a (cosh u sin v + cosh u cos v) = a cosh u.

Next,

~xu × ~xv = a(− cosh u cos v, − cosh u sin v, cosh u sinh u).

Then

~x × ~x a(− cosh u cos v, − cosh u sin v, cosh u sinh u) ~n = u v = 2 2 2 2 2 2 2 1/2 |~xu × ~xv| [a (cosh u cos v + cosh u sin v + cosh u sinh u)]

(− cosh u cos v, − cosh u sin v, cosh u sinh u) = q (cosh2 u)(1 + sinh2 u)

cos v sin v = − , − , tanh u . cosh u cosh u

Also,

~xuu = a(cosh u cos v, cosh u sin v, 0),

~xuv = a(− sinh u sin v, sinh u cos v, 0),

~xvv = a(− cosh u cos v, − cosh u sin v, 0).

Hence,

2 2 e = ~xuu · ~n = a(− cos v − sin v) = −a,

f = ~xuv · ~n = a(tanh u cos v sin v − tanh u cos v sin v) = (a)(0) = 0,

17 2 2 g = ~xvv · ~n = a(cos v + sin v) = a.

Finally, we see that the numerator of the mean curvature H equals

eG − 2fF + gE = −a3 cosh2 u − 0 + a3 cosh2 u = 0.

We conclude that the catenoid is a minimal surface. We note that the catenoid and the plane are the only minimal surfaces of revolution [6].

4 The Minimization of Area

To gain a better understanding of why a regular surface with mean curvature everywhere zero is said to be a minimal surface, we will demonstrate that if a certain condition is met, then the area of such a surface is the least among all surfaces that share the same boundary. We will also demonstrate that if a surface has the least area among all surfaces that share its boundary, then the surface must have mean curvature identically equal to zero and must therefore be a minimal surface. We shall derive a formula for the change in area in terms of mean curvature by utilizing normal variations of the surface. Here, we are considering surfaces in Euclidean n-space

En. We note that the results obtained in this section, along with those obtained in Sections 5, 6, and 7, have been drawn from Osserman [6].

We shall let ~x : U → M ⊂ En be the parametrization of a regular surface M, we shall let ~γ be a closed curve contained in a domain U that bounds a sub-domain ∆ ⊂ U, we shall let Σ be the surface deﬁned by ~x restricted to ∆, and we shall let

1 ~n(u) be a C function on U that is normal to M at ~x(u), where u = (u1, u2) ∈ U.

(Note, we are using (u1, u2) instead of (u, v) for convenience.) We shall let H(~n) denote the mean curvature of Σ at each point, with respect to ~n. We will also be

18 using the matrix entries aij and bij(~n) to denote the coefficients of the first and second fundamental forms of M as defined earlier in Section 2. It follows that

∂~x ~n(u) · ≡ 0, for i = 1, 2. (4.1) ∂ui

Diﬀerentiating both sides gives us

∂~n ∂~x ∂2~x · + ~n · = 0. ∂uj ∂ui ∂ui∂uj

Therefore, we have ∂~n ∂~x ∂2~x · = −~n · = −bij(~n). (4.2) ∂uj ∂ui ∂ui∂uj

Now, we shall let h(u) be an arbitrary C2 function on U. For each real number λ, let

n ~y : U → Mλ ⊂ E be the parametrization of a surface Mλ, such that

~y(u) = ~x(u) + λh(u)~n(u).

We call ~y a normal variation of ~x because we are varying the surface ~x along the

direction of ~n via the parameter λ. For each λ, we shall let Σλ denote the surface

deﬁned by restricting ~y to ∆, and we shall denote the area of the surface Σλ by A(λ). Now, we claim that ZZ A0(0) = −2 H(~n)h(u) dA, Σ

where the integral of an arbitrary continuous function f of u on the closure of ∆, denoted by ∆, with respect to surface area on Σ is deﬁned as

ZZ ZZ q f(u) dA = f(u) det(aij) du1 du2. Σ ∆

19 We note that A0(0) represents the rate of change of area as a function of λ at λ = 0. To prove the claim, we begin by diﬀerentiating ~y with respect to the domain

coordinates ui which gives us

∂~y ∂~x ∂~n ∂h = + λ h + ~n . ∂ui ∂ui ∂ui ∂ui

λ Let aij denote the entries for the ﬁrst fundamental form of the surface Mλ. Then it follows from (4.1) and (4.2) that

λ ∂~y ∂~y 2 aij = · = aij − 2λhbij(~n) + λ cij, ∂ui ∂uj

where cij is a continuous function of u on U. Thus,

λ 2 det(aij) = q0 + q1λ + q2λ , (4.3)

where q0 = det(aij), q1 = −2h(a11b22(~n) − 2a12b12(~n) + a22b11(~n)), and q2 is a continuous function of u1, u2, λ for u in U.

Since M is a regular surface and the determinant function is continuous, then q0 has a positive minimum on ∆ [6]. Therefore, with q1 and q2 continuous on U, there

λ exists an > 0 such that |λ| < implies det(aij) > 0 on ∆. Hence, for |λ| < , the

surfaces Σλ deﬁned by restricting ~y to ∆ are all regular surfaces. Now, considering (4.3), we can deduce from the Taylor series expansion of the determinant function that

q λ √ q1 2 det(aij) − q0 − √ λ < W λ (4.4) 2 q0

holds on ∆ for some positive constant W . By using the formula for the area of a

20 surface, we have that the area of the surface Σ (when λ = 0) is

ZZ ZZ q ZZ √ A(0) = dA = det(aij) du1 du2 = q0 du1 du2. Σ ∆ ∆

Integrating (4.4) yields

ZZ q1 2 A(λ) − A(0) − λ √ du1 du2 < Cλ ∆ 2 q0 for some positive constant C. Hence,

ZZ A(λ) − A(0) q1 − √ du1 du2 < Cλ. (4.5) λ ∆ 2 q0

Now, we can write H(~n) in terms of aij and bij(~n) as follows:

a b (~n) − 2a b (~n) + a b (~n) H(~n) = 22 11 12 12 11 22 . 2 det(aij)

Thus, we can write (4.5) as

ZZ A(λ) − A(0) q + 2 H(~n)h(u) det(aij) du1 du2 < Cλ. λ ∆

If we let λ go to zero, then we get

ZZ 0 q A (0) = −2 H(~n)h(u) det(aij) du1 du2, (4.6) ∆ so that when integrating with respect to surface area on Σ, we have

ZZ 0 A (0) = −2 H(~n)h(u) dA. Σ

21 Now, if it is the case that H(~n) ≡ 0, then A0(0) = 0. Thus, A has a critical point at λ = 0 when the mean curvature of Σ is everywhere zero. This indicates that the name minimal surface is sometimes misleading since we can only ensure that A has a critical point at λ = 0 when H(~n) ≡ 0. Hence, the area of Σ is the least among all surfaces that share the same boundary as long as the critical point of A at λ = 0 is an absolute minimum of A. We will now show that if the surface Σ does indeed minimize area, then the mean curvature of Σ vanishes everywhere. Suppose that Σ minimizes area and that the mean curvature of Σ does not vanish everywhere. Then there would exist a point p in ∆ and a unit normal ~n to Σ at the point ~x(p) such that H(~n) 6= 0. Note that we can orient ~n so that H(~n) > 0. Now, it is a proven fact that there would exist

1 a neighborhood V1 of p, and ~n(u) ∈ C on V1, such that ~n(u) is normal to M at

~x(u) for u ∈ V1 [6]. Therefore, there would exist a neighborhood V2 3 p contained in V1 such that H(~n(u)) > 0 at every point u of V2. We shall choose a function h(u) ∈ C2 on U with the following properties: h(p) > 0, h(u) ≥ 0 for all u ∈ U, and h(u) ≡ 0 for u∈ / V2. It follows from these properties of h(u) that the integral on the right-hand side of (4.6) will be strictly positive. With this being the case, then

0 A (0) will be strictly negative. Now, if V2 is small enough so that it is contained in ∆, then h(u) = 0 for all points u ∈ ~γ. Thus, ~y(u) = ~x(u) for all u ∈ ~γ. Therefore,

for each λ, the surface Σλ and the surface Σ will share the same boundary. Since we have assumed that Σ minimizes area, then A(λ) ≥ A(0) for all λ, which implies that A0(0) = 0. However, we just showed A0(0) to be strictly negative, so we have a

contradiction. We conclude that the mean curvature of Σ vanishes everywhere.

22 5 The Minimal-Surface Equation

For minimal surfaces with a particular parameterization in En, there exists an equation known as the minimal-surface equation for non-parametric minimal surfaces

in En. We shall ﬁrst consider a surface in E3 with the following parametrization: ~x(u, v) = (u, v, f(u, v)), where f is a C2 function on an open domain U ⊂ R2. Such a parametrization is called a Monge patch or Monge parametrization of a surface, and the surface is said to be given in non-parametric or explicit form. To

formulate the minimal-surface equation for non-parametric minimal surfaces in E3, we begin by noting that the ﬁrst fundamental form of the surface ~x is given by

2 E = a11 = 1 + fu ,

F = a12 = a21 = fufv,

2 G = a22 = 1 + fv , and the second fundamental form of ~x is given by

fuu e = b11(~n) = , p 2 2 1 + fu + fv

fuv f = b12(~n) = b21(~n) = , p 2 2 1 + fu + fv

fvv g = b22(~n) = . p 2 2 1 + fu + fv

We once again consider the mean curvature equation in terms of aij and bij(~n) given by a b (~n) − 2a b (~n) + a b (~n) H(~n) = 22 11 12 12 11 22 . 2 det(aij)

Plugging the corresponding values of the ﬁrst and second fundamental forms of ~x into

23 this equation gives us

(1 + f 2)f − 2f f f + (1 + f 2)f H(~n) = v uu u v uv u vv . p 2 2 2 det(aij) 1 + fu + fv

Now, a surface is minimal if and only if H(~n) is everywhere zero, that is, if and only if the numerator in the above expression for H(~n) is everywhere zero. Thus, ~x is a minimal surface if and only if

2 2 (1 + fv )fuu − 2fufvfuv + (1 + fu )fvv = 0. (5.1)

Equation (5.1) is the minimal-surface equation for non-parametric minimal surfaces in E3. We can deﬁne a surface in non-parametric or explicit form in En by letting i and j be any two ﬁxed distinct natural numbers from 1 to n, and letting U be a domain in the xi, xj-plane. The equations given by

xk = fk(xi, xj), k = 1, ..., n; k 6= i, j;(xi, xj) ∈ U

n deﬁne a surface M in E , where xi = u1 and xj = u2. If we relabel the coordinates in

n E , then we may assume that x1 = u1 and x2 = u2, so that the surface is deﬁned by

xk = fk(x1, x2), k = 3, ..., n.

Next, with x1 = u1 and x2 = u2, we introduce the vector notation

f(x1, x2) = (f3(x1, x2), ..., fn(x1, x2)).

Then using this notation, it can be shown that the minimal-surface equation for

24 non-parametric minimal surfaces in En is given by

2! 2 2 2! 2 ∂f ∂ f ∂f ∂f ∂ f ∂f ∂ f 1 + 2 − 2 · + 1 + 2 = 0, (5.2) ∂x2 ∂x1 ∂x1 ∂x2 ∂x1∂x2 ∂x1 ∂x2

which is a generalization of Equation (5.1) [6].

Now, it is a proven fact that if M is a surface in En parametrized by

n ~x : U → M ⊂ E , and u0 ∈ U is a point at which ~x is regular, then there exists

a neighborhood ∆ of u0 such that the surface Σ obtained by restricting ~x to ∆ has

a re-parametrization Σe in non-parametric form [6]. From this, it follows that every

regular minimal surface in En provides local solutions of Equation (5.2). We shall see in Section 7 that a corollary of Bernstein’s Theorem tells us that for n=3, there are

no non-trivial solutions of Equation (5.2) valid in the whole x1, x2-plane. In order to derive several other forms of the minimal-surface equation, we take a surface in non-parametric form:

xk = fk(x1, x2), k = 3, ..., n,

and we introduce the vector notation:

∂f ∂f f = (f3, ..., fn), p = , q = , ∂x1 ∂x2

∂2f ∂2f ∂2f r = 2 , s = , t = 2 . ∂x1 ∂x1∂x2 ∂x2

Then the minimal-surface equation (5.2) may be written as

∂p ∂p ∂q ∂q (1 + |q|2) − (p · q) + + (1 + |p|2) = 0, (5.3) ∂x1 ∂x2 ∂x1 ∂x2

25 or as (1 + |q|2)r − 2(p · q)s + (1 + |p|2)t = 0. (5.4)

We note that the ﬁrst fundamental form of the surface is given by

2 2 a11 = 1 + |p| , a12 = p · q, a22 = 1 + |q| ,

so that

2 2 2 2 2 det(aij) = 1 + |p| + |q| + |p| |q| − (p · q) .

p We deﬁne W = det(aij). Now, we shall make a variation in our surface by letting

fek = fk + λhk, k = 3, ..., n,

1 where λ is a real number, and hk ∈ C is deﬁned on the domain U for which fk is

deﬁned. In vector notation, setting h = (h3, ..., hn) gives us

∂h ∂h fe = f + λh, pe = p + λ , qe = q + λ , ∂x1 ∂x2 so that

Wf2 = W 2 + 2λX + λ2Y, where ∂h ∂h X = [(1 + |q|2)p − (p · q)q] · + [(1 + |p|2)q − (p · q)p] · ∂x1 ∂x2

and Y is continuous in x1, x2. Hence,

X Wf = W + λ + λ2Z W

where Z is also continuous.

26 Let ~γ be a closed curve in the domain for which f(x1, x2) is deﬁned, and let ∆ be the region bounded by ~γ. Suppose the surface xk = f(x1, x2) over ∆ minimizes the area among all surfaces with the same boundary. Thus, we have

ZZ ZZ Wf dx1 dx2 ≥ W dx1 dx2 (5.5) ∆ ∆ for every choice of h where h = 0 on ~γ. Since λ can be any real number, we deduce from (5.5) and the deﬁnition of Wf that

ZZ X dx1 dx2 = 0. (5.6) ∆ W

Now, if we substitute the deﬁnition of X into (5.6), integrate by parts, and use the fact that h = 0 on ~γ, we have

ZZ ∂ 1 + |q|2 p · q ∂ 1 + |p|2 p · q p − q + q − p h dx1 dx2 = 0. ∆ ∂x1 W W ∂x2 W W

Since this holds for every choice of h such that h = 0 on ~γ, then the equation

∂ 1 + |q|2 p · q ∂ 1 + |p|2 p · q p − q + q − p = 0 (5.7) ∂x1 W W ∂x2 W W must hold everywhere. To verify that (5.7) is a consequence of the minimal-surface equation (5.4), we shall write the left-hand side of (5.7) as the sum of three terms:

1 + |q|2 ∂p p · q ∂q ∂p 1 + |p|2 ∂q − + + W ∂x1 W ∂x1 ∂x2 W ∂x2

∂ 1 + |q|2 ∂ p · q ∂ 1 + |p|2 ∂ p · q + − p + − q. ∂x1 W ∂x2 W ∂x2 W ∂x1 W

27 Now, we note that the ﬁrst term vanishes by (5.3). Expanding out the coeﬃcient of p in the second term given by

∂ 1 + |q|2 ∂ p · q − ∂x1 W ∂x2 W yields 1 [(p · q)q − (1 + |q|2)p] · [(1 + |q|2)r − 2(p · q)s + (1 + |p|2)t]. W 3

We note that this expression vanishes by (5.4). If we interchange p and q, x1 and x2, then it is clear that the coeﬃcient of q in the third term vanishes as well. Hence, (5.7) is veriﬁed. In addition, we have shown that every solution of the minimal-surface equation (5.4) must satisfy the two equations

∂ 1 + |q|2 ∂ p · q = , (5.8) ∂x1 W ∂x2 W

∂ p · q ∂ 1 + |p|2 = . (5.9) ∂x1 W ∂x2 W

Equations (5.8) and (5.9) are very important and useful in the study of minimal surfaces.

6 Isothermal Parameters

When properties of a surface do not depend on the choice of its parameters, then it is convenient to choose parameters in such a way that the surface’s geometric properties are reﬂected in the parameter plane. The conformal mapping of the parameter plane onto the surface is an example of this because the angles between curves on the surface are equal to the angles between the corresponding curves in the parameter plane. This

28 condition can be expressed in terms of the ﬁrst fundamental form as follows:

a11 = a22, a12 = 0,

2 aij = λ δij, λ = λ(u) > 0,

where δij is the Kronecker delta. Parameters u1, u2 satisfying these conditions are called isothermal parameters. Using isothermal parameters can greatly simplify many of the basic quantities in surface theory, for instance,

2 2 4 det(aij) = a11a22 − a12 = a11 = λ , and the formula for mean curvature simpliﬁes to

a22b11(~n) − 2a12b12(~n) + a11b22(~n) b11(~n) + b22(~n) H(~n) = = 2 . 2 det(aij) 2λ

2 It can be shown that if M is a regular surface parametrized by ~x(u) ∈ C where u1, u2 are isothermal parameters, then

∆~x = 2λ2H,

2 2 2 2 where ∆ denotes the Laplace operator deﬁned as ∂ /∂u1 + ∂ /∂u2, and H is the mean curvature vector [6]. The following lemma is an immediate consequence of this fact.

2 LEMMA 6.1: Let M be a regular surface parametrized by ~x(u) ∈ C where u1, u2

are isothermal parameters. Then the coordinate functions xk(u1, u2) are harmonic if

and only if M is a minimal surface. Note that Lemma 6.1 gives us an alternative way of deﬁning minimal surfaces.

29 We shall now incorporate complex analysis into our study of minimal-surface theory. We introduce the following notation. Let ~x(u) be the parametrization of a surface M and consider the complex-valued functions given by

∂xk ∂xk √ φk(z) = − i , z = u1 + iu2, i = −1. (6.1) ∂u1 ∂u2

It follows from the deﬁnition of φk(z) that

n n X 2 X 2 φk(z) = a11 − a22 − 2ia12 and |φk(z)| = a11 + a22. k=1 k=1

Now, the functions φk(z) have the following properties [6]:

a) φk(z) is analytic in z if and only if xk is harmonic in u1, u2;

b) u1, u2 are isothermal parameters if and only if

n X 2 φk(z) ≡ 0; (6.2) k=1

c) if u1, u2 are isothermal parameters, then M is regular if and only if

n X 2 |φk(z)| 6= 0. (6.3) k=1

LEMMA 6.2: Let ~x(u) be the parametrization of a regular minimal surface M, with isothermal parameters u1, u2. Then the functions φk(z) deﬁned by (6.1) are analytic, and they satisfy equations (6.2) and (6.3). Conversely, let φ1(z), ..., φn(z) be analytic functions of z which satisfy (6.2) and (6.3) in a simply-connected domain U. Then there exists a regular minimal surface ~x(u) deﬁned over U, such that equations (6.1) are valid.

30 Proof: Since M is a regular minimal surface, then the coordinate functions xk are harmonic in u1, u2 by Lemma 6.1. Thus, by Property a), the functions φk(z) deﬁned by (6.1) are analytic. Since M is a regular surface with isothermal parameters u1, u2, then Properties b) and c) apply. For the converse, we shall deﬁne

Z xk = Re φk(z) dz.

Then, (6.1) is satisﬁed, and since the functions φ1(z), ..., φn(z) are analytic in z, then x1, ..., xn are harmonic in u1, u2 by Property a). Because the functions φ1(z), ..., φn(z) satisfy (6.2) and (6.3) on a simply-connected domain U, then we deduce from Prop- erties b) and c) that ~x(u) = (x1(u1, u2), ..., xn(u1, u2)) deﬁnes a regular surface over

U with isothermal parameters u1, u2. Since the coordinate functions xk(u1, u2) are harmonic, then we conclude from Lemma 6.1 that ~x(u) is a regular minimal surface.

The next lemma addresses the representation of minimal surfaces locally in terms of isothermal parameters.

LEMMA 6.3: Let M be a minimal surface. Every regular point of M has a neighborhood in which there exists a re-parametrization of M in terms of isothermal parameters.

Proof: We know from Section 5 that we may ﬁnd a neighborhood of the regular point in which M may be represented in non-parametric form. Hence, equations (5.8) and

2 2 2 (5.9) are satisﬁed on some disk D characterized by (x1 − c1) + (x2 − c2) < R . We deduce from these equations that there must exist functions F (x1, x2) and G(x1, x2) on D satisfying ∂F 1 + |p|2 ∂F p · q = , = , (6.4) ∂x1 W ∂x2 W

31 and ∂G p · q ∂G 1 + |q|2 = , = . (6.5) ∂x1 W ∂x2 W

We deﬁne a map (x1, x2) → (ξ1, ξ2) by

ξ1 = x1 + F (x1, x2), ξ2 = x2 + G(x1, x2), (6.6) so that ∂ξ 1 + |p|2 ∂ξ p · q 1 = 1 + , 1 = , ∂x1 W ∂x2 W ∂ξ p · q ∂ξ 1 + |q|2 2 = , 2 = 1 + , ∂x1 W ∂x2 W and ∂(ξ , ξ ) 2 + |p|2 + |q|2 det J = det 1 2 = 2 + > 0, ∂(x1, x2) W where det J is the Jacobian determinant. Since det J 6= 0, then the transformation given by (6.6) has a local inverse (ξ1, ξ2) → (x1, x2). Thus, we may represent the surface in terms of the parameters ξ1, ξ2 by setting xk = fk(x1, x2) for k = 3, ..., n. It can be shown that

∂x W + 1 + |q|2 ∂x p · q 1 = , 2 = − , ∂ξ1 (det J)W ∂ξ1 (det J)W

2 ∂xk W + 1 + |q| p · q = pk − qk, k = 3, ..., n; ∂ξ1 (det J)W (det J)W

∂x p · q ∂x W + 1 + |p|2 1 = − , 2 = , ∂ξ2 (det J)W ∂ξ2 (det J)W

2 ∂xk W + 1 + |p| p · q = qk − pk, k = 3, ..., n [6]. ∂ξ2 (det J)W (det J)W

32 Therefore, with respect to the parameters ξ1, ξ2, we have

2 2 2 ∂~x ∂~x W W a11 = a22 = = = = 2 2 (6.7) ∂ξ1 ∂ξ2 det J 2W + 2 + |p| + |q|

and ∂~x ∂~x a12 = · = 0. (6.8) ∂ξ1 ∂ξ2

We conclude that ξ1, ξ2 are isothermal parameters. Before moving on to the next section, we will need the following lemma, as it will be essential for completing the proof of Bernstein’s Theorem. We shall state it here without proof.

LEMMA 6.4: Let a surface M be deﬁned by ~x(u), where u1, u2 are isothermal

parameters, and let Mf be a re-parametrization of M deﬁned by a diﬀeomorphism

u(ue). Then ue1, ue2 are also isothermal parameters if and only if the map u(ue) is either conformal or anti-conformal [6].

7 Bernstein’s Theorem

In this section, we shall state and prove Bernstein’s Theorem and its corollaries. Bernstein’s Theorem was introduced by the Russian mathematician Sergei Natanovich Bernstein in 1915. It was one of the earliest theorems concerning solutions of the minimal-surface equation. It is global in nature, and it has provided the motivation for a number of results involving minimal surfaces. Because of its importance in the study of minimal surfaces, Bernstein’s Theorem has become a famous theorem over the years. Before delving into Bernstein’s Theorem, we will ﬁrst need to cover several preliminary lemmas.

33 LEMMA 7.1: Let E : D → R be a C2 function on a convex domain D, and suppose that the Hessian matrix ∂2E (7.1) ∂xi∂xj evaluated at any point is positive definite. (This means that the quadratic form it defines sends every nonzero vector to a positive number, or equivalently, that it is symmetric with positive eigenvalues.) Define a mapping φ : D → R2 by

∂E ∂E φ(x1, x2) = (x1, x2), (x1, x2) . ∂x1 ∂x2

Then if x and y are two distinct points of D, we have

(φ(y) − φ(x)) · (y − x) > 0. (7.2)

Proof: Let G(t) = E(ty + (1 − t)x), t ∈ [0, 1]. Then

2 X ∂E G 0(t) = (ty + (1 − t)x) (y − x ), ∂x i i i=1 i and 2 X ∂2E G 00(t) = (ty + (1 − t)x) (y − x )(y − x ). ∂x ∂x i i j j i,j=1 i j

Note that G 00(t) > 0 for t ∈ [0, 1] because the matrix (7.1) is positive deﬁnite and the vector y − x is nonzero. Thus, G 0(1) > G 0(0), or

2 2 X X φi(y)(yi − xi) > φi(x)(yi − xi), i=1 i=1 which implies (7.2).

34 LEMMA 7.2: Assume the hypotheses of Lemma 7.1. Deﬁne the map ξ : D → R2 by

ξi(x1, x2) = xi + φi(x1, x2), (7.3)

where φi(x1, x2) is deﬁned as in Lemma 7.1. Then for any two distinct points x and y of D, we have (ξ(y) − ξ(x)) · (y − x) > |y − x|2, and |ξ(y) − ξ(x)| > |y − x|. (7.4)

Proof: Since ξ(y) − ξ(x) = (y − x) + (φ(y) − φ(x)), we have

(ξ(y) − ξ(x)) · (y − x) = |y − x|2 + (φ(y) − φ(x)) · (y − x) > |y − x|2 by Lemma 7.1. Also, using the Cauchy-Schwarz inequality, we have

|y − x|2 < |(ξ(y) − ξ(x)) · (y − x)| ≤ |ξ(y) − ξ(x)||y − x|.

Therefore,

|y − x| < |ξ(y) − ξ(x)|.

LEMMA 7.3: Assume the hypotheses of Lemmas 7.1 and 7.2. If D is the disk

2 2 2 x1 + x2 < R , then the map (7.3) is a diﬀeomorphism of D onto a domain ∆ which includes a disk of radius R about ξ(0).

2 Proof: The map (7.3) is continuously diﬀerentiable since E(x1, x2) ∈ C . If x(t) is any diﬀerentiable curve in D, and ξ(t) is its image, then it follows from (7.4)

0 0 that |ξ (t)| > |x (t)|. Hence, denoting the Jacobian matrix of ξ by Jξ, the Jacobian

35 0 0 determinant of ξ is everywhere greater than one since ξ (t) = Jξ x (t) implies that

0 0 |ξ (t)| = (det Jξ)|x (t)|. Therefore, the map is a local diﬀeomorphism [6]. In addition, by (7.4), ξ(x) = ξ(y) implies ξ(y) − ξ(x) = 0 implies y − x = 0 implies x = y, so the map is one-to-one and is thus a global diﬀeomorphism onto a domain ∆ [6]. Now, we must show that ∆ includes all points ξ such that |ξ − ξ(0)| < R. If ∆ is the whole plane, then this is obvious. If ∆ is not the whole plane, then there is a point ξ in the complement of ∆ which minimizes the distance to ξ(0). Let ξ(k) be a sequence of points in ∆ which approach ξ, and let x(k) be the corresponding sequence of points in D. We note that the sequence x(k) cannot have an accumulation point in D because the image of such a point would have to be the point ξ, and this would contradict our assumption that ξ is not contained in ∆. Hence, |x(k)| → R as k → ∞. By (7.4), we have |ξ(k) − ξ(0)| > |x(k) − 0| = |x(k)|, so it follows that |ξ − ξ(0)| ≥ R. Therefore, every point within R of ξ(0) is in ∆.

LEMMA 7.4: Let f(x1, x2) be a solution of the minimal-surface equation (5.4)

2 2 2 for x1 + x2 < R . Then returning to the notation that we used in equations (5.4),

(6.4), and (6.5), it is the case that the map (x1, x2) → (ξ1, ξ2) deﬁned by (6.6) is a diﬀeomorphism onto a domain ∆ which includes a disk or radius R about the point ξ(0).

Proof: It follows from the deﬁning characteristics of the functions F and G described in (6.4) and (6.5) that there exists a function E(x1, x2) satisfying

∂E ∂E = F, = G, (7.5) ∂x1 ∂x2

36 2 for the same reason that F and G exist. Then E(x1, x2) ∈ C and

∂2E 1 + |p|2 ∂2E ∂(F,G) 2 = > 0, det = det ≡ 1 > 0 ∂x1 W ∂x1∂x2 ∂(x1, x2) by the definition of W given in Section 5 and the defining characteristics of F and G. Now, any matrix   a b     b c with a > 0 and ac − b2 > 0 must have c > 0, so its trace and determinant are both positive which implies that the sum and product of its eigenvalues are both positive, so it is positive definite. Hence, the function E(x1, x2) has a positive definite Hessian matrix so that Lemmas 7.1-7.3 may be applied to it. We note that it follows from

(7.5) that the map (6.6) is simply the map (7.3) applied to E(x1, x2). Therefore, by

Lemma 7.3, the map (x1, x2) → (ξ1, ξ2) deﬁned by (6.6) is a diﬀeomorphism onto a domain ∆ which includes a disk or radius R about the point ξ(0).

LEMMA 7.5: Let f(x1, x2) be a C1 real-valued function on a domain D. Then the

3 surface M in E deﬁned in non-parametric form by x3 = f(x1, x2) lies on a plane if and only if there exists a nonsingular linear transformation (u1, u2) → (x1, x2) such that u1, u2 are isothermal parameters on M.

Proof: Suppose such parameters u1, u2 exist. We consider the complex-valued functions (6.1), with 1 ≤ k ≤ 3, given by

∂xk ∂xk φk(z) = − i ; z = u1 + iu2; k = 1, 2, 3. ∂u1 ∂u2

Now, φ1 and φ2 are constant because x1 and x2 are linear functions of u1, u2. Since

37 u1, u2 are isothermal parameters, then we know from (6.2) that

3 X 2 φk(z) ≡ 0. (7.6) k=1

2 2 Since φ1 and φ2 are constant, then φ1 and φ2 are constant. We deduce from (7.6) that

2 φ3 must be constant, which limits φ3 to at most two values, and since φ3 must be continuous, it follows that it must be constant as well. This means that x3 has constant gradient with respect to u1, u2, and hence also with respect to x1, x2. Therefore,

f(x1, x2) = Ax1 + Bx2 + C for some constants A, B, and C, which is the equation of a plane. Conversely, suppose the surface M lies on a plane. Then

x3 = f(x1, x2) = Ax1 + Bx2 + C for some constants A, B, and C. We shall show that

u1 and u2 of the linear map (u1, u2) → (x1, x2) deﬁned by

2 2 2 −1 x1 = λAu1 + Bu2, x2 = λBu1 − Au2, where λ = (1 + A + B ) ,

are isothermal parameters on M. To show this, we see that

φ1 = λA − iB, φ2 = λB + iA,

so that

2 2 2 2 2 2 2 2 φ1 = λ A − B − 2λABi, φ2 = λ B − A + 2λABi.

Now,

x3 = Ax1 + Bx2 + C = A(λAu1 + Bu2) + B(λBu1 − Au2) + C,

2 2 2 2 2 2 2 φ3 = λ(A + B ) and thus φ3 = λ (A + B ) .

38 Therefore, we have

2 2 2 2 2 2 2 2 2 2 2 2 φ1 + φ2 + φ3 = λ (A + B ) − (A + B ) + λ (A + B )

= (A2 + B2)(λ2 − 1 + λ2(A2 + B2))

= (A2 + B2)(λ2(1 + A2 + B2) − 1)

= (A2 + B2)(1 − 1) = 0.

It follows from (6.2) that u1, u2 are isothermal parameters on M.

BERNSTEIN’S THEOREM: Let f(x1, x2) be a solution of the minimal-surface equation (5.2) on the whole x1, x2-plane. Then there exists a nonsingular linear transformation

x1 = u1, x2 = au1 + bu2, b > 0, (7.7)

such that (u1, u2) are (global) isothermal parameters for the surface M deﬁned by

xk = fk(x1, x2), k = 3, ..., n.

Proof: We take the map (6.6) to now be deﬁned on the entire x1, x2-plane. Then by

Lemma 7.4, this map is a diﬀeomorphism of the x1, x2-plane onto the entire ξ1, ξ2- plane. From (6.7) and (6.8), we know that (ξ1, ξ2) are isothermal parameters on the surface M deﬁned by xk = fk(x1, x2), k = 3, ..., n. By Lemma 6.2, the functions

∂xk ∂xk φk(z) = − i , k = 1, ..., n ∂ξ1 ∂ξ2 are analytic functions of z. Now,

∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 φ1φ2 = + + i − , ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ1 ∂ξ1 ∂ξ2

39 so we see that

∂(x1, x2) Im(φ1φ2) = − det = − det J. ∂(ξ1, ξ2)

Since the Jacobian determinant det J in the above expression is always positive, we deduce that φ1 6= 0, φ2 6= 0 everywhere and that Im(φ1φ2) < 0. We note that

φ2 φ1φ2 = 2 . φ1 |φ1|

Hence, φ2 1 Im = 2 Im(φ1φ2) < 0. φ1 |φ1|

We note that the function φ2/φ1 is analytic on the whole complex plane, or z-plane, and so it is an entire function. Also, the function has negative imaginary part everywhere. By Picard’s Little Theorem, an entire function whose range misses more than one value in the z-plane is constant, so

φ 2 = c, where c = a − ib, b > 0. φ1

This implies that φ2 = (a − ib)φ1. Thus, it follows from the deﬁnitions of φ1 and φ2 that

∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 − i 2 = (a − ib) 1 − i 1 = a 1 − b 1 − i b 1 + a 1 . ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2

From this equation, we ﬁnd that the real part gives us

∂x ∂x ∂x 2 = a 1 − b 1 , (7.8) ∂ξ1 ∂ξ1 ∂ξ2

40 and the imaginary part gives us

∂x ∂x ∂x 2 = b 1 + a 1 . (7.9) ∂ξ2 ∂ξ1 ∂ξ2

Applying the linear transformation (7.7) to equations (7.8) and (7.9) yields

∂u ∂u ∂u ∂u a 1 + b 2 = a 1 − b 1 , (7.10) ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ2

and ∂u ∂u ∂u ∂u a 1 + b 2 = b 1 + a 1 . (7.11) ∂ξ2 ∂ξ2 ∂ξ1 ∂ξ2

From equations (7.10) and (7.11), we deduce that

∂u ∂u ∂u ∂u 1 = 2 , 2 = − 1 , ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2

which we recognize as the Cauchy-Riemann equations. Because these equations are

satisﬁed, we conclude that u1 + iu2 is a complex-analytic function of ξ1 + iξ2. It follows from Lemma 6.4 that u1, u2 are isothermal parameters, and thus the theorem is proved.

COROLLARY 1: In the case n = 3, the only solution of the minimal-surface equation on the whole x1, x2-plane is the trivial solution, f, a linear function of x1, x2.

Proof: This follows immediately from Bernstein’s Theorem and Lemma 7.5.

COROLLARY 2: A bounded solution of equation (5.2) on the whole plane must be constant (for arbitrary n).

Proof: By Lemma 6.1, each coordinate function xk, for k = 3, ..., n, will be a bounded

41 harmonic function of (u1, u2) on the whole u1, u2-plane. This implies that each xk, for k = 3, ..., n, must be constant [6].

COROLLARY 3: Let f(x1, x2) be a solution of equation (5.2) on the whole x1, x2- plane, and let Mf be the surface deﬁned by

xk = fek(u1, u2), k = 3, ..., n (7.12) obtained by referring the surface M to the isothermal parameters given by (7.7). Then the functions

∂fek ∂fek φek = − i , k = 3, ..., n (7.13) ∂u1 ∂u2 are analytic functions of u1 + iu2 on the whole u1, u2-plane and satisfy

n X 2 2 φek ≡ −1 − c , where c = a − ib. (7.14) k=3 Conversely, given any complex constant c = a − ib with b > 0, and given any entire functions φe3, ..., φen of u1 + iu2 satisfying (7.14), equations (7.13) may be used to deﬁne harmonic functions fek(u1, u2), and substituting u1, u2 as functions of x1, x2 from (7.7) into equations (7.12) yields a solution of the minimal-surface equation (5.2) valid on the whole x1, x2-plane.

Proof: This follows immediately from Lemma 6.2, using the fact that

∂x1 ∂x1 ∂x2 ∂x2 φe1 = − i ≡ 1, φe2 = − i ≡ a − ib ∂u1 ∂u2 ∂u1 ∂u2 because of (7.7).

42 8 The Weierstrass-Enneper Representations of a

Minimal Surface in E3

The Weierstrass-Enneper representations of a minimal surface in E3 were developed by the German mathematicians Karl Weierstrass and Alfred Enneper in the 19th century, and they provide an interesting link between geometry and complex analysis. They allow for the creation of a minimal surface based on two functions in the ﬁrst case and one function in the second case, which we shall discuss shortly. The Weierstrass-

Enneper representations have been essential to the study of minimal surfaces in E3. The results presented in this section have been drawn from Oprea [5] and Osserman [6].

First, we let M represent a surface in E3 parametrized by ~x(u, v). We let z = u + iv be the corresponding complex coordinate. Thenz ¯ = u − iv. Now, we have that z +z ¯ = (u + iv) + (u − iv) = 2u and z − z¯ = (u + iv) − (u − iv) = 2iv. Hence, u = (z +z ¯)/2 and v = (z − z¯)/(2i). This means that ~x(u, v) may be written

as ~x(z) = (x1(z), x2(z), x3(z)), where ~x(z) takes real values. We let F (u, v) be an arbitrary function of u = (z +z ¯)/2 and v = (z − z¯)/(2i). Then by the chain rule, we have ∂F ∂F ∂u ∂F ∂v ∂F ∂ z +z ¯ ∂F ∂ z − z¯ = + = + ∂z ∂u ∂z ∂v ∂z ∂u ∂z 2 ∂v ∂z 2i

∂F 1 ∂F 1 1 ∂F ∂F = · + · = − i . ∂u 2 ∂v 2i 2 ∂u ∂v Similarly,

∂F ∂F ∂u ∂F ∂v ∂F ∂ z +z ¯ ∂F ∂ z − z¯ = + = + ∂z¯ ∂u ∂z¯ ∂v ∂z¯ ∂u ∂z¯ 2 ∂v ∂z¯ 2i

∂F 1 ∂F 1 1 ∂F ∂F = · − · = + i . ∂u 2 ∂v 2i 2 ∂u ∂v

43 Therefore, ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i and = + i . ∂z 2 ∂u ∂v ∂z¯ 2 ∂u ∂v

These are called Wirtinger derivatives. From these equations, we see that

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = and − = −i or = i − . ∂z ∂z¯ ∂u ∂z ∂z¯ ∂v ∂v ∂z ∂z¯

Hence, for any function f(u, v) = f(z), we have

∂f ∂f ∂ ∂ ∂ ∂ df = du + dv = + fdu + i − fdv ∂u ∂v ∂z ∂z¯ ∂z ∂z¯

∂f ∂f ∂f ∂f = (du + idv) + (du − idv) = dz + dz.¯ ∂z ∂z¯ ∂z ∂z¯

Now, we deﬁne

∂x 1 ∂x ∂x φ = k = k − i k , where k = 1, 2, 3, k ∂z 2 ∂u ∂v and we deﬁne ∂~x ∂x ∂x ∂x φ = = 1 , 2 , 3 = (φ , φ , φ ). ∂z ∂z ∂z ∂z 1 2 3

It follows from Lemma 6.1 and Property a) immediately following Lemma 6.1 that if M is a regular surface with isothermal parametrization ~x(u, v) = ~x(z) ∈ C2, and if

2 2 2 φ = ∂~x/∂z so that φ1 + φ2 + φ3 ≡ 0 since ~x is isothermal by Property b) immediately

following Lemma 6.1, then M is minimal if and only if each φk is complex analytic (holomorphic).

COROLLARY 8.1: Suppose ~x(u, v) = ~x(z) ∈ C2 is an isothermal parametriza-

3 tion for a minimal surface M in E and let φ = ∂~x/∂z, where φ1, φ2, and φ3 are R holomorphic. Then xk = 2 Re ( φk dz) + ck, where ck is a constant.

44 Proof: With z = u + iv, z¯ = u − iv, then dz = du + idv, dz¯ = du − idv. Thus,

1 ∂x ∂x φ dz = k − i k (du + idv) k 2 ∂u ∂v

1 ∂x ∂x ∂x ∂x = k du + k dv + i k dv − k du , 2 ∂u ∂v ∂u ∂v

1 ∂x ∂x φ dz¯ = k + i k (du − idv) k 2 ∂u ∂v

1 ∂x ∂x ∂x ∂x = k du + k dv − i k dv − k du . 2 ∂u ∂v ∂u ∂v

Hence, ∂x ∂x dx = k dz + k dz¯ = φ dz + φ dz¯ k ∂z ∂z¯ k k

1 ∂x ∂x 1 ∂x ∂x = k du + k dv + k du + k dv 2 ∂u ∂v 2 ∂u ∂v

∂x ∂x = k du + k dv = 2 Re φ dz. ∂u ∂v k R Therefore, integrating both sides yields xk = 2 Re ( φk dz) + ck. It follows from Lemma 6.2 and its proof that we can use Corollary 8.1 to construct

3 a minimal surface in E by ﬁnding a triple of holomorphic functions φ1, φ2, φ3 which

2 2 2 satisfy φ1 + φ2 + φ3 ≡ 0 on a simply-connected domain D. Now, let D be a domain in the complex plane. Let f be a holomorphic function on D and g be a meromorphic function on D such that fg2 is holomorphic on D. Let

1 i φ = f(1 − g2), φ = f(1 + g2), φ = fg. (8.1) 1 2 2 2 3

45 Then it is clear that φ1, φ2, and φ3 are holomorphic on D, and we note that

1 1 φ2 + φ2 + φ2 = f 2(1 − g2)2 − f 2(1 + g2)2 + f 2g2 1 2 3 4 4

1 1 = f 2(1 − 2g2 + g4) − f 2(1 + 2g2 + g4) + f 2g2 4 4

= −f 2g2 + f 2g2 = 0.

This brings us to the Weierstrass-Enneper representation I.

Weierstrass-Enneper Representation I: If D is a simply-connected domain in the complex plane, and if f is a holomorphic function on D and g is a meromorphic function on D such that fg2 is holomorphic on D, then a minimal surface is deﬁned by the parametrization ~x(z) = (x1(z), x2(z), x3(z)), where

Z z 2 x1(z) = Re f(1 − g ) dζ, z0

Z z 2 x2(z) = Re if(1 + g ) dζ, z0

Z z x3(z) = Re 2fg dζ, z0 with the integral being taken along an arbitrary path from the ﬁxed point z0 ∈ D to the point z. It can be shown that the catenoid and helicoid have representations of the form (f(z), g(z)) = (−(e−z)/2, −ez) and (f(z), g(z)) = (−(ie−z)/2, −ez), respectively [5]. It can also be shown that the minimal surface known as the Enneper surface has a representation of the form (f(z), g(z)) = (1, z) [5]. Now, if g is holomorphic on D with an inverse g−1 that is also holomorphic, then

46 we shall let τ = g(z). Then dτ = g0(z)dz. Hence,

f(z) f(g−1(τ)) f(z)dz = dτ = dτ. g0(z) g0(g−1(τ))

Deﬁne F (τ) = f(g−1(τ))/[g0(g−1(τ))], so that F (τ)dτ = f(z)dz. Note that F (τ) is holomorphic. Therefore, substituting τ for g and F (τ)dτ for fdz in the Weierstrass- Enneper representation I gives us the Weierstrass-Enneper representation II.

Weierstrass-Enneper Representation II: If U is a simply-connected domain in the complex plane, and if F (τ) is a holomorphic function on U, then a minimal surface

is deﬁned by the parametrization ~x(z) = (x1(z), x2(z), x3(z)), where

Z z 2 x1(z) = Re (1 − τ )F (τ) dτ, z0

Z z 2 x2(z) = Re i(1 + τ )F (τ) dτ, z0

Z z x3(z) = Re 2τF (τ) dτ, z0

with the integral being taken along an arbitrary path from the ﬁxed point z0 ∈ U to the point z. Note we can write φ in terms of τ and F (τ) as

1 i φ = (1 − τ 2)F (τ), (1 + τ 2)F (τ), τF (τ) . 2 2

What is remarkable about this representation is that it indicates that any holomorphic function F (τ) on a simply-connected domain deﬁnes a minimal surface. For instance, it can be shown using this representation that the catenoid and helicoid can be deﬁned by F (τ) = 1/(2τ 2) and F (τ) = i/(2τ 2), respectively [5]. It can also be

47 shown using this representation that the Enneper surface can be deﬁned by F (τ) ≡ 1 [5]. Now, the following two lemmas will lead us to an important theorem that tells us that every simply-connected regular minimal surface in E3 has a Weierstrass- Enneper representation (f, g). The second of these lemmas shall be stated without proof; however, we note that its proof relies on the fact that a regular minimal surface cannot be compact, and it also relies on the Koebe uniformization theorem [6].

LEMMA 8.1: Let D be a domain in the complex plane. Every triple of holomorphic functions φ1, φ2, φ3 satisfying

2 2 2 φ1 + φ2 + φ3 ≡ 0 (8.2)

may be represented in the form (8.1), except for the case when φ1 ≡ iφ2 and φ3 ≡ 0.

Proof: Choose holomorphic functions φ1, φ2, φ3 that satisfy (8.2). Let

φ3 f = φ1 − iφ2, g = . (8.3) φ1 − iφ2

Writing (8.2) in the form

2 (φ1 − iφ2)(φ1 + iφ2) = −φ3 (8.4) implies that 2 φ3 2 φ1 + iφ2 = − = −fg . (8.5) φ1 − iφ2

Now, since g = φ3/(φ1 − iφ2) = φ3/f, then φ3 = fg. Since f = φ1 − iφ2, then

φ1 = f + iφ2. Hence,

2 2 2 2 2 2 0 = φ1 + φ2 + φ3 = (f + iφ2) + φ2 + (fg)

48 2 2 2 2 2 2 2 2 = f + 2ifφ2 − φ2 + φ2 + f g = f + 2ifφ2 + f g

2 2 = f (1 + g ) + 2ifφ2.

This implies that f 2(1 + g2) i φ = − = f(1 + g2), 2 2if 2 which implies that

i f fg2 φ = f + i f(1 + g2) = f − − 1 2 2 2

f fg2 1 = − = f(1 − g2). 2 2 2

2 The condition that fg is holomorphic must hold, because otherwise φ1 + iφ2 would fail to be holomorphic by equation (8.5). We note that the only case in which this representation can fail is if the denominator in the equation for g in (8.3) is identically equal to zero, that is, if φ1 ≡ iφ2. In this case, it follows from (8.4) that φ3 ≡ 0 as expected.

LEMMA 8.2: Every simply-connected minimal surface M in En has a re- parametrization of the form ~x : D → M, where D is either the complex unit disk characterized by |z| < 1, or D is the entire complex plane [6].

THEOREM 8.1: Every simply-connected regular minimal surface in E3 can be represented in the form

Z z xk(z) = 2 Re φk(ζ) dζ + ck, k = 1, 2, 3, (8.6) 0

where each ck is a constant, and the functions φk are deﬁned by (8.1) with the domain D being either the complex unit disk or the entire complex plane, f being

49 holomorphic on D, g being meromorphic on D, fg2 being holomorphic on D, f having the additional property that it vanishes only at the poles of g, and the order of its zero at such a point is exactly twice the order of the pole of g, and where the integral is being taken along an arbitrary path from the origin to the point z.

Proof: Let M be a simply-connected regular minimal surface in E3. By Lemma 8.2, M has a re-parametrization of the form ~x : D → M, where D is either the complex unit disk or the entire complex plane. Assume that this re-parametrization is isothermal. Let 1 ∂x ∂x φ = k − i k , z = u + iv. k 2 ∂u ∂v

Note that it follows from Lemma 6.2 that the functions φk are holomorphic. Therefore, it follows from Corollary 8.1 that (8.6) holds, where the integral is independent of path since D is simply-connected. Note that (8.2) holds by Lemma 6.2. Hence, by

Lemma 8.1, φ1, φ2, and φ3 may be represented in the form (8.1). We note that in order

2 for all the φk to vanish simultaneously, either f = 0 where g is regular or fg = 0 where g has a pole. However, since M is a regular minimal surface with isothermal parameters, then all the φk cannot vanish simultaneously by Theorem 6.2. Thus, f vanishes only at the poles of g, and the order of its zero at such a point is exactly twice the order of the pole of g.

50 9 The Gaussian Curvature of a Minimal Surface

in E3

We recall from Section 2 that the Gaussian curvature K of a surface in E3 can be deﬁned in the form of Gauss’s Theorema Egregium given by

1 ∂ E ∂ G K = − √ √ v + √ u . 2 EG ∂v EG ∂u EG

One of the many interesting uses of the Weierstrass-Enneper representations is that the Gaussian curvature of a minimal surface in E3 can be defined in terms of the Weierstrass-Enneper representation I, and in terms of the Weierstrass-Enneper representation II, as indicated in Oprea [5]. We shall demonstrate the latter case first. Let ~x(u, v) be an isothermal parametrization of a regular minimal surface M, and let φ = ∂~x/∂z = (∂x1/∂z, ∂x2/∂z, ∂x3/∂z) = (φ1, φ2, φ3). We shall define

2 2 2 2 |φ| = |φ1| + |φ2| + |φ3| . Since ~x is an isothermal parametrization, then

2 2 2 2 2 E = |~xu| = |~xv| = G and F = ~xu · ~xv = 0. Let λ = |~xu| = |~xv| . Now,

2 2 2 2 2 2 2 ∂x1 ∂x2 ∂x3 |φ| = |φ1| + |φ2| + |φ3| = + + ∂z ∂z ∂z

2 2 2 1 ∂x1 ∂x1 1 ∂x2 ∂x2 1 ∂x3 ∂x3 = − i + − i + − i . 2 ∂u ∂v 2 ∂u ∂v 2 ∂u ∂v

We note that

2 1 ∂xk ∂xk 1 ∂xk ∂xk 1 ∂xk ∂xk − i = − i + i 2 ∂u ∂v 2 ∂u ∂v 2 ∂u ∂v

" # 1 ∂x 2 ∂x 2 = k + k , for k = 1, 2, 3. 4 ∂u ∂v

51 Thus,

3 " 2 2# X 1 ∂xk ∂xk 1 1 1 1 1 |φ|2 = + = |~x |2 + |~x |2 = λ2 + λ2 = λ2. 4 ∂u ∂v 4 u 4 v 4 4 2 k=1

Now, the Gaussian curvature K of M can be expressed in terms of λ2 as follows:

1 ∂ (λ2) ∂ (λ2) K = − √ √ v + √ u 2 λ2 · λ2 ∂v λ2 · λ2 ∂u λ2 · λ2

1 ∂ (λ2) ∂ (λ2) = − v + u 2λ2 ∂v λ2 ∂u λ2

1 ∂2 ∂2 = − (ln λ2) + (ln λ2) 2λ2 ∂v2 ∂u2

1 1 = − ∆ ln λ2 = − ∆ ln λ, 2λ2 λ2 where ∆ represents the Laplace operator.

Suppose M is deﬁned by the parametrization ~x(z) = (x1(z), x2(z), x3(z)), such that Z z 2 x1(z) = Re (1 − τ )F (τ) dτ, z0

Z z 2 x2(z) = Re i(1 + τ )F (τ) dτ, z0

Z z x3(z) = Re 2τF (τ) dτ, z0 where F (τ) is a holomorphic function on a simply-connected domain U in the complex plane, τ = u + iv, and the integral is being taken along an arbitrary path from the

52 ﬁxed point z0 ∈ U to the point z. We earlier showed that

1 i φ = (1 − τ 2)F (τ), (1 + τ 2)F (τ), τF (τ) , 2 2 and also that |φ|2 = (λ2)/2. Then, letting F = F (τ), we have

2 2 2 2 2 λ = 2|φ| = 2(|φ1| + |φ2| + |φ3| )

2 2 ! 1 2 i 2 2 = 2 (1 − τ )F + (1 + τ )F + |τF | 2 2

1 1 = 2|F |2 |1 − τ 2|2 + |1 + τ 2|2 + |τ|2 4 4

1 = |F |2(|1 − τ 2|2 + |1 + τ 2|2 + 4|τ|2) 2

1 = |F |2[(1 − τ 2)(1 − τ 2) + (1 + τ 2)(1 + τ 2) + 4|τ|2] 2

1 = |F |2[1 − (τ 2 + τ 2) + |τ|4 + 1 + (τ 2 + τ 2) + |τ|4 + 4|τ|2] 2

1 = |F |2(2 + 4|τ|2 + 2|τ|4) = |F |2(1 + 2|τ|2 + |τ|4) 2

= |F |2(1 + |τ|2)2.

This implies that 1 K = − ∆ ln λ λ2

∆ ln (|F |(1 + |τ|2)) = − |F |2(1 + |τ|2)2

∆ ln |F | + ∆ ln (1 + |τ|2) = − . |F |2(1 + |τ|2)2

53 We shall now show that if h = h(z) is a holomorphic function, then

∂ ∂h ∆h = 4 . (9.1) ∂z¯ ∂z

Now, since ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i and = + i , ∂z 2 ∂u ∂v ∂z¯ 2 ∂u ∂v then we have

∂ ∂h ∂ 1 ∂h ∂h = − i ∂z¯ ∂z ∂z¯ 2 ∂u ∂v

1 1 ∂2h i ∂2h i ∂2h 1 ∂2h = − + + 2 2 ∂u2 2 ∂u∂v 2 ∂v∂u 2 ∂v2

1 ∂2h ∂2h 1 = + = ∆h. 4 ∂u2 ∂v2 4

Thus, (9.1) immediately follows. With F = F (τ), this implies that

∂ ∂(ln |F |) ∂ ∂(ln(F F )1/2) ∆ ln |F | = 4 = 4 ∂τ ∂τ ∂τ ∂τ

∂ ∂(ln(F F )) ∂ ∂(ln F + ln F ) = 2 = 2 ∂τ ∂τ ∂τ ∂τ

∂ ∂(ln F ) ∂(ln F ) ∂ F (F ) = 2 + = 2 τ + τ . ∂τ ∂τ ∂τ ∂τ F F

Since F is holomorphic, then F cannot be holomorphic [5]. Thus, (F )τ = 0. This implies that ∂ F ∆ ln |F | = 2 τ = 0, ∂τ F

54 since F,Fτ , and, hence, Fτ /F are holomorphic. We also have that

∂ ∂(ln(1 + |τ|2)) ∂ ∂(ln(1 + τ τ)) ∆ ln(1 + |τ|2) = 4 = 4 ∂τ ∂τ ∂τ ∂τ

∂ τ 1 + |τ|2 − τ τ 4 = 4 = 4 = . ∂τ 1 + |τ|2 (1 + |τ|2)2 (1 + |τ|2)2

Therefore, 4 K = − |F (τ)|2(1 + |τ|2)4

is the Gaussian curvature of a minimal surface determined by the Weierstrass-Enneper representation II. To ﬁnd K in terms of the Weierstrass-Enneper representation I, we recall from Section 8 that F (τ) = f(g−1(τ))/[g0(g−1(τ))], where f, g, and g−1 are holomorphic functions with g(z) = τ, so that g−1(τ) = z. Hence, we can write F (τ) = f(z)/g0(z). Therefore, 4 4 |g0(z)|2 K = − = − |F (τ)|2(1 + |τ|2)4 |f(z)|2(1 + |g(z)|2)4

is the Gaussian curvature of a minimal surface in terms of the Weierstrass-Enneper representation I. We conclude from these expressions for K that the Gaussian curvature of a minimal surface is non-positive everywhere.

10 The Gauss Map of a Minimal Surface in E3

In Section 2, we deﬁned the Gauss map of a regular parametrized surface M in E3 to be the function ~n : M → S2 that assigns to each point P ∈ M the unit normal ~n(P ) ∈ S2, where S2 represents the unit 2-sphere. In this section, we shall consider

the Gauss map of a regular minimal surface in E3, and we shall delineate a number of its properties by consolidating results from Oprea [5], Osserman [6], Fujimoto [1]

55 and [2], and Kawakami [4].

Now, let ~x(u, v) be the parametrization of a regular minimal surface M in E3, and let ~n : M → S2 deﬁne the Gauss map of M. We can write ~n(u, v) = ~n(~x(u, v)), so that ~n(u, v) may be regarded as a parametrization of S2 for a small piece of M, as

described in Oprea [5]. We let {~xu, ~xv} be the basis for the tangent plane of M at

the point P on M, denoted by TP M. We note that the vector ~n(P ) is normal to M at the point P , and is also normal to S2 at the point ~n(P ). Therefore, as subspaces

3 2 2 2 of E , we have that T~n(P )S = TP M, where T~n(P )S is the tangent plane of S at the

2 point ~n(P ). Now, with P = ~x(u0, v0), we deﬁne (~n∗)P : TP M → T~n(P )S = TP M to be the induced linear mapping of the tangent planes which, for {~xu, ~xv}, is given by

(~n∗)P (~xu) = ~nu(u0, v0) = D~xu~n(P ) = −SP (~xu)

and

(~n∗)P (~xv) = ~nv(u0, v0) = D~xv ~n(P ) = −SP (~xv),

where SP is the shape operator deﬁned in Section 2 [5]. Letting M and N be surfaces with parametrizations ~x(u, v) and ~y(u, v), respectively, we consider a mapping f : M → N in terms of these parametrizations such that f(~x(u, v)) = ~y(u, v), where the same parameters u and v are being used in both ~ parametrizations. Thus, we have (f∗)P (V ) ∈ Tf(P )N for all V ∈ TP M. Now, f is a conformal map if

2 2 E~x = ~xu · ~xu = λ(u, v) (f∗)P (~xu) · (f∗)P (~xu) = λ(u, v) E~y,

2 2 F~x = ~xu · ~xv = λ(u, v) (f∗)P (~xu) · (f∗)P (~xv) = λ(u, v) F~y,

2 2 G~x = ~xv · ~xv = λ(u, v) (f∗)P (~xv) · (f∗)P (~xv) = λ(u, v) G~y,

56 where λ(u, v) is called the scaling factor [5].

PROPOSITION 10.1: Let M be a regular minimal surface with isothermal parametrization ~x(u, v). Then the Gauss map ~n of M is a conformal map.

Proof: Our objective is to show the map ~n : M → S2 to be conformal, so we just need to show that

2 (~n∗)P (~xu) · (~n∗)P (~xu) = ω(u, v) ~xu · ~xu,

2 (~n∗)P (~xu) · (~n∗)P (~xv) = ω(u, v) ~xu · ~xv,

2 (~n∗)P (~xv) · (~n∗)P (~xv) = ω(u, v) ~xv · ~xv,

where ω(u, v) is a scaling factor. We recall that isothermal parameters have E = G

and F = 0. In Section 2, we saw that SP (~xu) = ea~xu + eb~xv and SP (~xv) = ec~xu + d~xe v, where

eG − fF fE − eF fG − gF gE − fF a = , eb = , c = , and de= . e EG − F 2 EG − F 2 e EG − F 2 EG − F 2

Here, e, f, and g are the coeﬃcients of the second fundamental form. Thus, we have

eE fE e f (~n∗)P (~xu) = −SP (~xu) = −(a~xu + eb~xv) = − ~xu + ~xv = − ~xu − ~xv e E2 E2 E E and

fE gE f g (~n∗)P (~xv) = −SP (~xv) = −(c~xu + d~xe v) = − ~xu + ~xv = − ~xu − ~xv. e E2 E2 E E

57 This implies that

e2E 2efF f 2G 1 (~n ) (~x ) · (~n ) (~x ) = + + = (e2 + f 2), ∗ P u ∗ P u E2 E2 E2 E

efE egF f 2F fgG f (~n ) (~x ) · (~n ) (~x ) = + + + = (e + g), ∗ P u ∗ P v E2 E2 E2 E2 E

f 2E 2fgF g2G 1 (~n ) (~x ) · (~n ) (~x ) = + + = (f 2 + g2). ∗ P v ∗ P v E2 E2 E2 E

In Section 2, we saw that the mean curvature H of a surface is given by

eG − 2fF + gE H = . 2(EG − F 2)

Therefore, the mean curvature of a surface with isothermal parametrization is given by eE + gE e + g H = = . 2E2 2E

If the surface is also minimal, then we have

e + g = H ≡ 0. 2E

This implies that e = −g when the surface is also minimal. Hence, with M being a minimal surface with isothermal parametrization ~x(u, v), we have

1 1 (~n ) (~x ) · (~n ) (~x ) = (e2 + f 2) = ((−g)2 + f 2) ∗ P u ∗ P u E E

1 = (g2 + f 2) = (~n ) (~x ) · (~n ) (~x ), E ∗ P v ∗ P v

f f (~n ) (~x ) · (~n ) (~x ) = (e + g) = (−g + g) = 0. ∗ P u ∗ P v E E

58 Now, since ~xu · ~xu = E = G = ~xv · ~xv and ~xu · ~xv = 0, then

1 (e2 + f 2) (~n ) (~x ) · (~n ) (~x ) = (e2 + f 2) = E = ω(u, v)2~x · ~x , ∗ P u ∗ P u E E2 u u

(e2 + f 2) (~n ) (~x ) · (~n ) (~x ) = 0 = · 0 = ω(u, v)2~x · ~x , ∗ P u ∗ P v E2 u v

1 (e2 + f 2) (~n ) (~x ) · (~n ) (~x ) = (e2 + f 2) = E = ω(u, v)2~x · ~x , ∗ P v ∗ P v E E2 v v where ω(u, v) = pe2 + f 2/E. Therefore, the Gauss map ~n of M is conformal with p scaling factor ω(u, v) = e2 + f 2/E. In Section 2, we saw that the Gaussian curvature of a surface can be written as

eg − f 2 K = . EG − F 2

In the case above, the Gaussian curvature of M can be written as

−e2 − f 2 K = . E2

Hence, the scaling factor above can be expressed in terms of the Gaussian curvature K of M, that is, ω(u, v) = pe2 + f 2/E = p|K|. The stereographic projection from the north pole of the unit 2-sphere S2 will be of use to us so we shall now deﬁne it. In Cartesian coordinates, the stereographic projection from the north pole is the mapping St : S2 \{(0, 0, 1)} → R2 deﬁned by x1 x2 St(x1, x2, x3) = , , 0 . (1 − x3) (1 − x3)

This mapping allows us to picture the unit 2-sphere S2 as a plane, in a sense. We note that the stereographic projection from the north pole is a bijective, conformal

59 mapping where it is deﬁned [5]. We may identify R2 with C and extend the map St to a bijective mapping St : S2 → C ∪ {∞} deﬁned by

x1 + ix2 (x1, x2, x3) 7→ and (0, 0, 1) 7→ ∞. (10.1) 1 − x3

We can consider the Gauss map ~n as a mapping from a surface M to C ∪ {∞} via St : S2 → C ∪ {∞}. The resulting map will still be conformal since ~n and St are both conformal. This brings us to the following theorem.

THEOREM 10.1: Let M be a regular minimal surface with isothermal parametrization ~x(u, v) and Weierstrass-Enneper representation (f, g). Then the Gauss map of

M, given by ~n : M → C ∪ {∞}, may be identiﬁed with the meromorphic function g deﬁned in (8.3).

Proof: (For convenience, we shall use superscripts instead of subscripts to denote all components.) Let φ = ∂~x/∂z, φ = ∂~x/∂z¯, and recall that φ = (φ1, φ2, φ3), where φ1 = f(1 − g2)/2, φ2 = if(1 + g2)/2, and φ3 = fg. We shall describe the Gauss map

1 2 3 in terms of φ , φ , and φ . Now, the cross product of ~xu and ~xv is given by

1 2 3 ~xu × ~xv = ((~xu × ~xv) , (~xu × ~xv) , (~xu × ~xv) )

2 3 3 2 3 1 1 3 1 2 2 1 = (xu xv − xu xv, xu xv − xu xv, xu xv − xu xv).

1 Consider the ﬁrst component (~xu × ~xv) . Note that

1 2 3 3 2 2 3 2 3 2 3 3 2 (~xu × ~xv) = xu xv − xu xv = Im [xu xu + xv xv + i(xu xv − xu xv)]

∂x2 ∂x3 = Im [(x2 − ix2)(x3 + ix3)] = Im 2 · 2 u v u v ∂z ∂z¯

= 4 Im (φ2φ 3).

60 Using this technique for the other two components, we get

2 3 1 3 1 2 (~xu × ~xv) = 4 Im (φ φ ) and (~xu × ~xv) = 4 Im (φ φ ).

Thus,

2 3 3 1 1 2 ~xu × ~xv = 4 Im (φ φ , φ φ , φ φ ).

Now, the cross product of φ and φ is given by

φ × φ = (φ2φ 3 − φ3φ 2, φ3φ 1 − φ1φ 3, φ1φ 2 − φ2φ 1)

= (φ2φ 3 − φ2φ 3, φ3φ 1 − φ3φ 1, φ1φ 2 − φ1φ 2)

= (2i Im (φ2φ 3), 2i Im (φ3φ 1), 2i Im (φ1φ 2))

= 2i Im(φ2φ 3, φ3φ 1, φ1φ 2).

Hence, 2 ~x × ~x = 4 Im (φ2φ 3, φ3φ 1, φ1φ 2) = −2i(φ × φ) = (φ × φ). u v i

Since ~x(u, v) is isothermal, then

√ √ 2 2 |~xu × ~xv| = EG − F = E = E.

2 2 2 2 2 In Section 9, we showed that |φ| = (λ )/2, where λ = |~xu| = E. Thus, E = 2|φ| .

2 This implies that |~xu × ~xv| = 2|φ| . Therefore, considering the Gauss map ~n initially as a mapping from M to S2, we have

~xu × ~xv 2(φ × φ) φ × φ ~n = = 2 = 2 |~xu × ~xv| 2i|φ| i|φ|

2i Im(φ2φ 3, φ3φ 1, φ1φ 2) = i|φ|2

61 2 Im(φ2φ 3, φ3φ 1, φ1φ 2) = . |φ|2

Combining this result with the mapping St : S2 → C ∪ {∞} given by (10.1), we may now deﬁne the Gauss map ~n as a mapping from M to C ∪ {∞} in the following way:

2 Im(φ2φ 3) + 2i Im(φ3φ 1) ~n = |φ|2[1 − 2|φ|−2 Im(φ1φ 2)]

2 Im(φ2φ 3) + 2i Im(φ3φ 1) = . |φ|2 − 2 Im(φ1φ 2)

Letting N denote the numerator of this fraction, we have

N = 2 Im(φ2φ 3) + 2i Im(φ3φ 1)

1 = [φ2φ 3 − φ3φ 2 + iφ3φ 1 − iφ1φ 3] i

= φ3(φ 1 + iφ 2) − φ 3(φ1 + iφ2).

Since ~x(u, v) is isothermal, then by Lemma 6.2, we have

0 ≡ (φ1)2 + (φ2)2 + (φ3)2 = (φ1 − iφ2)(φ1 + iφ2) + (φ3)2.

This implies that (φ3)2 φ1 + iφ2 = − . φ1 − iφ2

Making the substitution gives us

(φ3)2 N = φ3(φ 1 + iφ 2) + φ 3 φ1 − iφ2

φ3[(φ1 − iφ2)(φ 1 + iφ 2) + |φ3|2] = φ1 − iφ2

62 φ3 = [|φ1|2 + |φ2|2 + |φ3|2 + i(φ1φ 2 − φ2φ 1)] φ1 − iφ2

φ3 = [|φ|2 − 2 Im(φ1φ 2)]. φ1 − iφ2

Therefore, we have

N φ3 |φ|2 − 2 Im(φ1φ 2) ~n = = |φ|2 − 2 Im(φ1φ 2) φ1 − iφ2 |φ|2 − 2 Im(φ1φ 2)

φ3 = = g, φ1 − iφ2 where g was deﬁned in (8.3). Returning now to the notion of the Gauss map ~n as a mapping from a regular parametrized surface M to the unit sphere S2, we can show that ~n may be written in terms of the meromorphic function g when M is a regular minimal surface with isothermal parametrization ~x(u, v). To do this, we use the following expression for ~n that was obtained in the previous proof. (Note that we are reverting to the use of subscripts.) 2 Im(φ φ , φ φ , φ φ ) ~n = 2 3 3 1 1 2 . (10.2) |φ|2

It follows from (8.1) that

i 1 φ φ = f(1 + g2)(fg) = |f|2(ig + i|g|2g). 2 3 2 2

Hence,

1 1 Im(φ φ ) = Im |f|2(ig + i|g|2g) = |f|2(Re(g) + |g|2 Re(g)) 2 3 2 2

1 = |f|2 Re(g)(1 + |g|2). 2

63 Also, 1 1 φ φ = fg[f(1 − g2)] = |f|2(g − |g|2g). 3 1 2 2

Hence,

1 1 Im(φ φ ) = Im |f|2(g − |g|2g) = |f|2(Im(g) + |g|2 Im(g)) 3 1 2 2

1 = |f|2 Im(g)(1 + |g|2). 2

Also,

i i φ φ = − f(1 − g2)[f(1 + g2)] = − |f|2(1 + g 2 − g2 − |g|4) 1 2 4 4

1 = |f|2(i|g|4 − ig 2 + ig2 − i). 4

Hence,

1 1 Im(φ φ ) = Im |f|2(i|g|4 − ig 2 + ig2 − i) = |f|2(|g|4 − 1) 1 2 4 4

1 = |f|2(|g|2 − 1)(|g|2 + 1). 4

Therefore, the numerator of ~n is given by

1 2 Im(φ φ , φ φ , φ φ ) = |f|2(1 + |g|2) (2 Re(g), 2 Im(g), |g|2 − 1). 2 3 3 1 1 2 2

Using (8.1), we can determine the denominator of ~n in terms of f and g.

2 2 2 2 |φ| = |φ1| + |φ2| + |φ3|

2 2 1 2 i 2 2 = f(1 − g ) + f(1 + g ) + |fg| 2 2

64 1 1 = |f|2 |1 − g2|2 + |1 + g2|2 + |g|2 4 4

1 = |f|2(|1 − g2|2 + |1 + g2|2 + 4|g|2) 4

1 = |f|2[(1 − g2)(1 − g2) + (1 + g2)(1 + g2) + 4|g|2] 4

1 = |f|2[1 − (g2 + g2) + |g|4 + 1 + (g2 + g2) + |g|4 + 4|g|2] 4

1 1 = |f|2(2 + 4|g|2 + 2|g|4) = |f|2(1 + 2|g|2 + |g|4) 4 2

1 = |f|2(1 + |g|2)2. 2

We conclude that

2 Im(φ φ , φ φ , φ φ ) 2 Re(g) 2 Im(g) |g|2 − 1 ~n = 2 3 3 1 1 2 = , , . (10.3) |φ|2 |g|2 + 1 |g|2 + 1 |g|2 + 1

We note that E = G = λ2 = 2|φ|2 = |f|2(1 + |g|2)2. (10.4)

We shall now explore several more important results involving the Gauss map of

a minimal surface in E3.

LEMMA 10.1: Let M be a regular minimal surface in E3 with parametrization ~x(z): D → M, where D is the entire complex plane. Then M either lies on a plane or has an image under the Gauss map ~n that omits at most two points.

Proof: Let g be the function deﬁned in (8.3) which will correspond to M. We note

that the only case in which g is not deﬁned is when φ1 ≡ iφ2 and φ3 ≡ 0. In this case,

since 0 ≡ φ3 = ∂x3/∂z, then x3 is constant so that M lies on a plane. If it is not the

65 case that φ1 ≡ iφ2 and φ3 ≡ 0, then g is meromorphic on the entire complex plane. By Picard’s theorem, g either takes on all values with at most two exceptions, or else g is constant [6]. Therefore, we deduce from (10.3) that the Gauss map ~n of M either takes on all values except for perhaps two or else is constant. If ~n is constant, then

M must lie on a plane. In order to proceed, we will need to introduce a few deﬁnitions from Osserman [6]. For the sake of brevity, we will assume that the reader is familiar with the deﬁnition of a smooth 2-manifold.A Riemannian 2-manifold is a real smooth 2-manifold Σ endowed with an inner product h· , ·iP on the tangent space TP Σ at each point P ∈ Σ that varies smoothly from point to point. A Riemannian metric on Σ is the family

of inner products h· , ·iP . It is a fact from the theory of 2-manifolds that each 2-manifold Σ has a universal covering surface which consists of a simply-connected 2-manifold Σb and a map π : Σb → Σ, having the property that each point of Σ has a neighborhood U such that the restriction of π to each component of π−1(U) is a homeomorphism onto U [6]. We note that the map π is a local homeomorphism, and because of this, any structure

on Σ, such as a Riemannian metric, induces a corresponding structure on Σb [6]. A divergent path (or divergent curve) on a Riemannian 2-manifold Σ is a continuous map ~α : [0, ∞) → Σ with the property that for every compact subset Q of Σ, there exists a t0 such that ~α(t) ∈/ Q for t > t0.

Suppose a divergent path ~α : [0, ∞) → Σ is diﬀerentiable. For any t1 > 0, let k · k

be the norm induced by the inner product h· , ·i~α(t1) on the tangent space T~α(t1)Σ, so

0 0 that the length of the tangent vector ~α (t1) ∈ T~α(t1)Σ is given by k~α (t1)k. By taking the integral of these lengths, we can deﬁne the length of the divergent path ~α on Σ as follows: Z ∞ k~α 0(t)k dt. (10.5) 0

66 A Riemannian 2-manifold Σ is said to be complete with respect to a given Rie- mannian metric if the integral (10.5) diverges for every diﬀerentiable divergent path on Σ. It can be shown that Σ is complete with respect to a given Riemannian metric if and only if its universal covering surface Σb is complete with respect to the induced Riemannian metric [6]. Now, it is a fact that there is a Riemannian 2-manifold that corresponds to each regular surface in En [6]. Suppose M is a regular minimal surface in En deﬁned by a map ~x :Σ → M, where Σ is the Riemannian 2-manifold that corresponds to M.

Let Σb and π : Σb → Σ constitute the universal covering surface of Σ. Then there is an associated simply-connected regular minimal surface Mc in En deﬁned by the composite map ~x ◦ π : Σb → Mc, where Mc is called the universal covering surface of M [6]. We say that M is complete if its corresponding Riemannian 2-manifold Σ is complete with respect to the appropriate Riemannian metric. It is the case that Mc is complete if and only if M is complete [6]. Next, we will need the following lemma.

LEMMA 10.2: Let f(z) be a holomorphic function on the complex unit disk D which has at most a ﬁnite number of zeros. Then there exists a divergent path C on D such that Z |f(z)| |dz| < ∞. (10.6) C

Proof: We suppose ﬁrst that f(z) 6= 0 on D. We deﬁne

Z z w = F (z) = f(ζ) dζ. 0

Then F (z) maps |z| < 1 onto a complex manifold of complex dimension one, known as a Riemann surface, which has no branch points [6]. Letting z = G(w) be that

67 branch of the inverse function satisfying G(0) = 0, then because |G(w)| < 1, there is a largest disk |w| < R < ∞ in which G(w) is deﬁned [6]. With this being the case,

there must exist a point w0 with |w0| = R such that G(w) cannot be extended to a

neighborhood of w0. Deﬁne L to be the line segment given by w = tw0, 0 ≤ t < 1, and let C denote the image of L under G(w). It follows that C must be a divergent path

because if it were not, there would be a sequence tn → 1 such that the corresponding

sequence of points zn on C would converge to a point z0 ∈ D. However, this would

0 imply that F (z0) = w0, and since F (z0) = f(z0) 6= 0, the function G(w) could be

extended to a neighborhood of w0, which would be a contradiction. Therefore, the path C is divergent, and we have

Z Z 1 Z 1 dz dw |f(z)| |dz| = |f(z)| dt = dt = R < ∞. C 0 dt 0 dt

Hence, when f(z) has no zeros, (10.6) holds. Now, suppose f(z) has a ﬁnite number

of zeros of order σk at the points zk. Then the function given by

σk Y 1 − z¯kz f1(z) = f(z) z − zk never vanishes, and so by the above result, there exists a divergent path C such that

Z |f1(z)| |dz| < ∞. C

Since |f(z)| < |f1(z)| throughout D, then we deduce that (10.6) must hold. We will now investigate three signiﬁcant theorems regarding the Gauss map of a complete regular minimal surface in E3.

THEOREM 10.2: Let M be a complete regular minimal surface in E3. Then either M is a plane or else the image of M under the Gauss map ~n is everywhere dense in

68 the unit sphere.

Proof: Suppose that the image of M under the Gauss map ~n is not everywhere dense in the unit sphere. Then there must exist an open set on the unit sphere which is not intersected by the image of M under the Gauss map. By a rotation, we may assume that the point (0, 0, 1) is in this open set [6]. Then the Gauss map ~n = (n1, n2, n3) satisﬁes n3 ≤ < 1. This also holds true for the universal covering surface Mc of

M. Note that since Mc is a simply-connected regular minimal surface in E3, then by Theorem 8.1, it may be represented in the form ~x : D → E3 of (8.6), where D is either the entire complex z-plane or else the complex unit disk. Note that by (10.3), n3 ≤ < 1 if and only if |g(z)| ≤ δ < ∞. Also, since Mc is regular, then by Theorem 8.1, f cannot vanish. Now, since F = 0 and E = G = |f|2(1 + |g|2)2 by (10.4), then we know from Section 2 that the length of any path C on Mc, with u = u(t), v = v(t), u0 = u0(t), and v0 = v0(t), would be

Z [E(u, v)(u0)2 + 2F (u, v)u0v0 + G(u, v)(v0)2 ]1/2 dt C

1 " !# 2 Z du2 dv 2 = 2 |f|2(1 + |g|2)2 + dt C dt dt

Z √ 2 dz = 2 |f|(1 + |g| ) dt C dt

√ Z = 2 |f|(1 + |g|2) |dz| C

√ Z ≤ 2 (1 + δ2) |f| |dz|. C

Thus, if D were the complex unit disk, then by Lemma 10.2, there would exist a divergent path C for which this integral converges, and so the surface Mc would not

69 be complete. Since M was assumed to be complete, then Mc must be complete, so we conclude that D is the entire complex plane. Since the Gauss map of Mc omits more than two points, then Mc must lie on a plane by Lemma 10.1. This implies that M must also lie on a plane. Therefore, with M being complete, we deduce that M must be the whole plane. We note that an immediate consequence of Theorem 10.2 is Bernstein’s Theorem.

3 This is because a non-parametric minimal surface in E deﬁned over the whole x1, x2- plane is a complete regular surface whose image under the Gauss map is contained in a hemisphere [6]. Thus, it must be a plane.

THEOREM 10.3: Let W be an arbitrary set of k points on the unit sphere, where k ≤ 4. Then there exists a complete regular minimal surface in E3 whose image under the Gauss map omits precisely the set W .

Proof: By a rotation, we may assume that the set W contains the point (0, 0, 1). Suppose (0, 0, 1) is the only point that W contains. We recall from Section 8 that the minimal surface referred to as the Enneper surface has the representation given by f(z) = 1, g(z) = z on C. It follows from (8.1) and Lemma 6.2 that the Enneper surface is regular, and it is easy to show that the Enneper surface is complete using the fact from the proof of Theorem 10.2 that the length of any path C on the surface is given by √ Z 2 |f|(1 + |g|2) |dz|. (10.7) C

Since g takes on all values of C, and since g = St◦~n by Theorem 10.1, then we deduce that ~n takes on all values of S2 \{(0, 0, 1)}. Since g(z) 6= ∞ for any z ∈ C, then we conclude that the Gauss map ~n of the Enneper surface omits precisely the set W . Now, suppose (0, 0, 1) is not the only point that W contains. Let the other points

of W correspond to the complex points wj, j = 1, ..., k − 1, under the stereographic

70 projection from the north pole given by (10.1). Now, let

1 f(z) = , g(z) = z, Qk−1 j=1 (z − wj) and use the Weierstrass-Enneper representation I on the domain C \{w1, ..., wk−1} to form a minimal surface. Since g takes on all values of C except for the points wj, and since g(z) 6= ∞ for any z ∈ C, then the same argument from above can be used to conclude that the Gauss map of the surface omits precisely the set W . To show that the surface is complete, we note that the length of any path C on the surface is given by (10.7). Now, any divergent path must tend either to ∞ or to one of the points wj. If C is a divergent path that tends to ∞, then, with the degree of |f|(1 + |g|2) being at least −1 because there are at most three terms in the denominator of f, we have

√ Z 2 |f|(1 + |g|2) |dz| = ∞. C

If C is a divergent path that tends to one of the points wj, then |g| goes to a constant and |f| becomes unbounded. Hence,

√ Z 2 |f|(1 + |g|2) |dz| = ∞. C

Thus, every divergent curve on the surface has unbounded length, so we conclude that the surface is complete. We note that the integrals given in Weierstrass-Enneper representation I may not be single-valued, but by passing to the universal covering surface, we get a single-valued map deﬁning a regular surface having the same

properties [6]. The American mathematician Robert Osserman proved in 1961 that the Gauss

map of a complete non-ﬂat minimal surface in E3 cannot omit a set of positive loga-

71 rithmic capacity [1]. It was later proved by Frederico Xavier that the Gauss map of a complete non-flat minimal surface in E3 can omit at most six points of the sphere [1]. This result by Xavier was an improvement of Osserman’s earlier result. Hirotaka Fujimoto further improved upon this by proving that the Gauss map of a complete non-flat minimal surface in E3 can omit at most four points of the sphere [1]. Our final mission of this project will be to prove that the Gauss map of a complete non-flat regular minimal surface in E3 can omit at most four points of the sphere. We shall follow the approach of Kawakami [4]. (Note that the remaining subject matter requires a knowledge of certain elements of manifold theory.)

To begin, we let z1 and z2 be two distinct points in C ∪ {∞} and we deﬁne

|z1 − z2| |z1, z2| = p 2p 2 1 + |z1| 1 + |z2|

if z1 6= ∞ and z2 6= ∞, and

1 |z1, ∞| = |∞, z1| = . p 2 1 + |z1|

Next, we will need two lemmas that are essential for our proof. The ﬁrst lemma given here shall be stated without proof. The proof of the second lemma given here is drawn from Fujimoto [2].

LEMMA 10.3: Let g be a non-constant meromorphic function on a disk

∆(R) = {z ∈ C : |z| < R}, 0 < R ≤ ∞, which omits k distinct points α1, ..., αk. If k > 2, then for each positive with < (k − 2)/k, there exists a positive constant B, depending on k and L = mini

|g0| R ≤ B , 2 Qk 1− R2 − |z|2 (1 + |g| ) j=1 |g, αj|

72 where g0 = dg/dz [4].

LEMMA 10.4: Let dσ2 be a conformal ﬂat metric on an open Riemann surface Σ. Then, for each point p ∈ Σ, there exists a local diﬀeomorphism Φ of a disk

∆(R) = {w ∈ C : |w| < R}, 0 < R ≤ ∞, onto an open neighborhood of p with Φ(0) = p such that Φ is a local isometry, that is, the pull-back Φ∗dσ2 is equal to

2 the standard Euclidean metric dsEuc on ∆(R) and, for a point a0 with |a0| = 1, the

Φ-image Γa0 of the curve Ca0 = {w = a0t : 0 ≤ t < R} is divergent in Σ.

Proof: Let Σb and π : Σb → Σ constitute the universal covering surface of Σ, and let dσd2 = π∗dσ2 represent the pull-back of the original dσ2, so that dσd2 is a conformal

ﬂat metric on the simply-connected open Riemann surface Σ.b Thus, instead of Σ and dσ2, we may consider Σb and dσd2, respectively [2]. Since Σb is simply-connected, then it follows from Koebe’s theorem that Σb is biholomorphic with either the entire complex z-plane or else the complex unit disk ∆(1), and so Σb has a global coordinate z [2]. Let

dσd2 = λ2|dz|2, (10.8)

where λ is a positive C∞ function, and let v = log λ. Since dσd2 is ﬂat, then v is

a harmonic function [2]. Because Σb is simply-connected, we can take a harmonic function v∗ on Σb such that v + iv∗ is holomorphic [2]. Let ψ = exp(v + iv∗), so that ψ is a holomorphic function that satisﬁes the condition |ψ| = λ [2]. Then we deﬁne

Z z w = F (z) = ψ(ζ) dζ, (10.9) z0

where F is a holomorphic function, and the integral is being taken along an arbitrarily

chosen piecewise smooth path joining a point zo with z in Σ.b We note that F (z0) = 0

and dF (z0) 6= 0. Thus, F maps an open neighborhood U of z0 biholomorphically

73 onto an open disk ∆(R) = {w ∈ C : |w| < R}, 0 < R ≤ ∞ [2]. Let R0 denote the least upper bound of R > 0 such that F biholomorphically maps some open

neighborhood of z0 onto ∆(R). By deﬁnition, there exists a sequence Rn → R0

such that F |Un : Un → ∆(Rn) is biholomorphic for open neighborhoods Un of z0 [2].

−1 Letting U0 = ∪nUn, then it follows that F maps U0 onto ∆(R0). Let Φ = (F |U0 ) be the inverse map of F |U0 . When R = ∞, then Σb = U0 since an open Riemann surface which includes an open set that is biholomorphic with C is itself biholomorphic with

C [2]. Hence, when R = ∞, then every curve Γa0 is divergent in Σ.b Now, assume

R < ∞ and suppose the curve Γa0 is not divergent in Σb for some a0 with |a0| = 1.

Then there is some sequence {tn}, where 0 ≤ tn < R0, such that tn → R0 and Φ(tna0) converges to a point p ∈ Σ.b Let w0 = R0a0. Because dF (p) = ψ(p) 6= 0, then F biholomorphically maps some open neighborhood of p onto an open neighborhood of w0 [2]. Thus, Φ is holomorphically extended to a neighborhood of w0 [2]. If there is

no curve Γa0 which diverges in Σ,b then Φ has a holomorphic extension to ∆(R) for some R > R0 and the map F : Φ(∆(R)) → ∆(R) is biholomorphic [2]. However,

this contradicts the deﬁnition of R0. We conclude that some Γa0 diverges in Σ.b In addition, it follows from (10.8), (10.9), and the deﬁnition of v that

2 ∗ 2 dz 2 2v ◦Φ 1 2 2 2 Φ dσd2 = (λ ◦ Φ) |dw| = e |dw| = |dw| = ds , dw |ψ|2 Euc

and therefore the lemma is proved.

THEOREM 10.4: The Gauss map of a complete non-ﬂat regular minimal surface

in E3 can omit at most four points of the sphere.

Proof: Let M be a complete non-ﬂat regular minimal surface in E3. Suppose that the Gauss map of M omits k ≥ 5 distinct points of the unit 2-sphere S2. Then the Gauss

74 map ~n of the universal covering surface Mc of M also omits k ≥ 5 distinct points of S2. We note that Mc is a complete non-ﬂat simply-connected regular minimal

surface in E3. Since Mc is a simply-connected regular minimal surface in E3, then by Theorem 8.1, it may be represented in the form ~x : D → E3 of (8.6), where D is either the entire complex z-plane or else the complex unit disk ∆(1). Let (f, g)

be the corresponding Weierstrass-Enneper representation of Mc. Consider the Gauss

map of Mc as a mapping from Mc to C ∪ {∞} via the mapping St : S2 → C ∪ {∞} as described earlier. Then the Gauss map of Mc omits k ≥ 5 distinct points of C ∪ {∞}. By Theorem 10.1, the Gauss map of Mc, given by ~n : Mc → C∪{∞}, can be identified with the meromorphic function g. Thus, g omits k ≥ 5 distinct points of C ∪ {∞}. Note that g is defined on D. Since Mc is non-flat, then the Gauss map ~n of Mc must be non-constant. This implies that g is non-constant. Now, Picard’s theorem tells us that any meromorphic function on C which omits three or more distinct values is a constant [1]. We deduce that D must be the complex unit disk ∆(1). We note that ∆(1) is an open Riemann surface [1]. We also note that by (10.4) and [1], the conformal metric on D induced from E3 is given by

ds2 = λ2|dz|2 = 2|φ|2|dz|2 = |f|2(1 + |g|2)2|dz|2.

For each p ∈ D, we shall let d(p) denote the geodesic distance from p to the boundary of D, that is, the inﬁmum of the lengths of the divergent curves in D emanating from

p [4]. Since Mc is complete, so that the metric ds2 is complete, then it follows that

d(p) = ∞ for all p ∈ D. Now, let α1, ..., αk be the distinct points omitted by g. By

using an appropriate M¨obiustransformation, we may assume that αk = ∞ [4]. Take

75 a positive number such that

k − 6 k − 4 < < . k k

1 Let δ = 2/(k − 2 − k). Because k ≥ 5, then it is the case that 2 < δ < 1. We shall now deﬁne a new conformal ﬂat metric by

2δ/(1−δ)  k−1 !1− 2 2/(1−δ) 1 Y |g − αj| 2 dσ = |f|   |dz| (10.10) |g0| p 2 j=1 1 + |αj| on the set D0 = {p ∈ D : g0(p) 6= 0}, where g0 = dg/dz with respect to the local complex coordinate z [4]. We note that D0 is also an open Riemann surface [4]. Suppose p ∈ D0. Now, because dσ2 is a conformal ﬂat metric on an open Riemann surface, then Lemma 10.4 tells us that there exists a local diﬀeomorphism Φ of a

disk ∆(R) = {z ∈ C : |z| < R}, 0 < R ≤ ∞, onto an open neighborhood of p with Φ(0) = p such that Φ is a local isometry, that is, the pull-back Φ∗dσ2 is equal to

2 the standard Euclidean metric dsEuc on ∆(R) and, for a point a0 with |a0| = 1, the

0 Φ-image Γa0 of the curve Ca0 = {w = a0t : 0 ≤ t < R} is divergent in D . Deﬁne h = g ◦ Φ on ∆(R), so that h is a non-constant meromorphic function on ∆(R) which

omits the points α1, ..., αk. Then by Lemma 10.3, we have

|h0(0)| BR B ≤ = . 2 Qk 1− R2 R (1 + |h(0)| ) j=1 |h(0), αj|

This implies that

k (1 + |h(0)|2) Y R ≤ B |h(0), α |1− < ∞. (10.11) |h0(0)| j j=1

76 2 Thus, the length of Γa0 with respect to the metric dσ is as follows:

Z dσ = R < ∞.

Γa0

0 Now, suppose that the Φ-image Γa0 converges to a point p0 ∈ D \ D as t → R. Take

0 a local complex coordinate ζ = h in a neighborhood of p0 with ζ(p0) = 0 so that we have dσ2 = |ζ|−2δ/(1−δ)ν|dζ|2 for some positive smooth function ν [4]. However, because δ/(1 − δ) > 1, we have

Z Z |dζ| R = dσ ≥ η = ∞ |ζ|δ/(1−δ) Γa0 Γa0

for some positive constant η [4]. Since this contradicts (10.11), we conclude that Γa0 diverges outside any compact subset of D as t → R. Now, because

∗ 2 2 2 Φ dσ = dsEuc = |dz| , we deduce from (10.10) that

 1−δ k−1 p 2 ! Y 1 + |αj| |f ◦ Φ| = |h0| .  |h − α |  j=1 j

Using Lemma 10.3, we have

Φ∗ds = |f ◦ Φ| (1 + |h|2) |dz|

 1−δ k−1 p 2 ! Y 1 + |αj| = |h0| (1 + |h|2)1/δ |dz|  |h − α |  j=1 j

 1−δ k−1 p 2 ! Y 1 + |αj| = |h0| (1 + |h|2)(k−2−k)/2 |dz|  |h − α |  j=1 j

77 !δ |h0| = |dz| 2 Qk 1− (1 + |h| ) j=1 |h, αj|

R δ ≤ B δ |dz|. R2 − |z|2 Hence, we have

Z Z Z R δ R 1−δ d(p) ≤ ds = Φ∗ds ≤ B δ |dz| ≤ B δ < ∞ R2 − |z|2 1 − δ Γa0 Ca0 Ca0 since 0 < δ < 1 [4]. However, this contradicts the fact that d(p) = ∞ for all p ∈ D as noted earlier. We conclude that the Gauss map of M cannot omit k ≥ 5 distinct points of the unit 2-sphere S2. Therefore, the Gauss map of a complete non-ﬂat

3 regular minimal surface in E can omit at most four points of the sphere. In the proof of Theorem 10.3, we constructed a complete regular minimal surface in E3 whose Gauss map omitted an arbitrary set of four points of the unit 2-sphere. We note that the surface was also non-ﬂat since the function g of its Weierstrass- Enneper representation was non-constant. Thus, Theorem 10.4 is optimal.

11 Conclusion

Arising originally from the problem of determining surfaces that minimize area, the subject of minimal surfaces has broadened over the years into a rich ﬁeld of study with exciting results that are often unrelated to notions of area, as we have seen. Throughout this project, we have explored quite extensively a number of fundamental and profound results pertaining to minimal surfaces. After introducing the concept of a minimal surface from a mathematical perspective, we then examined several classical examples of minimal surfaces, explained how minimal surfaces relate

78 to the minimization of area, derived several forms of the minimal-surface equation, defined isothermal parameters, expounded Bernstein’s Theorem, demonstrated a link between differential geometry and complex analysis via the Weierstrass-Enneper representations of a minimal surface in E3, illustrated how the Gaussian curvature of a minimal surface in E3 can be expressed in terms of its Weierstrass-Enneper representations, and concluded with a detailed inspection of some fascinating discoveries involving the Gauss map of a minimal surface in E3. The field of minimal surfaces has made the usual mathematical journey from a theoretical question to an indispensable tool used in a variety of research and applications, and it was the goal of this project to elucidate much of the early part of that journey in a comprehensive manner.

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