A Study of Particular Coordinate Systems

A STUDY OF PARTICULAR COORDINATE SYSTEMS

A Thesis by

Jennifer D. Ragan

Bachelor of Science, Elon University, 2005

Submitted to the Department of Mathematics and Statistics and the faculty of the Graduate School of Wichita State University in partial fulﬁllment of the requirements for the degree of Masters of Science

The following faculty members have examined the ﬁnal copy of this thesis for form and content, and recommend that it be accepted in partial fulﬁllment of the requirement for the degree of Masters of Science with a major in Applied Mathematics.

Kirk Lancaster, Committee Chair

Thalia Jeﬀres, Committee Member

Elizabeth Behrman, Committee Member

iii DEDICATION

To my husband, parents, sister, and my wonderful family and friends

iv ACKNOWLEDGEMENTS

I want to thank my thesis advisor Dr. Kirk Lancaster for his time, encouragement, and patience. Without Dr. Lancaster’s support, enthusiasm and guidance this thesis would not of been possible. I want to thank Dr. Jeﬀres for her time and contributions that she provided as I was revising. I am grateful to Dr. Behrman for all her support and ﬂexibility.

I want to give a special thank you to Dr. Christopher Earles, Dr. Zhiren Jin and Dr. Diana Palenz for the time that they kindly gave to this thesis. Lastly, a big thank you to all my family and friends for their constant support and love.

v ABSTRACT

This thesis is a study of the properties and relationships between isothermal, harmonic and characteristic coordinate systems. The proof of the existence of isothermal and characteristic coordinates on a manifold which is a graph is given using the Uniformization theorem. Equations of prescribed mean curvature are discussed and the relationship between equations of minimal surface type and mean curvature type are shown. It is also proven that the map from a domain parameterized by characteristic coordinates to the domain parameterized by isothermal coordinates is quasiconformal.

vi PREFACE

Let Ω ⊂ IR2 be a connected, open set and consider the prescribed mean curvature

problem in a cylinder, which consists of ﬁnding a solution f of the equation

Nf = H(·, f(·)) in Ω, (1)

which satisﬁes one of the following boundary conditions:

(i) f = φ (a.e.) on ∂Ω,

(ii) T f · ν = cos γ (a.e.) on ∂Ω;

here T f = √ ∇f , Nf = ∇ · T f, ν is the exterior unit normal on ∂Ω,H(x, t) is a weakly 1+|∇f|2 increasing function of t for each x ∈ Ω, φ : ∂Ω → IR and γ : ∂Ω → [0, π].

Starting with [23], Lancaster and his coauthors used isothermal parameterizations to investigate the existence and behavior of the radial limits of bounded solutions of (1) which satisfy the Dirichlet boundary condition (i). In [25] and [24], Lancaster and Siegel used isothermal parameterizations to investigate the existence and behavior of the radial limits of bounded solutions of (1) which satisfy the contact angle boundary condition (ii) (e.g. capillary surfaces.)

The goal of this thesis is to begin to create the infrastructure required to further investigate the behavior at corners of solutions of boundary value problems for equations of mean curvature type. The focus here is on the role of special types of local (and global) coordinates. Chapter 1 will consist of deﬁnitions and concepts from diﬀerential geometry.

In chapter 2, we introduce isothermal coordinates followed by discussion of harmonic coordinates and then consider the relationships between isothermal and harmonic coordinates. In chapter 3, we will focus on characteristic coordinates. Chapter 4 will consist of looking at the structure conditions of mean curvature type equations. In chapter 5, we prove the

vii global existence of isothermal and characteristic coordinates on a surface utilizing the Uni- formization theorem. We will then prove that the map between a domain parameterized by characteristic coordinates and a domain parametrized by isothermal coordinates is quasiconformal. Chapter 6 discusses the applications of this thesis to Lancaster’s work and provides a theorem and conjecture for further study.

viii TABLE OF CONTENTS

Chapter Page

1 DIFFERENTIAL GEOMETRY AND RIEMANNIAN MANIFOLDS ...... 1

1.1 Notation ...... 1 1.2 Riemannian Manifolds ...... 1 1.3 Diﬀerential Operators ...... 4 1.4 Beltrami Equation ...... 6

2 ISOTHERMAL AND HARMONIC COORDINATES ...... 9

2.1 Isothermal Coordinates ...... 9 2.2 Harmonic Coordinates ...... 10 2.3 Relationships between Isothermal and Harmonic Coordinates ...... 13

3 CHARACTERISTIC COORDINATES ...... 16

3.1 Quasilinear Operators ...... 16 3.2 Normal Form ...... 18 3.3 Coordinate Changes ...... 22 3.4 An Example ...... 26

4 STRUCTURE CONDITIONS FOR MEAN CURVATURE TYPE ...... 29

4.1 Prescribed Mean Curvature ...... 29 4.2 Structure Conditions ...... 30

5 CHARACTERISTIC COORDINATES ARE QUASICONFORMAL FUNCTIONS OF ISOTHERMAL COORDINATES ...... 35

5.1 Structure Class ...... 35 5.2 Uniformization Theorem ...... 36 5.3 Global Existence of Characteristic Coordinates ...... 36 5.4 Global Existence of Isothermal Coordinates ...... 40 5.5 Quasiconformal Mappings ...... 42

6 APPLICATIONS ...... 50

7 CONCLUSION ...... 53

REFERENCES ...... 55

APPENDIX ...... 58

A ...... 59

ix TABLE OF CONTENTS (continued)

Chapter Page

A.1 Connection and Covariant Derivative ...... 59 A.2 Mean Curvature ...... 60

x CHAPTER 1

DIFFERENTIAL GEOMETRY AND RIEMANNIAN MANIFOLDS

1.1 Notation

Here we provide a list of notation used throughout this thesis.

• x : U ⊂ IRn → M is a parametrization of the manifold M

• (u, v) represent isothermal coordinates

• (σ, ρ) represent characteristic coordinates

• ∆g represents the Laplace-Beltrami Operator for a manifold M with metric g.

• ∆o is the n-dimensional Laplace Operator in n-dimension Euclidean space

• If g is a Riemannian metric deﬁned as a family of inner products, we will use hw1, w2ip

to represent the inner product of w1, w2 on TpM at p.

3 • If w1, w2 ∈ TpM ⊂ R , then hw1, w2io will be used to represent the inner product of

3 w1, w2 as vectors in R .

1.2 Riemannian Manifolds

We will begin with a discussion on diﬀerential geometry mostly using [6] and [13] as references. In n-dimensions, we deﬁne a manifold of class Ck to be a set M and a family

n n of injective mappings x : Uα ⊂ IR → M of open sets Uα of IR into M such that

S (a) α xα(Uα) = M

−1 −1 (b) for any pair α and β with xα(Uα) ∩ xβ(Uβ) = W 6= ∅, the sets xα (W ) and xβ (W )

n −1 k are open sets in IR and the mappings xβ ◦ xα are C .

1 The mapping xα with p ∈ xα(Uα) is called a parameterization of M at p and xα(Uα) is called a coordinate neighborhood or coordinate system at p.

Let the smooth mapping γ :(−, ) → M be a curve with γ(0) = p with p ∈ M. For

∂ −1 a parameterization x at p, the tangent vector to the curve is w = ∂t (x ◦ γ)(0). The set

of all tangent vectors at p is called the tangent space, denoted TpM.

∂x ∂x The choice of parameterization determines a basis { , ..., } of TpM. In two ∂u1 ∂un ∂x ∂x dimensions we find it convenient to write xu = ∂u and xv = ∂v to represent the basis. If w is a tangent vector at p and f a Ck function defined near p, then differentiating f along any curve gives the directional derivative along w to be dfp(w): TpM → IR where

d df (w) = (f ◦ γ)(t) p dt

∂ −1 if w = ∂t (x ◦ γ)(0). Note that this is independent of the chosen path γ. Returning to the case of n-dimensions, consider a parameterization x : U → M and

the mapping ai : U → IR. A vector ﬁeld X on M is a relation that associates to each point

p ∈ M a vector w(p) ∈ TpM. We write

n X ∂ X(p) = a (p) i ∂x i=1 i where { ∂ , ..., ∂ } is a local basis for the tangent space at p. ∂x1 ∂xn A Ck manifold M equipped with a Riemannian metric g is called a Riemannian

manifold, which we denote (M, g). The metric g : TpM × TpM → IR is a family of inner

k products h, ip on the tangent space which vary C smoothly from point to point. In other

k n words < w, v >p(t) is a C function for all w, v ∈ IR where {p(t) : 0 ≤ t ≤ 1} is a smooth curve.

Now consider a 2-dimensional manifold M ⊂ IR3 (i.e. a surface). The natural inner

3 3 product of IR induces on each tangent plane TpM an inner product; if w1, w2 ∈ TpM ⊂ IR ,

3 then hw1, w2io is equal to the inner product of w1 and w2 as vectors in IR ([12]). The 3 ﬁrst fundamental form of a surface in IR , expressed in the basis {xu, xv} associated to a

2 parameterization x(u, v) at p, is

2 2 g11du + 2g12dudv + g22dv (1.1) where

g11 = hxu, xuio , g12 = g21 = hxu, xvio , g22 = hxv, xvio . (1.2)

The positive deﬁnite, symmetric matrix   EF [gij] =   FG provides alternate notation for (1.1) given by Gauss.

We can ﬁnd the induced metric for a manifold immersed in R3. As an example, consider a graph M = {(x, y, f(x, y): x, y ∈ U} ⊂ IR3. M is a manifold of class Ck if and

k 3 only if f ∈ C and we can ﬁnd the Riemannian metric [gij] induced by IR . Consider a

2 3 parameterization x : IR(u,v) → IR(x,y,z) such that x(u, v) = (x(u, v), y(u, v), z(u, v)). Set

x(u, v) = u, y(u, v) = v, z = f(u, v). (1.3)

∂ ∂ We ﬁnd dx ∂u and dx ∂v to be:

∂ ∂x ∂ ∂y ∂ ∂z ∂ dx = + + ∂u ∂u ∂x ∂u ∂y ∂u ∂z ∂ ∂ ∂ = 1 + 0 + f ∂x ∂y u ∂z

∂ ∂x ∂ ∂y ∂ ∂z ∂ dx = + + ∂v ∂v ∂x ∂v ∂y ∂v ∂z ∂ ∂ ∂ = 0 + 1 + f . ∂x ∂y v ∂z.

3 Now ﬁnd each component gij by taking the inner product of the basis vectors ∂ ∂ 2 g11 = dx , dx = 1 + fu ∂u ∂u o ∂ ∂ g12 = g21 = dx , dx = fufv ∂u ∂v o ∂ ∂ 2 g22 = dx , dx = 1 + fv . ∂v ∂v o

Thus on the graph M, the local representation of the metric induced by IR3 with respect

to the given parametrization (1.3) is   2 1 + fu fufv [gij] =   . (1.4) 2 fufv 1 + fv

ij The inverse of [gij] is written [g ] and equals   1 + f 2 −f f ij 1 v u v g =   ; (1.5) 1 + f 2 + f 2 2 u v −fufv 1 + fu

i ij i then δj = [g ][gij] where δj is the Kronecker delta deﬁned by ( 1 if i = j δi = (1.6) j 0 if i 6= j.

1.3 Diﬀerential Operators

We will now provide some deﬁnitions for diﬀerential operators on a manifold M. The

motivation of this thesis is from Lancaster’s work where he focused the local behavior at a corner. To expand on his results, one wishes to consider local coordinates at the corner and

therefore we will provide the needed deﬁnitions in local coordinates. Global deﬁnitions have been given in appendix A.

We start by considering a particular connection that is compatible with the metric,

called the Levi-Civita Connection, but for a more general deﬁnition of connection refer to

appendix A. One interpretation of the connection ∇V W is that it allows us to define the directional derivative of a vector field W in the direction of another vector field V .

4 From [13], given a Riemannian manifold M there exists a unique aﬃne connection ∇

called the Levi-Civita connection on M satisfying the conditions:

X(hY,Zi) = h∇X Y,Zi + hY, ∇X Zi and ∇X Y − ∇Y X = [X,Y ]. (1.7)

n Consider a parameterization x : U ⊂ IR → M and a curve γ(t) = (x1(t), ..., xn(t)). Consider the vector ﬁeld V deﬁned by

n X ∂ V (t) = v (t) | i ∂x γ(t) i=1 i

dxi where vi = dt (t). Then the covariant derivative of a tangent vector ﬁeld V on M along a curve γ can be written in local coordinates as follows

n " n # DV X dvi ∂ X 0 ∂ = + vi(t)xi(t)∇ ∂ . (1.8) dt dt ∂x ∂xj ∂x i=1 i j=1 i

In appendix A, we provide a invariant deﬁnition of the covariant derivative and from that

k we derive (1.8). We deﬁne the Christoﬀel symbols Γij to be

n ∂ X k ∂ ∇ ∂ = Γij (1.9) ∂xj ∂xi ∂xk k=1 and a consequence of (1.9) is that Christoﬀel symbols can be written as k 1 X kl ∂ ∂ ∂ Γij = g gil + gjl − gij . (1.10) 2 ∂xj ∂xi ∂xl l Since the Levi-Civita connection is metric compatible by (1.7), we can use (1.9) and then by [15], the covariant derivative of the metric tensor [gij] is

ij ij ∂ ij X il j X jl i g;k = ∇kg = g + g Γlk + g Γlk. (1.11) ∂xk l l from [6]

The gradient of f as a vector ﬁeld gradf on TpM is deﬁned to be

hgradf(p), vi = dfp(v)

5 for p ∈ M, v ∈ TpM. In local coordinates (x1, ..., xn), the gradient of the function f is X ∂f ∂ grad f = gij · (1.12) ∂x ∂x i,j j i The divergence of X is a function divX : M → IR, where div(p) is the trace of

the linear mapping of a vector Y (p) to ∇Y X(p) with p ∈ M ([13]). In local coordinates,

P ∂ (x1, ..., xn), the divergence of a vector ﬁeld X = ak is k ∂xk " # X ∂ak X k div X = + aiΓik . (1.13) ∂xk k i

The Laplacian (or Laplace-Beltrami Operator) is deﬁned to be ∆gf = div(gradf), where f is a Ck, k ≥ 2, function on M. Using (1.13)-(1.12) the Laplacian can then be written as n " # X ki ∂ ∂f X ij ∂f l ∆gf = g − g Γkj (1.14) ∂xk ∂xj ∂xl k,j=1 l

in local coordinates (x1, ..., xn).

The process of equating two indices of a mixed tensor, one being an upper index and

the other a lower index, and then summing with respect to this pair of indices, is called contraction ([22]). We use this process in the proof of Theorem 1, section 2.2.

1.4 Beltrami Equation

Now we will introduce the Beltrami equation. Consider the partial derivatives of w

with respect to z andz ¯ to be 1 w = ∂w¯ = (w + iw ) z¯ 2 x y and 1 w = ∂w = (w − iw ). z 2 x y Let D be a domain in C and µ : D → C a measurable function where |µ| < k < 1. The

Beltrami equation for µ is

wz¯ = µ(z)wz. (1.15)

6 Notice that if µ ≡ 0, then (1.15) becomes the Cauchy-Riemann equation.

A known relationship between Beltrami equations and quasiconformal mappings is

that every homeomorphic solution of (1.15) is k-quasiconformal, provided µ satisﬁes |µ(z)| ≤

k−1 k+2 and conversely, every k-quasiconformal mapping is a solution of some Beltrami equation k−1 satisfying (1.15) with |µ(z)| ≤ k+2 ([4]). We will deﬁne and discuss quasiconformal mappings in section 5.5.

Another theorem regarding Beltrami equations is as follows: If w1(z) is a solution

of (1.15) and f(w) an analytic function, then w2(z) = f(w1(z)) is also a solution of (1.15).

Conversely, if w1(z) and w2(z) are two solutions of the same Beltrami equation deﬁned in

the same domain and w1(z) is a homeomorphism, then w2(z) = f(w1(z)) where f(w) is an analytic function ([4]).

One can think of the Beltrami equation as a change of variables from IR2 → IR2, while

a parameterization IR2 → IR3 is a mapping from the parameter domain to the manifold.

Gauss ﬁrst studied the Beltrami equation in the 1820’s while exploring the existence

of isothermal parameters on a surface which will be deﬁned in section 2.1. In sections 5.3 and 5.4 we will prove the existence of a single normalizing parameter system for isothermal and

characteristic coordinates, where we will use the Beltrami equations corresponding to each coordinate system. In the late 1930’s Morrey studied the complex Beltrami equation and established the existence of homeomorphic solutions for measurable µ, but it took another

20 years before Bers found that quasiconformal maps can be regarded as homeomorphic solutions of the Beltrami equation ([19]).

We can associate µ with the Riemannian metric

Edx2 + 2F dxdy + Gdy2 = |dz + µdz¯|2.

Then in real coordinates (1.15) has the form,

W ux = F vx + Gvy (1.16)

−W uy = Evx + F vy (1.17)

7 with W 2 = EG−F 2. Beltrami equations are found in the study of differential geometry, complex function theory and differential equations ([3]). We will use different parameterizations to find local representations of the metric and the corresponding Beltrami equations.

8 CHAPTER 2

ISOTHERMAL AND HARMONIC COORDINATES

2.1 Isothermal Coordinates

Consider (M, g) with a parameterization x : U ⊂ Rn → M. Local coordinates

that make the metric a multiple of the identity are called Isothermal Coordinates, also

called conformal coordinates. If (x1, ..., xn) are local isothermal coordinates, then the local representation of the metric in these coordinates is of the form

φ 2 2 g = e (dx1 + ··· + dxn), (2.1)

where φ is a smooth function.

Now consider a surface (M, g) in R3 with the local parameterization x(u, v). The local representation of the metric in isothermal coordinates on a surface is   λ2 0 [gij] =   (2.2) 0 λ2

2 where E = G = λ = hxu, xuio = hxv, xvio and F = 0, where h, io is the standard inner product from IR3. In other words, in isothermal coordinates the tangential coordinate vectors at each point are perpendicular and have the same length.

Starting with [23], Lancaster and his coauthors used an isothermal parametrization to investigate the existence and behavior of the radial limits of bounded solutions of (1) which satisfy the Dirichlet boundary condition mentioned in the preface. In [25] and [24], Lancaster and Siegel used an isothermal parametrization to investigate the existence and behavior of the radial limits of bounded solutions of (1) which satisfy the contact angle boundary condition. Isothermal coordinates exist in a neighborhood of any point on a two-dimensional Riemannian manifold as long as the metric has certain regularity assumptions as given in [7]. Isothermal coordinates do not exist in higher dimensions unless the manifold is conformally

9 ﬂat ([28]). A proof of the global existence of isothermal coordinates on a surface deﬁned by a

graph of a function will be given in Theorem 6 found in section 5.4. A proof with a diﬀerent approach can be found in [7].

2.2 Harmonic Coordinates

As mentioned in the previous section, isothermal coordinates do not exist in higher

dimensions unless the manifold is conformally ﬂat. Therefore, we now consider harmonic co-

ordinates. For a manifold (M, g) with a parameterization x : U → M then local coordinates

(x1, ..., xn) are called harmonic coordinates if each coordinate function xi is harmonic. In other words,

∆gxi = 0, (2.3)

for i = 1, ..., n where ∆g represents the Laplace-Beltrami operator, as given by (1.14). The existence of harmonic coordinates is a consequence of existence theory for elliptic partial diﬀerential equations ([8, p. 91]).

Harmonic coordinates in higher dimensions were ﬁrst used by Einstein in 1916 in the study of general relativity ([2]). Moreover, DeTurck and Kazdan proved in 1981, that “a metric has optimal regularity in any harmonic chart, i.e., that it is no smoother in any other coordinates” ([11]). In [11, p. 252] is the following

Theorem 1. If a [local representation of a] metric g ∈ Ck,α, 1 ≤ k ≤ ∞ (or Cω) in some coordinate chart, then it is also of class Ck,α (or Cω) in harmonic coordinates, while it is of at least class Ck−2,α (or Cω) in geodesic normal coordinates.

As an example provided in [11], consider p(x, y) > 0, p ∈ Ck,α(Ω) in an open set

Ω ⊂ IR2. The metric h = p(x, y)(dx2 + dy2) is then of class Ck,α in that set. Note that this metric h is of the form (2.1) and therefore is isothermal. We claim that the “metric’s

“diﬀerentiability cannot be increased by changing coordinates” ([11]). This claim is veriﬁed

10 because isothermal coordinates are harmonic in two dimensions and utilizing the above theorem h must be of at least class Ck,α. We will prove isothermal coordinates are harmonic in two-dimensions in Theorem 3.

Returning to the deﬁnition of harmonic coordinates, we elaborate on (2.3) using

(1.14).

n " # X jk ∂ ∂xi X jk ∂xi l 0 = ∆gxi = g − g Γjk ∂xj ∂xk ∂xl k,j=1 l jk X i jk l = (0)g − δl g Γjk l,j,k X jk i = − g Γjk jk

Therefore,

X jk i ∆gxi = − g Γjk = 0. (2.4) jk

An alternate (yet, equivalent) deﬁnition for harmonic coordinates is given in [18]: “A system of coordinates xi in an arbitrary ﬁxed n-dimensional Riemannian manifold will be called harmonic if in these coordinates the components of the metric tensor of the space satisfy the equations for i = 1, 2, ..., n,

X ∂ (p|g|gik) = 0, (2.5) ∂xk k where |g| represents the determinant of [gij].” We will show that (2.4) and (2.5) are equivalent, but ﬁrst we provide the following lemma.

i P 1 ∂|g| Lemma 1. In two dimensions, if |g| is the determinant of the metric, then Γli = l . 2|g| ∂xl

Proof. Let |g| be the determinant of the metric [gij].

∂|g| ∂ 2 ∂g11 ∂g22 ∂g12 = (g11g22 − g12) = g22 + g11 − 2g12 . ∂xl ∂xl ∂xl ∂xl ∂xl

11 g g g Using g11 = 22 , g12 = g21 = − 12 , g22 = 11 , we can write |g| |g| |g| ∂|g| ∂g ∂g ∂g = |g| g11 11 + g22 22 + 2g12 12 ∂xl ∂xl ∂xl ∂xl X ∂gij = |g|gij ∂xl ijl X i j = |g|(Γil + Γjl). ijl Replacing j by i yields

∂|g| X i = 2|g|Γil ∂xl l which implies

i X 1 ∂|g| ∂ p Γil = = (log |g|). 2|g| ∂xl ∂xl l

Theorem 2. The deﬁnitions (2.4) and (2.5) are equivalent.

Proof. Recall (1.11): ∂gij gij = + gilΓj + gjlΓi = 0. ;k ∂xk lk lk Contracting the indices k, j in the above equation gives,

ij ∂g il j ij i + g Γij + g Γlj = 0. (2.6) ∂xj Consider (2.5), with |g| representing the determinant of the metric

∂ 1 ij 0 = |g| 2 g ∂xj ij 1 ∂g 1 ij ∂|g| = |g| 2 + g ∂xj 2|g| ∂xj ij 1 ∂g 1 ∂|g| ij = |g| 2 + g ∂xj 2|g| ∂xj ij 1 ∂g j 2 il = |g| + g Γlj [by Lemma 1] ∂xj 1 il i 2 = −g Γlj |g| [by (2.6)]

1 = |g| 2 ∆gxj.

12 ∂ p ik ∂ p ik 1 Recall that ( |g|g ) = 0 and the previous steps showed that ( |g|g ) = |g| 2 ∆gxj, ∂xk ∂xk 1 therefore |g| 2 ∆gxj = 0. This implies that ∆gxj = 0.

2.3 Relationships between Isothermal and Harmonic Coordinates

Theorem 3. Isothermal coordinates are harmonic on a 2-dimensional manifold, M 2.

Proof. Let x(u, v) be a parametrization of a surface M, where u and v are local isothermal

coordinates. By the deﬁnition of isothermal coordinates,

2 ij −2 ij gij = λ δij and g = λ δ ,

2 where λ = hxu, xuio = hxv, xvio. To show that u is harmonic, let x1 = u and x2 = v. Now use (2.4) to calculate ∆gu. 2 X jk 1 ∆gu = ∆gx1 = g Γjk j,k=1 2 X −2 jk 1 = λ δ Γjk j,k=1 −2 1 1 = λ (Γ11 + Γ22) 1 ∂g ∂g ∂g ∂g ∂g ∂g = λ−2 g11 11 + 11 − 11 + g11 21 + 21 − 22 2 ∂u ∂u ∂u ∂v ∂v ∂u 1 ∂g ∂g = (λ−2)2 11 − 22 2 ∂u ∂u 1 ∂ ∂ = (λ−2)2 (λ2) − (λ2) = 0. 2 ∂u ∂u Similarly, we can ﬁnd ∆v. 2 X jk i ∆gv = ∆gx2 = g Γjk j,k=1 −2 2 2 = λ (Γ11 + Γ22) 1 ∂g ∂g ∂g ∂g ∂g ∂g = λ−2 g22 12 + 12 − 11 + g22 22 + 22 − 22 2 ∂u ∂u ∂v ∂v ∂v ∂v 1 ∂g ∂g = (λ−2)2 11 + 22 2 ∂v ∂v 1 ∂ ∂ = (λ−2)2 − (λ2) + (λ2) 2 ∂v ∂v = 0.

13 2 Since ∆gxi = 0 for i = 1, 2, an isothermal parameterization of M is harmonic.

Now using (2.5) instead of (2.4), we ﬁnd the following alternate proof of Theorem 3.

Proof. Let (x1, x2) be isothermal coordinates. Consider (2.5) with i = 1:

2 X ∂ ∂ ∂ p|g|g2k = p|g|g11 + p|g|g12 . ∂xk ∂x1 ∂x2 k=1

12 21 11 22 −2 Since (x1, x2) is an isothermal coordinate system, then g = g = 0 and g = g = λ . Thus,

2 X ∂ ∂ p|g|g1k = (λ2(λ−2)) ∂xk ∂x1 k=1 ∂ = (1) = 0. ∂x1 Similarly, for i = 2:

2 X ∂ ∂ ∂ p|g|g2k = p|g|g21 + p|g|g22 ∂xk ∂x1 ∂x2 k=1 ∂ = (λ2(λ−2)) ∂x2 ∂ = (1) = 0. ∂x2

P2 ∂ p ik We have found that the system of coordinates (x1, x2) is harmonic, since k=1 |g|g = ∂xk 0 for i = 1, 2.

Next we will consider the converse: Is a harmonic coordinate system isothermal? We

find that a harmonic coordinate system is not necessarily isothermal. One can see that the metric  √  2λ2 λ2 [gij] =  √  λ2 2λ2 represents a harmonic coordinate system using the requirement of (2.5). To see this, first we √ √ 2 2 2 2 4 4 4 p 2 find the determinant of [gij] is |g| = ( 2λ )( 2λ ) − (λ ) = 2λ − λ = λ . So, |g| = λ Then,  √  −2 −2 ij 2λ −λ [g ] =  √  −λ−2 2λ−2

14 and so, (√ 2 for i 6= j p|g|gij = . −1 for i = j

Obviously, the derivative of p|g|gij for either case is 0. So, we have a harmonic coordinate

system. The metric is not isothermal, since g12 = g21 6= 0.

In higher dimensions, isothermal coordinates do not have to be harmonic. Consider the metric g with local representation   λ2 0 0    2  [gij] =  0 λ 0  .   0 0 λ2

Since it is a multiple of the identity matrix, it represents an isothermal coordinate system. The inverse metric will be,     λ2 0 0 λ−2 0 0 1     [gij] =  2  =  −2  . 6  0 λ 0   0 λ 0  λ     0 0 λ2 0 0 λ−2

Now use (2.5) to see this is not a harmonic coordinate system. Notice that ( λ for i = j p|g|gij = 0 for i 6= j

∂λ and therefore this is not a harmonic coordinate system provided 6= 0. ∂xk

15 CHAPTER 3

CHARACTERISTIC COORDINATES

In this chapter we will introduce a third type of coordinates, characteristic coor-

dinates, that are often used in the study of partial diﬀerential equations. We will then derive and discuss the normal form given by these coordinates using [10]. Next the Beltrami

equations are derived for characteristic coordinates. In section 3.3, we will ﬁnd the local representation of the metric of a surface in characteristic coordinates in terms of Carte- sian coordinates. Then with an example we show that characteristic coordinates are not

harmonic, unless we are working with a minimal surface.

3.1 Quasilinear Operators

We now look at characteristic coordinates, also called characteristic parameters.

Characteristic coordinates allow us to write a partial diﬀerential equation in a canonical form. The normal form given by characteristic coordinates may provide insight into expand-

ing the results in [25] and [23].

We consider a partial diﬀerential operator of the form

Q[f] = afxx + 2bfxy + cfyy + d (3.1)

with p = fx, and q = fy and coeﬃcients a, b, c, d being functions of the quantities x, y, p, q. This is called a quasilinear diﬀerential operator because it is linear in the derivatives of highest order ([10]). We will be focusing on the elliptic case which occurs when b2 −ac < 0.

Utilizing [10] and [5] we transform the elliptic partial diﬀerential equation

afxx + 2bfxy + cfyy + d = 0 (3.2) into a canonical form

∆f + ··· = fρρ + fσσ + ··· = 0. (3.3)

16 Introducing local coordinates σ = φ(x, y) and ρ = ψ(x, y) where φ and ψ are C2 and

φxψy − φyψx 6= 0. By the chain rule, we ﬁnd the following:

p = fx = fσσx + fρρx

q = fy = fσσy + fρρy

2 2 px = fxx = fσσσx + 2fρσσxρx + fρρρx + fσσxx + fσσxx (3.4)

2 2 qy = fyy = fσσσy + 2fρσσyρy + fρρρy + fσσyy + fσσyy

py = qx = fσσσxσy + fσρ(σx + ρy + ρxσy) + fρρρxρy + uσσxy + uρρxy.

Substituting (3.4) into (3.2) we get

∗ ∗ ∗ ∗ a (σ, ρ)fσσ + b (σ, ρ)fσρ + c (σ, ρ)fρρ = Φ (σ, ρ, f, fρ, fσ) (3.5)

∗ where Φ (σ, ρ, f, fρ, fσ) includes terms involving d from (3.2) as well as other terms and

∗ 2 2 ∗ 2 2 a = aσx + 2bσxσy + cσy c = aρx + 2bρxρy + cρy (3.6)

∗ b = aρxσx + b(σxρy + σyρx) + cσyρy.

Now we stipulate that a∗ = c∗ and b∗ = 0, which can be written explicitly using (3.6) as

2 2 2 2 aρx + 2bρxρy + cρy = aσx + 2bσxσy + cσy, (3.7)

aρxσx + b(ρxσy + ρyσx) + cρyσy = 0. (3.8)

Using these conditions and dividing (3.5) by a∗, we obtain the canonical form for an elliptic equation:

fσσ + fρρ = Ψ(σ, ρ, f, fσ, fρ) (3.9) where a, b, c and d are speciﬁed real-valued functions of σ and ρ. Notice that if Ψ = 0, then we obtain Laplace’s equation

fσσ + fρρ = 0.

17 From here we can ﬁnd the Beltrami equation for characteristic coordinates. Recall σ

and ρ transformed (3.2) into the canonical form (3.9), if σ and ρ satisfy (3.7) and (3.8). In

order to solve for σ(x, y) or ρ(x, y), we multiply equation (3.8) by 2i and then add the result

to (3.7) to get

2 2 a(σx + iρx) + 2b(σx + iρx)(σy + iρy) + c(σy + iρy) = 0.

We solve this equation for σx+iρx and recall that since we are considering the elliptic case σy+iρy then b2 − ac < 0 which yields √ σ + iρ −b ± i b2 − ac x x = . σy + iρy a

Multiplying by σy + iρ and then solving for the real and imaginary parts we get 1 √ σ = (−bσ − ±ρ b2 − ac) x a y y 1 √ ρ = (−bρ − ±σ b2 − ac) x a y y which can be written

√ 2 aσx + bσy = ±ρy b − ac √ 2 aρx + bρy = ±σy b − ac

Then we ﬁnd the Beltrami equation for characteristic coordinates to be

bρ + cρ aρ + bρ σ = x y , −σ = x y , (3.10) x ω y ω where ω2 = ac − b2. (3.11)

The problem of locally reducing L[u] + ··· = 0 to normal form (3.3) by a transformation

σ(x, y) and ρ(x, y) is equivalent to ﬁnding a solution of the Beltrami equation (3.10) for

which σxρy − σyρx 6= 0 ([10]).

3.2 Normal Form

Consider a certain neighborhood on a surface with an equation of elliptic form. As

shown previously, we can then ﬁnd a normal form of the elliptic diﬀerential equation. We

18 now want to ﬁnd a system of three diﬀerential equations for x, y, f in terms of σ and ρ. To

ﬁnd these equations we perform a change of variables. Let φ(x, y) = σ and ψ(x, y) = ρ.

Then we have

∂φ ∂φ ∂x ∂φ ∂y 1 = = + ∂σ ∂x ∂σ ∂y ∂σ ∂φ ∂φ ∂x ∂φ ∂y 0 = = + ∂ρ ∂x ∂ρ ∂y ∂ρ ∂ψ ∂ψ ∂x ∂ψ ∂y 0 = = + ∂σ ∂x ∂σ ∂y ∂σ ∂ψ ∂ψ ∂x ∂ψ ∂y 1 = = + . ∂ρ ∂x ∂ρ ∂y ∂ρ

∂φ In matrix notation (with σx = ∂x , etc) we have       σx σy xσ xρ 1 0     =   . ρx ρy yσ yρ 0 1

Setting β = σxρy − ρxσy and solving the matrix equation to complete a change of variables, we ﬁnd the following       xσ xρ 1 ρy −σy 1 0 = .   β     yσ yρ −ρx σx 0 1

This yields

1 1 x = ρ x = − σ σ β y ρ β y 1 1 y = − ρ y = σ . σ β x ρ β x

We can then write (3.7)-(3.8) using the inverse relations we just found as the following system of three diﬀerential equations for the quantities x, y, f or the position vector x as function of the parameters σ and ρ

2 2 2 2 ayσ − 2byσxσ + cxσ = ayρ − 2byρxρ + cxρ, (3.12)

ayσyρ − b(yσxρ + yρxσ) + cxσxρ = 0. (3.13)

19 It follows that we have equation (34a) of [10], which is

∆ox ∆oy ∆oz 2 d h∆ox, (xσ × xρ)i = x y z = (xσyρ − xρyσ) √ , (3.14) o σ σ σ 2 ac − b

xρ yρ zρ √ 2 where ∆ox = xσσ + xρρ denotes the Laplace operator on the vector x. Let ω = ac − b and set a b b c M = + and N = + . ω x ω y ω x ω y A (tedious) calculation by Professor Lancaster concludes that

∆ox = (Myσ − Nxσ)xρ + (Nxρ − Myρ)xσ (3.15)

∆oy = (Myσ − Nxσ)yρ + (Nxρ − Myρ)yσ (3.16) d ∆ z = (My − Nx )z + (Nx − My )z + (x y − x y ) . (3.17) o σ σ ρ ρ ρ σ σ ρ ρ σ ω

Notice that the ﬁrst two equations simplify to

∆ox = M(xρyσ − xσyρ)

∆oy = N(xρyσ − xσyρ).

Now suppose the vector x is a function of isothermal coordinates (u, v) and d = 0 in

(3.1); then ∆ox = αxu + βxv. By the deﬁnition of isothermal coordinates

hxu, xuio = hxv, xvio (3.18)

hxu, xvio = 0. (3.19)

Now diﬀerentiate (3.18) with respect to u and then diﬀerentiate (3.19) with respect to v which yields

hxuu, xuio = hxvu, xvio and

hxuv, xvio + hxvv, xuio = 0.

20 Combining these we get

hxuu, xuio = hxvu, xvio = − hxvv, xuio and algebra produces

h(xuu + xvv), xuio = hxvu, xvio .

This can be rewritten as

h∆x, xuio = 0 (3.20)

which implies that α = 0. Similarly, if we diﬀerentiate (3.18) with respect to v and (3.19)

with respect to u we ﬁnd

h∆x, xvio = 0 (3.21)

and thus β = 0. Then from (3.20) and (3.21) we conclude

∆ox = 0.

In general in isothermal coordinates when d 6= 0 in (3.1), the previous paragraph

shows that (3.20) and (3.21) still hold and so

∆ox = γxu × xv.

It can be shown ([27]) that γ is in fact equal to twice the mean curvature of the surface; that

∆x = 2H(x)xu × xv (3.22)

where H(x) is the mean curvature of the surface as deﬁned in section 4.1 and is discussed

in more detail in appendix A.2. Notice that if H(x) = 0 then this implies that ∆ox = 0. In other words, if the mean curvature is zero, (i.e., we have a minimal surface) then the

coordinates x(σ, ρ) = (x(σ, ρ), y(σ, ρ), z(σ, ρ)) are harmonic in the sense that xσσ + xρρ = 0.

21 Now, if d = 0 in (3.14), then it is independent of a, b, c and it has the form

h∆ox, (xσ × xρ)io = 0 (3.23)

with ∆ox = xσσ + xρρ and so ∆ox = αxσ + βxρ. Professor Lancaster’s calculation shows that

∆ox = (Myσ − Nxσ)xσ + (Nxρ − Myρ)xρ.

3.3 Coordinate Changes

Here we want to ﬁnd a local representation of the metric in characteristic coordinates.

Consider a given function f : U ⊂ IR2 → IR and a given operator Q such that Qf = 0. Let x : U → IR3 be a parameterization of the graph of f. We want to ﬁnd a parametrization where (ρ, σ) are characteristic coordinates. Now give x(U) the metric induced by that of

IR3. Since x(U) is most easily parameterized in terms of (x, y), we can calculate the local components of the metricg ¯ij(x, y) to be   2 1 + fx fxfy g¯ij =   . (3.24) 2 fxfy 1 + fy

The following calculation allows us to express the local representation of the metric (3.24) in ∂ui ∂uj terms of (ρ, σ). Now utilizingg ¯ = g from ([22]) we can ﬁnd the following system αβ ij ∂u¯α ∂u¯β of equations by letting u1 = σ, u2 = ρ,u ¯1 = x andu ¯2 = y. Thus we can write

2 2 2 g¯11 = 1 + fx = g11σx + 2g12σxρx + g22ρx (3.25)

g¯12 =g ¯21 = fxfy = g11σxσy + g12(σxρy + σyρx) + g22ρxρy (3.26)

2 2 2 g¯22 = 1 + fy = g11σy + 2g12σyρy + g22ρy. (3.27)

Solving ﬁrst for g11, we begin by eliminating g22. Combining equations (3.25) and (3.27) to get

2 2 2 2 2 2 2 2 ρy + fx ρy = g11σxρy + 2g12σxρxρy + g22ρxρy

2 2 2 2 2 2 2 2 −ρx − fy ρx = −g11σyρx − 2g12σyρyρx − g22ρxρy.

22 Adding these we have

2 2 2 2 2 2 2 2 2 2 2 2 ρy + fx ρy − ρx − fy ρx = g11(σxρy − σyρx) + 2g12(σxρxρy − σyρyρx). (3.28)

Then we multiply (3.26) by ρy and (3.27) by −ρx to obtain

2 fxfyρy = g11σxσyρy + g12ρy(σxρy + ρxσy) + g22ρyρx

2 2 2 −ρx − fy ρx = −g11σyρx − 2g12σyρyρx − g22ρxρy.

Then adding these then gives

2 2 2 fxfyρy − ρx − ρxfy = g11(σxσyρy − σyρx) + g12(σxρy − σyρyρx). (3.29)

We can then multiply (3.29) by −2ρx to continue solving for g11 by eliminating g12:

2 2 2 2 2 2 2 2 2 2 2 ρy + fx ρy − ρx − fy ρx = g11(σxρy − σyρx) + 2g12ρx(σxρy − σyρyρx)

2 2 2 2 −2fxfyρyρx + 2ρx + 2ρxfy = −2g11ρxσy(σxρy − σyρx) − 2g12ρx(σxρy − σyρxρy).

Now adding the two above equations, we have

2 2 2 2 2 2 2 2 2 2 ρy + ρyfx − 2ρxρyfxfy + ρx + ρxfy = g11(σxρy − 2ρxρyσxσy + ρxσy).

Therefore,

2 2 2 2 2 2 2 2 2 2 ρx + ρxfy − 2ρxρyfxfy + ρy + ρyfx (1 + fy )ρx − 2fxfy(ρxρy) + (1 + fx )ρy g11 = 2 = 2 . (σxρy − ρxσy) (σxρy − ρxσy)

2 2 Similarly, we solve for g22 by multiplying (3.25) by σy and multiplying (3.27) by −σx and then adding them to eliminate g11.

2 2 2 2 2 2 2 2 g11σxσy + 2g12σxρxσy + g22ρxσy = σy + σyfx

2 2 2 2 2 2 2 2 −g11σxσy − 2g12σxρyσy − g22ρyσx = −σx + σxfy .

Addition produces

2 2 2 2 2 2 2 2 2 2 2g12σxσy(ρxσy − σxρy) + g22(σyρx − ρyσx) = σy + fx σy − σx − fy σx. (3.30)

23 Continue eliminating g11 by multiplying (3.25) by σy and (3.26) by −σx

2 2 2 g11σxσy + 2g12σxρxσy + g22ρxσy = σy + σyfx

2 −g11σxσy − g12σx(σxρy + σyρx) − g22ρyρxσx = −σxfyfx.

Simplifying we now have

2 g12σx(σyρx − σxρy) + g22ρx(ρxσy − ρyσx) = σy + fx σy − σxfxfy. (3.31)

Next, we eliminate g12 by multiplying (3.30) by −2σy and then add this (3.31),

2 2 2 2 2 2 2 2 2 2 2g12σxσy(ρxσy − σxρy) + g22(σyρx − ρyσx) = σy + fx σy − σx − fy σx

2 2 2 −2g12σxσy(σyρx − σxρy) − 2g22σyρx(ρxσy − ρyσx) = −2σy − 2fx σy + 2σxσyfxfy to get

2 2 2 2 2 2 2 2 2 2 g22(−ρxσy + 2ρyρxσyσx − ρyσx) = −σy − fx σy + 2σxσyfxfy − σx − fy σx.

Thus we ﬁnd

2 2 2 2 2 2 2 2 2 2 σx + σxfy − 2σxσyfxfy + σy + σyfx (1 + fy )σx − 2fxfy(σxσy) + (1 + fx )σy g22 = 2 = 2 . (σyρx − ρyσx) (σxρy − ρxσy)

Then solve for g12 to obtain

2 2 −σxρx − σxρxfy + fxfy(σxρy + σyρx) − σyρy − σyρyfx g12 = 2 (σxρy − ρxσy)

2 2 (1 + fy )σxρx − fxfy(σxρy + σyρx) + (1 + fx )σyρy = − 2 . (σxρy − ρxσy) Thus, in characteristic coordinates (σ, ρ) the components of the local representation of the metric for a surface in IR3 are

24 2 2 2 2 (1 + fy )ρx − 2fxfyρxρy + (1 + fx )ρy g11 = 2 (σxρy − ρxσy)

2 2 (1 + fy )σxρx − fxfy(σxρy + σyρx) + (1 + fx )σyρy g12 = g21 = − 2 (3.32) (σxρy − ρxσy)

2 2 2 2 (1 + fy )σx − 2fxfyσxσy + (1 + fx )σy g22 = 2 . (σxρy − ρxσy)

2 Now if f is given, we can deﬁne an operator Q by setting a = 1 + fy , b = −fxfy,

2 c = 1 + fx and d = 0. Then suppose that (ρ, σ) were characteristic coordinates for this f, Q, now depending only on f. Then we get   2 2 1 aρx + 2b(ρxρy) + cρy −[aσxρx + b(σxρy + σyρx) + cσyρy] [gij] =  (3.33), β 2 2 −[aσxρx + b(σxρy + σyρx) + cσyρy] aσx + 2b(σxσy) + cσy

2 where β = (σxρy − ρxσy) . Now if the graph of f is a minimal surface, then (3.7)-(3.7), which we recall to be

2 2 2 2 aσx + 2bσxσy + cσy = aρx + 2bρxρy + cρy

aσxρx + b(σxρy + σyρx) + cσyρy = 0, would hold. Then if we deﬁne Q as in (3.33), the expression for the metric shows that the same coordinates (ρ, σ), which were characteristic for Q, are also isothermal for the metric.

We can easily see that g11 = g22 and g12 = g21 = 0. Thus, characteristic coordinates are

2 2 isothermal if the surface is minimal, i.e., 1 + fy = a, −fxfy = b and 1 + fx = c. Next one might ask “Is (σ, ρ) a system of harmonic coordinates?” Now since we previously showed the isothermal coordinates are a harmonic system, we know that if the surface is minimal then (σ, ρ) are harmonic. So now we consider the case where the function f is such that the surface formed by its graph is not minimal.

25 3.4 An Example

We will use the following example to show that characteristic coordinates need not

be harmonic if the surface formed by the graph of the function f is not minimal.

2 2 Let f(y, x) = y − x . First notice that fyy + fxx = 0. Since this is in canonical form, we have a characteristic parameterization. Alternatively, we can write the Beltrami equations (1.16) as

σx = ρy and σy = −ρx.

Notice these are in the form of the Cauchy-Riemann equation and one can obtain σ = x and

ρ = y as solutions. Moreover, we again conclude that σ and ρ are characteristic parameters.

We want to consider the example f(x, y) = y2 − x2 for the case where the surface is

not minimal, so ﬁrst we must show that the graph of f is not a minimal surface. Recall the

second fundamental form for the case where Σ ⊂ IR3 is the graph of f   2 2 1 (1 + fy )fxx − fxfyfxy (1 + fy )fxy − fxfyfyy S = 3   . (3.34) (1 + f 2 + f 2) 2 2 2 x y (1 + fx )fxy − fxfyfxx (1 + fx )fyy − fxfyfxy For more information on the second fundamental from see appendix A.2. The mean curvature is the trace of (3.34) ([26]). Using f(x, y) = y2 − x2, we ﬁnd

fx = −2x, fy = 2y

fxx = −2 fyy = 2

fxy = fyx = 0.

Thus   1 −2(1 + 4y2) 8xy 3   (1 + 4x2 + 4y2) 2 −8xy 2(1 + 4x2)   2 −(1 + 4y2) 4xy = 3   . (3.35) (1 + 4x2 + 4y2) 2 −4xy (1 + 4x2) Now ﬁnd the trace of the matrix (3.35) to be

2 2 2 2 2 8(x − y ) T r(S) = 3 (−4y + 4x ) = 3 6= 0. (1 + 4x2 + 4y2) 2 (1 + 4x2 + 4y2) 2

26 Since the trace of S is not equal to zero thus the mean curvature is not zero. Thus this is not a minimal surface. Now that we have established that the surface deﬁned by the graph of f is not minimal, we will show that the characteristic coordinates are not harmonic.

Let σ = x and ρ = y and we ﬁnd each of the components of the metric (3.33)

2 2 2 2 (1 + f )(0) − 2fyfx(0)(1) + (1 + fx) (1) g = y = 1 + f 2. 11 1 x Similarly, we can calculate

2 g22 = 1 + fy and g12 = fyfx.

2 2 Letting |g| represent the determinant of [gij] such that |g| = 1 + fy + fx we ﬁnd the inverse components to be 1 + f 2 2 11 y 12 −fyfx 22 1 + fx g = 2 2 g = 2 2 g = 2 2 1 + fy + fx 1 + fy + fx 1 + fy + fx

Note since g11 6= g22 and g12 6= 0 this is not an isothermal coordinate system. Recall if (2.5) is satisﬁed, then we have a harmonic coordinate system. We ﬁrst

p ij consider i = 1. Substituting |g|, the components of [g ] into (2.5) and letting x1 = x and x2 = y we ﬁnd q 2 q ∂ 2 2 1 + fx ∂ 2 2 −fyfx 1 + fy + fx 2 2 + 1 + fy + fx 2 2 . ∂x 1 + fy + fx ∂y 1 + fy + fx

Now substitute fy = 2y and fx = −2x. Applying the product rule to take the derivative and then simpliﬁcation yields ! ! ∂ 1 + 4y2 ∂ 4yx + = ∂x p1 + 4y2 + 4x2 ∂y p1 + 4y2 + 4x2

1 ∂ ∂ 1 2 2 − 2 2 2 2− 2 = (1 + 4y + 4x ) 2 1 + 4y + (1 + 4y ) 1 + 4y + 4x ∂x ∂x 1 ∂ ∂ 1 2 2 − 2 2− 2 +(1 + 4y + 4x ) 2 (4yx) + (4yx) 1 + 4y + 4x ∂y ∂y 2 2 − 3 2 2 2 − 1 = 0 + −4x(1 + 4y + 4x ) 2 (1 + 4y ) + 4x(1 + 4y + 4x ) 2

2 2 − 3 −4y(4xy)(1 + 4y + 4x ) 2

2 2 − 3 2 2 2 2 = (1 + 4y + 4x ) 2 4x(1 + 4y ) + 4x(1 + 4y + 4x ) − 16xy )

2 2 − 3 2 3 = (1 + 4y + 4x ) 2 −16xy + 16x 6= 0.

27 Similarly ! ! ! ! ∂ −f f ∂ 1 + f 2 ∂ 4yx ∂ 1 + 4x2 y x + x = + p 2 2 p 2 2 p 2 2 p 2 2 ∂x 1 + fy + fx ∂y 1 + fy + fx ∂x 1 + 4y + 4x ∂y 1 + 4y + 4x

1 ∂ ∂ 1 2 2 − 2 2− 2 = (1 + 4y + 4x ) 2 (4yx) + (4yx) 1 + 4y + 4x ∂x ∂x 1 ∂ ∂ 1 2 2 − 2 2 2 2− 2 + (1 + 4y + 4x ) 2 1 + 4x + (1 + 4x ) 1 + 4y + 4x ∂y ∂y 2 2 − 1 2 2 2 − 3 = 4y(1 + 4y + 4x ) 2 − 16x y(1 + 4y + 4x ) 2

2 2 2 − 3 + 0 − 4y(1 + 4x )(1 + 4y + 4x ) 2

2 2 − 3 2 2 2 2 = (1 + 4y + 4x ) 2 4y(1 + 4y + 4x ) − 16x y − 4y(1 + 4x )

2 2 − 3 3 2 = (1 + 4y + 4x ) 2 16y − 16x y 6= 0.

P ∂ p ik Therefore, since k ( |g|g ) 6= 0 for i = 1, 2 this is not a harmonic coordinate system. ∂xk

28 CHAPTER 4

STRUCTURE CONDITIONS FOR MEAN CURVATURE TYPE

4.1 Prescribed Mean Curvature

The mean curvature H of the surface at the point under consideration is the arith-

metic mean of the principal curvatures κi,

1 H = (κ + κ ), 2 1 2

where κ1 and κ2 are the maximum and minimum of the normal curvature at a given point on a surface. Note that the principal curvatures are also the eigenvalues of the shape operator ([26]). A minimal surface is a surface for which the mean curvature is zero. A

nonparametric minimal surface (i.e. S = {(x, y, z): z = f(x, y)}) satisﬁes

2 2 (1 + fy )fxx + 2fxfyfxy + (1 + fx )fyy = 0. (4.1)

Examples of minimal surfaces include planes, catenoids and helicoids. Prescribed mean

curvature occurs when you set the left-hand side of (4.1) equal to a prescribed function as follows

3 2 2 2 2 (1 + fy )fxx − 2fxfyfxy + (1 + fx )fyy = 2H(x, y, f(x, y))(1 + |Df| ) . (4.2)

For dimension n ≥ 2, the prescribed mean curvature is given by

div(T f) = nH(x, f(x))

∇f with x = (x1, ..., xn), where the vector ﬁeld T f is deﬁned to be T f = √ . We can then 1+|∇f|2 write n ∂f ! X ∂ ∂x i = nH. ∂x p 2 i=1 i 1 + |Df| The equations (4.1) and (4.2) serve as a prototypes for equations of minimal surface

type and equation of mean curvature type respectively which we will deﬁne here.

29 Let Ω be a bounded domain in IR2. An equation of the form

a(p, q)fxx + 2b(p, q)fxy + c(p, q)fyy = 0

2 p = fx, q = fy, with f ∈ C (Ω)

0,τ 2 is called an equation of minimal surface type, if and only if, a, b, c ∈ Cloc (IR ) for some τ ∈ (0, 1), ac − b2 = 1 and there is a positive number ≥ 1 such that

1 + p2 1 + q2 pq p a(p, q) + c(p, q) + 2b(p, q) ≤ 2, W = 1 + p2 + q2 (4.3) W W W for all p, q ∈ IR ([14]).

An equation of the form

2 X 2 aij(x, f, ∇f)Dijf = h(x, f, ∇f), f ∈ C (Ω), (4.4) i,j=1 is called an equation of mean curvature type, if and only if, there exists constants γ

2 and µ such that aij, with i, j = 1, 2 are real valued functions on Ω × IR × IR which satisfy

(p · ξ)2 (p · ξ)2 |ξ|2 − ≤ a (x, z, p)ξ ξ ≤ γ |ξ|2 − 1 + |p|2 ij i j 1 + |p|2

2 2 for all (x, z, p) ∈ Ω × IR × IR and ξ = (ξ1, ξ2) ∈ IR and

|h(x, z, p)| ≤ µp1 + |p|2 for all (x, z, p) ∈ Ω × IR × IR2.

4.2 Structure Conditions

Quoting from [17], “The pioneering work on two dimensional equations of mean curvature type was done by Finn who treated the case aij(x, z, p) ≡ aij(p) and b ≡ 0. Finn called his equations ‘equations of minimal surface type’ and stated the structure conditions for the coeﬃcients somewhat diﬀerently (but equivalently to)

(p · ξ)2 (p · ξ)2 |ξ|2 − ≤ a (x, z, p)ξ ξ ≤ γ |ξ|2 − (4.5) 1 + |p|2 ij i j 1 + |p|2

30 2 2 where aij, with i, j = 1, 2, for all (x, z, p) ∈ Ω × IR × IR and ξ = (ξ1, ξ2) ∈ IR .” This deﬁnition is given by Simon in [29]. In the following proof, we will show the second implies

the ﬁrst by ﬁnding the particular epsilon in (4.3)

P2 Theorem 4. Given i,j=1 aij(x, u, Du)Diju = h(x, u, Du) an equation of mean curvature

2 2 a11 a12 a22 type (4.5) and setting ω = a11a22 − a12 and a = ω , b = ω and c = ω then a, b, c satisfy 1 + p2 1 + q2 pq a(x, y, z, p, q) + c(x, y, z, p, q) + 2b(x, y, z, p, q) ≤ 2, W W W p W = 1 + p2 + q2 (4.6)

√ for all p, q ∈ IR and where = 2γ2.

Proof. Let W 2 = 1 + p2 + q2. We can rewrite (4.5) as

(1 + q2)ξ2 − 2pqξ ξ + (1 + p2)ξ2 (1 + q2)ξ2 − 2pqξ ξ + (1 + p2)ξ2 1 1 2 2 ≤ a ξ2 + 2a ξ ξ + a ξ2 ≤ γ 1 1 2 2 . W 2 11 1 12 1 2 22 2 W 2

Notice in (4.5) if ξ1 = 1 and ξ2 = 0 then

1 + q2 1 + q2 ≤ a ≤ γ . (4.7) W 2 11 W 2

If ξ1 = 0 and ξ2 = 1 in (4.5) then

1 + p2 1 + p2 ≤ a ≤ γ . (4.8) W 2 22 W 2 √ √ Now let ξ1 = a22 and ξ2 = a11 in (4.5) then

2 √ 2 (1 + q )a22 − 2pq a11a22 + (1 + p )a11 √ 2 ≤ a11a22 + 2a12 a11a22 + a22a11 W √ (1 + q2)a − 2pq a a + (1 + p2)a ≤ γ 22 11 22 11 . W 2 (4.9) √ √ Similarly, we let ξ1 = a22 and ξ2 = − a11 in (4.5) then

2 √ 2 (1 + q )a22 + 2pq a11a22 + (1 + p )a11 √ 2 ≤ a11a22 − 2a12 a11a22 + a22a11 W √ (1 + q2)a + 2pq a a + (1 + p2)a ≤ γ 22 11 22 11 . W 2 (4.10)

31 Now we want to multiply (4.9) and (4.10) together. To simplify this we introduce the

following notation. Let √ (1 + q2)a + (1 + p2)a 2pq a a G = 22 11 and J = 11 22 . W 2 W 2

Notice that in order to multiply (4.9) and (4.10), we must have G − J ≥ 0 and G + J ≥ 0

otherwise the inequality signs would change. We consider the left side of (4.9) and factoring yields √ √ √ (1 + q2)a − 2pq a a + (1 + p2)a a + a + (p a − q a )2 G − J = 22 11 22 11 = 11 22 11 22 ≥ 0. W 2 W 2

Similarly, √ √ √ (1 + q2)a + 2pq a a + (1 + p2)a a + a + (p a + q a )2 G + J = 22 11 22 11 = 11 22 11 22 ≥ 0. W 2 W 2

Therefore (G − J)(G + J) ≥ 0. Implementing the new notation, we multiply (4.9) and (4.10)

to obtain

2 2 2 (G − J)(G + J) ≤ 4(a11a22 − a12a11a22) ≤ γ(G − J)γ(G + J) which simpliﬁes to

2 2 2 2 2 2 G − J ≤ 4a11a22(a11a22 − a12) ≤ γ (G − J ).

2 2 Next dividing by 4a11a22 and substituting ω = a11a22 − a12 gives

G2 − J 2 G2 − J 2 ≤ ω2 ≤ γ2 . 4a11a22 4a11a22

Multiplying through by W 2 we ﬁnd

G2 − J 2 G2 − J 2 W 2 ≤ ω2W 2 ≤ γ2 W 2. (4.11) 4a11a22 4a11a22

Now utilizing (4.7) - (4.8) we observe that

4(1 + q2)(1 + p2) 4γ2(1 + q2)(1 + p2) ≤ 4a a ≤ . (4.12) W 4 11 22 W 4

32 Recall G2 − J 2 from (4.11): 1 G2 − J 2 = (1 + 2q2 + q4)a2 + 2(1 + p2)(1 + q2)a a + (1 + 2p2 + p4)a2 − 4p2q2a a W 4 22 11 22 11 11 22 1 = (p2a − q2a )2 + (a + a )2 + 2p2(a2 + a a ) + 2q2(a2 + a a ) W 4 11 22 11 22 11 11 22 22 11 22 1 1 = (p2a − q2a )2 + (a + a ) a + a + 2p2a + 2q2a W 4 11 22 W 4 11 22 11 22 11 22 a + a ≥ 11 22 (1 + 2p2)a + (1 + 2q2)a . (4.13) W 4 11 22 Considering (4.11), and then using (4.7) - (4.8) and (4.12) -(4.13),

a11 + a22 2 2 2 2 2 ((1 + 2p )a11 + (1 + 2q )a22)W 2 2 G − J 2 W 4 (ω W ) ≥ W ≥ 2 2 2 4a11a22 4γ (1 + p )(1 + q ) W 4 (a W 2 + a W 2)((1 + 2p2)a + (1 + 2q2)a ) = 11 22 11 22 4γ2(1 + p2 + q2 + p2q2)

1 + q2 1 + p2 (1 + q2 + 1 + p2) (1 + 2p2) + (1 + 2q2) W 2 W 2 ≥ 4γ2(1 + p2 + q2 + p2q2)

2 + p2 + q2 1 2 + 3p2 + 3q2 + 4p2q2 = · · 1 + p2 + q2 4γ2 1 + p2 + q2 + p2q2

1 1 p2 + q2 + 2 = 1 + 4 − W 2 4γ2 1 + p2 + q2 + p2q2

1 p2 + 1 1 + q2 ≥ (1) 4 − + 4γ2 (1 + p2)(1 + q2) (1 + p2)(1 + q2)

1 1 1 = 4 − − 4γ2 1 + q2 1 + p2

2 1 ≥ = . 4γ2 2γ2 Therefore, 1 ωW ≥ √ = c1. (4.14) γ 2

33 Let ξ1 = p and ξ2 = q then (4.5) becomes,

p2 + q2 p2 + q2 ≤ a p2 + a q2 + 2a pq ≤ γ . W 2 11 22 12 W 2

Now use (4.7)-(4.8) which yields

1 + q2 1 + p2 p2 + q2 1 + q2 + 1 + p2 + p2 + q2 + + ≤ a + a + a p2 + a q2 + 2a pq ≤ γ . W 2 W 2 W 2 11 22 11 22 12 W 2

a a a From the hypothesis a = 11 , b = 12 , and c = 22 . Dividing by W ω, ω ω ω

1 + q2 + 1 + p2 + p2 + q2 a(1 + p2) + (1 + q2)c + 2bpq 1 + q2 + 1 + p2 + p2 + q2 ≤ ≤ γ . W 3ω W W 3ω

Notice that 1 + q2 + 1 + p2 + p2 + q2 = 2W 2 and so

2 a(1 + p2) + (1 + q2)c + 2bpq 2γ ≤ ≤ . ωW W ωW

From (4.14), 1 ωW ≥ c1 = √ > 0. γ 2 Then √ 2γ 2γ 2 ≤ = 2γ 2 = c2 ≤ ∞ ωW c1 and therefore 2γ √ ≤ 2 2γ2. ωW √ We would like 2 to equal 2 2γ2 and so we set

√ = 2γ2.

Therefore an equation of mean curvature type (4.5) implies (4.6).

34 CHAPTER 5

CHARACTERISTIC COORDINATES ARE QUASICONFORMAL FUNCTIONS OF ISOTHERMAL COORDINATES

5.1 Structure Class

In order to show that we have global solvablity (or the global existence of ) characteristic coordinates and later isothermal coordinates, we will introduce the following deﬁnitions from [1].

Let S be a connected Hausdorﬀ space and Φ be a family of local homeomorphisms

such that

(i) Each h ∈ Φ is a topological mapping of an open set V ⊂ S onto an open set in C.

(ii) If h ∈ Φ has domain V , then the restriction of h to any open set V 0 ⊂ V is also in Φ.

(iii) Let h be a mapping of an open set V ⊂ S onto an open set in the complex plane and

suppose that V is covered by open subsets V 0. If the restriction of h to each V 0 ∈ Φ,

then the same will be true of h with domain V .

(iv) The domains of all h ∈ Φ form a covering of S.

A family Ψ of local homeomorphisms with domain and range on C is called a struc-

ture class if the following conditions are satisﬁed:

(i) The identity mapping belongs to Ψ,

(ii) If g ∈ Ψ, then g−1Ψ

(iii) If g1, g2 ∈ Ψ, then g1 ◦ g2 ∈ Ψ provided it is deﬁned.

Now we say Φ deﬁnes a structure of class of Ψ if it satisﬁes the following:

(i) If h(1), h(2) ∈ Φ have the same domain V then h(1) ◦ (h(2))−1 ∈ Ψ.

35 (ii) If h ∈ Φ and g ∈ Ψ, then g ◦ h ∈ Φ provided that it is deﬁned.

A structure class which is formed by all analytic mappings will be called a conformal structure.A Riemann surface is deﬁned by [1] to be a connected Hausdorﬀ space W

together with a conformal structure deﬁned by a family Φ of local homeomorphisms on W .

A complex valued function h = s + it is said to be analytic on the Riemann surface (W, Φ)

if and only if h ◦ f −1 is analytic on f(V ) for every f ∈ Φ with domain V .

5.2 Uniformization Theorem

The Uniformization Theorem as given by [1] is “the universal covering surface

of any Riemann surface is conformally equivalent to a disk, to the complex plane or to the

sphere.”

The following results are found in [16] and we will state them without proof here. For

each Riemann surface R, there is a covering map φ : S → R of a simply connected Riemann

surface S onto R. The surface S is called the universal covering surface of R. The only

Riemann surface having the Riemann sphere as its universal covering surface is the sphere itself. The only Riemann surfaces having the complex plane as universal covering surface

are the complex plane, the punctured complex plane and tori. All other connected, simply connected Riemann surfaces have the open unit disk as universal covering surface.

5.3 Global Existence of Characteristic Coordinates

In the following section we will use a footnote in [10, p. 160] as motivation; note

that much of the terminology is from [1]. We will show the existence of a single normalizing system of characteristic coordinates, (σ(x, y), ρ(x, y)) in the entire domain. We will ﬁrst

show that the change of coordinates is conformal and therefore S is a Riemann surface and apply the Uniformization theorem. We then conclude that the characteristic coordinates are

global because the Riemann surface is conformally equivalent to the unit disk.

Theorem 5. Given a surface S = {(x, y, f(x, y)) : (x, y) ∈ Ω} ⊂ IR3 with an equation of

36 elliptic form (3.1) then there exists a single normalizing parameter system of characteristic

coordinates with the mapping (x, y) → (σ, ρ).

Proof. Consider two solutions (σ(1), ρ(1)) and (σ(2), ρ(2)) of the Beltrami equations,

bρ(k) + cρ(k) aρ(k) + bρ(k) σ(k) = x y and − σ(k) = x y (5.1) x ω y ω ω2 = ac − b2 that are deﬁned in a small parametrized neighborhood on the surface S .

We set ω2 = ac − b2 = 1 by replacing a by √ a , b by √ b , c by √ c and d by ac−b2 ac−b2 ac−b2 √ d . We assume the coeﬃcients a(x, y, z, p, q), b(x, y, z, p, q), and c(x, y, z, p, q) are Holder ac−b2 continuous. Therefore, we have local solvability of the Beltrami equations ([4]).

Locally we are working in a plane with the induced metric, so we will use the following

h(1)(x, y) = σ(1)(x, y) + iρ(1)(x, y) and h(2)(x, y) = σ(2)(x, y) + iρ(2)(x, y).

Let H(s+it) = σ(1) (h(2))−1(s + it)+iρ(1) (h(2))−1(s + it) = m(s+it)+in(s+it). Notice

that m(s + it) = σ(1)(x, y) if and only if s + it = h(2)(x, y). ∂H We will show that = 0 which implies ∂z¯ ∂m ∂n ∂m ∂n − + i + = 0. ∂s ∂t ∂t ∂s

Here we will give a brief outline the proof. First, we will to show the Cauchy-Riemann equations

∂m ∂n ∂m ∂n = and = − . (5.2) ∂s ∂t ∂t ∂s

are satisﬁed. Then H(s + it) will be analytic in a neighborhood on the surface. Moreover,

the mapping H(s+it) = m(s+it)+in(s+it) deﬁned in any two neighborhoods is conformal

in the intersection of these two neighborhoods by (i) in the deﬁnition of structure of class Ψ

and by (ii) from the deﬁnition of Φ. Lastly, we will apply the Uniformization Theorem to get global existence of characteristic coordinates.

37 Let m(s+it) = σ(1)(x, y) and n(s+it) = ρ(1)(x, y). We begin by using the chain rule:

∂m ∂σ(1) ∂x ∂σ(1) ∂y = + (5.3) ∂s ∂x ∂s ∂y ∂s (1) (1) = σx xs + σy ys.

Similarly,

∂m = σ(1)x + σ(1)y (5.4) ∂t x t y t ∂n = ρ(1)x + ρ(1)y ∂s x s y s ∂n = ρ(1)x + ρ(1)y . ∂t x t y t

Notice that ∂σ(2) ∂σ(2) ∂x ∂σ(2) ∂y ∂σ(2) ∂σ(2) ∂x ∂σ(2) ∂y 1 = = + 0 = = + ∂s ∂x ∂s ∂y ∂s ∂t ∂x ∂t ∂y ∂t

∂ρ(2) ∂ρ(2) ∂x ∂ρ(2) ∂y ∂ρ(2) ∂ρ(2) ∂x ∂ρ(2) ∂y 0 = = + 1 = = + . ∂s ∂x ∂s ∂y ∂s ∂t ∂x ∂t ∂y ∂t Writing the equations above in matrix form gives

 (2) (2)      σx σy xs xt 1 0 = .  (2) (2)      ρx ρy ys yt 0 1

(2) (2) (2) (2) Solving the above equation and setting β = σx ρy − σy ρx yields

   (2) (2)    xs xt 1 ρy −σy 1 0   =     β (2) (2) ys yt −ρx σx 0 1 and therefore 1 (2) 1 (2) xs = β ρy xt = − β σy

1 (2) 1 (2) ys = − β ρx yt = β σx . Consider the ﬁrst Cauchy-Riemann Equation (5.2). Using (5.4) and substituting the Beltrami

38 (k) (k) equations (5.1) for σx and σy we ﬁnd ∂m ∂n βω − = βω σ(1)x + σ(1)y − ρ(1)x + ρ(1)y ∂s ∂t x s y s x t y t (1) (2) (1) (2) (1) (2) (1) (2) = ω σx ρy − σy ρx + ρx σy − ρy σx

(2) (1) (1) (2) (1) (1) (1) (2) (2) = ρy bρx + cρy + ρx aρx + bρy + ρx −aρx − bρy

(1) (2) (2) − ρy bρx + cρy

(1) (2) (1) (2) (1) (2) (1) (2) = ρx ρx (a − a) + ρx ρy (b − b) + ρy ρx (b − b) + ρy ρy (c − c)

= 0. ∂m ∂n Therefore, = . ∂s ∂t

Now consider the second Cauchy-Riemann equation. Substituting the Beltrami equations

(k) (k) (5.1) for σx and σy yeilds ∂m ∂n βω2 + = βω2 σ(1)x + σ(1)y + ρ(1)x + ρ(1)y ∂t ∂s x t y t x s y s 2 (1) (2) (1) (2) (1) (2) (1) (2) = ω −σx σy + σy σx + ρx ρy − ρy ρx

(1) (1) (2) (2) (1) (1) (2) (2) = (bρx + cρy )(aρx + bρy ) + (−aρx − bρy )(bρx + cρy )

2 (1) (2) 2 (1) (2) + ω ρx ρy − ω ρy ρx

(1) (2) (1) (2) 2 2 (1) (2) 2 2 = ρx ρx (ba − ab) + ρx ρy (b − ac + ω ) + ρy ρx (ca − b − ω )

(1) (2) + ρy ρy (cb − bc)

= 0 ∂m ∂n Therefore = − and hence the Cauchy-Riemann equations are satisﬁed. So we have ∂t ∂s shown H(s + it) = σ(1) (h(2))−1(s + it) + iρ(1) (h(2))−1(s + it) = m(s + it) + in(s + it) is analytic. Therefore h(1) ◦ (h(2))−1 ∈ Ψ. Thus the transformation given by the local normalizing parameters σ and ρ deﬁned in two neighborhoods is conformal in the intersection of these neighborhoods. These neighborhoods covering S form a Riemann Surface. Using the Uniformization theorem, S is a simply connected surface that is conformally equivalent to a disk and thus we have that characteristic coordinates are global parameters.

39 5.4 Global Existence of Isothermal Coordinates

Using the same method as in Theorem 5, we will now prove the global existence of

isothermal coordinates represented locally by (u, v). From [7], we see that in a domain of the

(x, y) plane, if the functions E,F,G satisfy a Holder condition then isothermal coordinates

exist.

Theorem 6. Given a smooth surface S such that the functions E,F,G are Holder continuous, then there exists a parameter system of isothermal coordinates, (u(x, y), v(x, y)).

Proof. Every smooth surface can be written locally as the graph over some plane and we

recall the local representation of the metric induced by IR3 to be   2 1 + fx fxfy [gij] =   . 2 fxfy 1 + fy Consider two solutions (u(1), v(1)) and (u(2), v(2)) of the Beltrami equations. In order to have

local solvability we assume that the functions E,F,G must be Holder continuous [7].

As before, Ψ is a conformal structure class such that if g(1), g(2) ∈ Φ have the same

domain V then g(1) ◦ (g(2))−1 ∈ Ψ. So let

g(1)(x, y) = u(1)(x, y) + iv(1)(x, y) and g(2)(x, y) = u(2)(x, y) + iv(1)(x, y).

Furthermore, let

−1 −1 G(s + it) = u(1) g(2) (s + it) + v(1) g(2) (s + it) = m(s + it) + in(s + it)

We will show that ∂m ∂n ∂m ∂n = and = − . ∂s ∂t ∂t ∂s Let m(s + it) = u(1)(x, y) and n(x, y) = v(1)(x, y). Using the chain rule, ∂m = u(1)x + u(1)y ∂s x s y s ∂m = u(1)x + u(1)y ∂t x t y t ∂n = v(1)x + v(1)y ∂s x s y s ∂n = v(1)x + v(1)y . ∂t x t y t

40 Notice that ∂u(2) ∂u(2) ∂x ∂u(2) ∂y ∂u(2) ∂u(2) ∂x ∂u(2) ∂y 1 = = + 0 = = + ∂s ∂x ∂s ∂y ∂s ∂t ∂x ∂t ∂y ∂t

∂v(2) ∂v(2) ∂x ∂v(2) ∂y ∂v(2) ∂v(2) ∂x ∂v(2) ∂y 0 = = + 1 = = + . ∂s ∂x ∂s ∂y ∂s ∂t ∂x ∂t ∂y ∂t Writing the previous equations in matrix form yields

 (2) (2)      ux uy xs xt 1 0 = .  (2) (2)      vx vy ys yt 0 1

(2) (2) (2) (2) Solving this matrix equation and letting δ = ux vy − vx uy , 1 1 x = v(2) x = − u(2) s δ y t δ y (5.5) 1 1 y = − v(2) y = u(2). s δ x t δ x Now we ﬁnd the Beltrami equations for isothermal coordinates

−pq 1 + p2 u(k) = v(k) + v(k) (5.6) x W x W y 1 + q2 −pq −u(k) = v(k) + v(k) (5.7) y W x W y

where W 2 = 1 + p2 + q2. Substituting (5.5) and (5.6) into the Cauchy-Riemann equation ∂m ∂n = we have ∂s ∂t ∂m ∂n δW − = δW u(1)x + u(1)y − v(1)x − v(1)y ∂s ∂t x s y s x t x t (1) (2) (1) (2) (1) (2) (1) (2) = W ux vy − uy vx + vx uy − vy ux

(1) 2 (1) (2) 2 (1) (1) (2) = (−pqvx + (1 + p )vy )vy + ((1 + q )vx − pqvy )vx

2 (2) (2) (1) (2) 2 (2) (1) − ((1 + q )vx − pqvy )vx − (−pqvx + (1 + p )vy )vy

(1) (2) (1) (2) 2 2 = vx vy [−pq + pq] + vy vy [(1 + p ) − (1 + p )]

(1) (2) 2 2 (1) (2) + vx vx [(1 + q ) − (1 + q )] + vy vx [−pq + pq]

= 0.

41 ∂m ∂n 2 2 2 Now consider ∂t + ∂s , recall that W = 1 + p + q and use (5.5)-(5.6), to simplify

∂m ∂n δW 2 + = δW 2 u(1)x + u(1)y + v(1)x + v(1)y ∂t ∂s x t y t x s y s 2 (1) (2) (1) (2) (1) (2) (1) (2) = W −ux uy + uy ux + vx vy − vy vx

(1) 2 (1) 2 (2) (2) = (−pqvx + (1 + p )vy )((1 + q )vx − pqvy )

2 (1) (1) (2) 2 (2) 2 (1) (2) 2 (1) (2) + (−(1 + q )vx + pqvy )(−pqvx + (1 + p )vy ) + W vx vy − W vy vx

2 (1) (2) 2 2 (1) (2) 2 2 (1) (2) = −pq(1 + q )vx vx + p q vx vy + (1 + p )(1 + q )vy vx

2 (1) (2) 2 (1) (2) 2 2 (1) (2) − (1 + p )pqvy vy + (1 + q )pqvx vx − (1 + q )(1 + p )vx vy

2 2 (1) (2) 2 (1) (2) (1) (2) (1) (2) − p q vy vx + pq(1 + p )vy vy + vx vy − vy vx

(1) (1) 2 2 (1) (2) 2 2 2 2 2 = vx vx [−pq(1 + q ) + pq(1 + q )] + vx vy [p q − (1 + q )(1 + p ) + W ]

(1) (2) 2 2 2 2 2 (1) (2) 2 2 + vy vx [(1 + p )(1 + q ) − p q − W ] + vy vy [pq(1 + p ) − pq(1 + p )]

= 0.

Thus the Cauchy-Riemann equations are satisﬁed. Therefore the transformation G : G(s +

it) = u(1) g(2)−1 (s + it) + v(1) g(2)−1 (s + it) = m(s + it) + in(s + it) deﬁned in two

neighborhoods is conformal in the intersection of these neighborhoods. These neighborhoods

form a Riemann Surface and we can now apply the Uniformization theorem to ﬁnd that isothermal coordinates are global parameters, as we did with characteristic coordinates.

5.5 Quasiconformal Mappings

Informally, one can think of a conformal map as one that maps inﬁnitesimal circles

to infinitesimal circles. A quasiconformal map takes infinitesimal circles to infinitesimal

ellipses. In the following proof, we will show that we have a quasiconformal mapping from the isothermal parameter domain to the characteristic parameter domain. For a mapping w(z) = u(x, y) + iv(x, y) having continuous partial derivatives and a non-vanishing Jacobian then the informal description of a quasiconformal mapping can be expressed by any of these

42 three equivalent diﬀerential equations as given in [10]

2 max |wx cos θ + wy sin θ| ≤ Q(uxvy − uyvx) 0≤θ≤2π 2 2 2 2 ux + uy + vx + vy ≤ 2K(uxvy − uyvx), (5.8)

|wz¯| ≤ k|wz|.

Here Q ≥ 1,K ≥ 1, 0 ≤ k < 1 and the constants are linked by the following:

1 1 Q − 1 K = Q + , k = . 2 Q Q + 1

From ([1]), we note that if T is k-quasiconformal, then T −1 is also k-quasiconformal. Consider

the following equation from [14] of the form

a(x, y, z, p, q)φxx + 2b(x, y, z, p, q)φxy + c(x, y, z, p, q)φyy = 0 (5.9)

2 with ac − b = 1 with p = φx and q = φy. It is shown in lemma 8 of [14] that

(1 + p2)a 2bpq (1 + q2)c + + ≤ 2 (5.10) W W W

for each (x, y) ∈ Ω, where ≥ 1 with = 1 if and only if (5.9) is the minimal surface

equation.

Theorem 7. Let E be the unit disk parametrized by characteristic coordinates (σ, ρ) and isothermal coordinates (u, v). Let K = E → Ω where K is a one-to-one and onto mapping such that K(u, v) = (x(u, v), y(u, v)). Let Ke = E → Ω be a one-to-one and onto mapping such that Ke(σ, ρ) = (˜x(σ, ρ), y˜(σ, ρ)). If T = K˜ −1 ◦ K, then T is quasiconformal.

Proof. We will outline the proof before we begin. In Step 1, we perform a change of variables for σx, σy, ρx and ρy. Step 2 will then consist of ﬁnding the Beltrami equations in characteristic coordinates in terms of isothermal coordinates. Step 3 will be algebraic calculations to show in Step 4 that those Beltrami equations satisfy the deﬁnition of quasiconformal coordinate systems from [10].

43 Step 1: We begin by using the chain rule to write

∂σ ∂σ ∂u ∂σ ∂v = + ∂x ∂u ∂x ∂v ∂x

= σuux + σvvx

∂σ ∂σ ∂u ∂σ ∂v = + ∂y ∂u ∂y ∂v ∂y

= σuuy + σvvy.

In matrix form       σx ux vx σu   =     . σy uy vy σv We solve the matrix equation and complete a change of variables       σu 1 vy −vx σx = .   u v − u v     σv x y y x −uy ux σy

Rewriting and let δ = uxvy − uyvx to obtain

v v σ = y σ − x σ u δ x δ y −u u σ = y σ + x σ . (5.11) v δ x δ y

Similarly we can perform a change of variables for ρx and ρy:

∂ρ ∂ρ ∂u ∂ρ ∂v = + ∂x ∂u ∂x ∂v ∂x

= uxρu + vxρv (5.12) ∂ρ ∂ρ ∂u ∂ρ ∂v = + ∂y ∂u ∂y ∂v ∂y

= uyρu + vyρv.

Then using a similar matrix equation as above, we can ﬁnd corresponding equations for ρu and ρv to be

v v ρ = y ρ − x ρ u δ x δ y −u u ρ = y ρ + x ρ . (5.13) v δ x δ y

44 Step 2: We recall the Beltrami equations (3.10) for characteristic coordinates (ρ, σ) in terms of Cartesian coordinates (x, y) to be bρ + cρ aρ + bρ σ = x y and σ = − x y x ω y ω

2 2 where ω = ac − b . We can substitute the Beltrami equations for σx and σy into (5.11) and then replace ρx and ρy with (5.12) to obtain v bρ + cρ v aρ + bρ σ = y x y − x − x y u δ ω δ ω

v b v c v a v b = y ρ + y ρ + x ρ + x ρ ωδ x ωδ y ωδ x ωδ y

v b v c v a v b = y (u ρ + v ρ ) + y (u ρ + v ρ ) + x (u ρ + v ρ ) + x (u ρ + v ρ ) ωδ x u x v ωδ y u y v ωδ x u x v ωδ y u y v

v u b + v u c + v u a + v u b v v b + v v c + v v a + v v b = y x y y x x x y ρ + x y y y x x x y ρ ωδ u ωδ v

2 2 v u a + (v u + v u )b + v u c v a + 2vxvyb + v c = x x y x x y y y ρ + x y ρ . ωδ u ωδ v

Similarly for σv −u u σ = y σ + x σ v δ x δ y

−u bρ + cρ u aρ + bρ = y x y + x − x y δ ω δ ω

u b u c u a u b = − y (u ρ + v ρ ) + y (u ρ + v ρ ) + x (u ρ + v ρ ) + x (u ρ + v ρ ) ωδ x u x v ωδ y u y v ωδ x u x v ωδ y u y v

u u b + u u c + u u a + u u b v u b + v u c + v u a + v u b = − y x y y x x x y ρ − x y y y x x y x ρ ωδ u ωδ v

2 2 u a + 2uxuyb + u c v u a + (u v + u v )b + u v c = − x y ρ − x x y x x y y y ρ . ωδ u ωδ v

45 Thus the Beltrami equations in characteristic coordinates (σ, ρ) in terms of isothermal coordinates (u, v) are:

2 2 v u a + (v u + v u )b + v u c v a + 2vxvyb + v c σ = x x y x x y y y ρ + x y ρ (5.14) u ωδ u ωδ v 2 2 u a + 2uxuyb + u c v u a + (u v + u v )b + u v c −σ = x y ρ + x x y x x y y y ρ , (5.15) v ωδ u ωδ v

2 2 where ω = ac − b and δ = uxvy − uyvx. Step 3: From [10], every solution w = φ + iψ of an elliptic system

σu = γ11ρu + γ12ρv

−σv = γ21ρu + γ22ρv

will satisfy (5.8) if the system is uniformly elliptic, that is if a12, and

2 (γ12 + γ21) 0 < 2 < constant. (5.16) 4γ12γ21 − (γ11 + γ22)

One can then conclude that the coordinate systems are quasiconformal and the constant Q of quasiconformality depends only on the constant in (5.16). We need to show that the system (5.14) satisﬁes (5.16). Utilizing (5.14)-(5.15), we ﬁnd the γij’s in (5.16)

v u a + (v u + v u )b + v u c γ = x x y x x y y y , 11 ωδ

2 2 v a + 2vxvyb + v c γ = x y , (5.17) 12 ωδ

2 2 u a + 2uxuyb + u c γ = x y , 21 ωδ

v u a + (u v + u v )b + u v c γ = x x y x x y y y . 22 ωδ

46 Now we substitute (5.17) into (5.16) and simplify

2 2 2 2 2 (v + u )a + 2b(vxvy + uxuy) + (u + v )c (γ + γ )2 = x x y y 12 21 ω2δ2 2 2 2 2 2 2 2 2 4v u a + 8v uyuxab + 4v u ac + 8vxvyu ab + 16uxuyvxvyb 4γ γ = x x x x y x (5.18) 12 21 ω2δ2 2 2 2 2 2 2 2 8vxvyu bc + 4v u ac + 8uxuyv bc + 4v u c + y y x y y y ω2δ2 [2u v a + 2b(u v + u v ) + 2u v c]2 (γ + γ )2 = x x y x x y y y (5.19) 11 22 ω2δ2 2 2 2 2 2 2 2 2 4u v a + 4b (uyvx + uxvy) + 4u v c + 8abuxvx(uyvx + uxvy) = x x y y ω2δ2 8acu u v v + 8bcu v (u v + u v ) + x y x y y y y x x y . ω2δ2

Using (5.18), we simplify to ﬁnd the denominator in (5.16)

4(ac − b2)(u v − u v )2 4γ γ − (γ + γ )2 = x y y x (5.20) 12 21 11 22 ω2δ2 4ω2δ2 = = 4. ω2δ2

Since (u, v) represent an isothermal coordinate system, we can utilize the Beltrami equations

−pq 1 + p2 u = v + v (5.21) x W x W y 1 + q2 −pq −u = v + v (5.22) y W x W y where W = p1 + p2 + q2.

47 Substituting (5.21)-(5.22), we can further simplify parts of (5.16).

δ = uxvy − uyvx −pq 1 + p2 1 + q2 −pq = v + v v + v + v v W x W y y W x W y x 2 2 2 (pvy − qvx) + v + v = x y W

−pq 1 + p2 1 + q2 pq u u + v v = v + v − v + v + v v x y x y W x W y W x W y x y pq = (qv − pv )2 + v2 + v2 W 2 x y x y (5.23) −pq 1 + p2 2 v2 + u2 = v2 + v + v x x x W x W y 1 + p2 = (qv − pv )2 + v2 + v2 W 2 x y x y

1 + q2 −pq 2 u2 + v2 = v + v + v2 y y W x W y y 1 + q2 = (qv − pv )2 + v2 + v2 . W 2 x y x y

Using (5.23), we can ﬁnd

2 2 2 2 2 (v + u )a + 2b(vxvy + uxuy) + (u + v )c (γ + γ )2 = x x y y 12 21 ω2δ2 1 + p2 pq = (qv − pv )2 + v2 + v2 a + 2b (qv − pv )2 + v2 + v2 ω2δ2W 4 x y x y ω2δ2W 4 x y x y 1 + q2 + (qv − pv )2 + v2 + v2 c ω2δ2W 4 x y x y 1 h i = (qv − pv )2 + v2 + v22 (1 + p2)a + 2bpq + (1 + q2)c2 (5.24) W 4ω2δ2 x y x y 1 = (1 + p2)a + 2bpq + (1 + q2)c2 . (5.25) W 4ω2

Substituting (5.20) and (5.25) into (5.16), and using (5.23) to simplify, we obtain

2 1 2 2 2 (γ12 + γ21) W 4ω2 [(1 + p )a + 2bpq + (1 + q )c] . 2 = (5.26) 4γ12γ21 − (γ11 + γ22) 4 1 = (1 + p2)a + 2bpq + (1 + q2)c2 . (5.27) 4ω2W 2

48 Step 4: Using the hypothesis given in the statement of Theorem 7 and given ≥ 1, then

2 2 2 2 2 (γ12 + γ21) 1 (1 + p )a + 2bpq + (1 + q )c 4 2 2 = 2 ≤ = . 4γ12γ21 − (γ11 + γ22) 4(ac − b ) W 4

Lemma 8 in [14], states if

1 + q2 −pq 1 + p2 α = , β = , γ = W W W

and αγ − β2 = 1 with aγ + cα − 2bβ = 2 where ≥ 1 then,

(1 + p2)a 2bpq (1 + q2)c 2 ≤ + + . W W W

Therefore, 2 2 (γ12 + γ21) 4 2 2 ≤ 2 ≤ = . 4γ12γ21 − (γ11 + γ22) 4 Recall from [10] that we require Q ≥ 1 and “the constant Q of quasiconformality depends

only on the constant” in (5.16) which is equal to 2 as shown. Therefore, we conclude that the mapping T from the parameter domain in isothermal coordinates (u, v) to the domain parameterized by characteristic coordinates (σ, ρ) is quasiconformal.

Note that the inverse mapping of T which maps from the domain in characteristic

coordinates into the domain parameterized by isothermal coordinates is also quasiconformal ([1]).

49 CHAPTER 6

APPLICATIONS

Let α ∈ (0, π) and Ω be a bounded, locally Lipschitz domain in IR2 with O = (0, 0) ∈

∂Ω such that θ = ±α are the tangent rays to ∂Ω at O and {r(cos(θ), sin(θ)) : r ≥ 0, −α ≤

θ ≤ α} is the cone obtained by blowing up Ω about O. Let

2 X 2 aij(x, f, ∇f)Dijf = h(x, f, ∇f), f ∈ C (Ω), (6.1) i,j=1 be a speciﬁc equation of mean curvature type on Ω and let φ be a piecewise continuous

function from ∂Ω to IR; we will consider f to be a ﬁxed bounded solution of (6.1) which

satisﬁes f = φ (a.e.) on ∂Ω. Since our interest is in the limiting behavior of f at O, we shall

assume f is discontinuous at O.

We may parametrize the graph of f in isothermal coordinates using the techniques

in [23] and [25]. The idea in these papers is to estimate the modulus of continuity of the isothermal parametrization X : E → IR3 (E = {(u, v): u2 + v2 < 1}) of

3 Sf = {(x, f(x)) ∈ IR : x ∈ Ω}

using Courant’s Lemma 3.1 ([9]) and the comparison principle and to prove that this

parametrization extends continuously to E. In the case here, these techniques yield an

3 isothermal, parametric description Y : E → IR of the closure S of Sf = {(x, f(x)) : x ∈ Ω},

Y (u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ E,

such that

(i) Y ∈ C2(E : IR3) ∩ W 1,2(E : IR3).

(ii) Y is a homeomorphism of E onto Sf .

(iii) Y maps ∂E onto {(x, f(x)) : x ∈ ∂Ω} ∪ ({O} × [z1, z2]) , where z1 = lim infΩ3x→O f(x)

and z2 = lim supΩ3x→O f(x).

50 (iv) Y is conformal on E: Yu · Yv = 0, |Yu| = |Yv| on E. ˜ (v) Let H(u, v) = H(Y (u, v)) denote the prescribed mean curvature of Sf at Y (u, v). Then ˜ 4Y := Yuu + Yvv = HYu × Yv. (vi) Y ∈ C0(E).

(vii) Writing G(u, v) = (x(u, v), y(u, v)),G(cos(t), sin(t)) moves clockwise about ∂Ω as t

increases, 0 ≤ t ≤ 2π, and G is an orientation reversing homeomorphism from E onto Ω; G

maps E onto Ω and, if f is continuous at O, then G is a homeomorphism from E onto Ω.

We note that when h 6= 0, the proof of (vi) depends on an extension of Theorem 2

of [21] whose correctness Professor Jin has veriﬁed. Since we assumed f is discontinuous

at O, the map G in (vii) cannot be a homeomorphism from E onto Ω. In fact, if G−1(O)

is a single point of ∂E, then f must be continuous at O; since this conclusion is false,

there must be a nonempty, open arc σ0 of ∂E whose closure σ is mapped by G to O. Now set B = {(u, v): u2 + v2 < 1, v > 0}, ∂0B = {(u, v): u2 + v2 = 1, v > 0} ∪ {(±1, 0)}

and ∂00B = {(u, v): u2 + v2 < 1, v = 0} and then let ψ be the conformal map from B

0 00 to E which maps ∂ Ω to ∂Ω \ σ0 and ∂ Ω to σ0. Finally, deﬁne X = Y ◦ ψ. Notice that X ∈ C2(B : IR3) ∩ W 1,2(B : IR3) and satisﬁes (i)-(vii) above (with E replaced by B and ∂E

by ∂B.)

If (6.1) is the prescribed mean curvature equation with bounded mean curvature, then

one can use (v) to show that X ∈ C1,δ(E ∪ ∂00B : IR3) and the gradient of X has a speciﬁc

00 ﬁrst-order expansion at any point of ∂ B at which Xu and Xv vanish; this information allows one to conclude that the radial limits of f at O,

Rf(θ) = lim f(r cos(θ), r sin(θ)), r↓0 exist for all (or almost all) θ ∈ (−α, α). In the general case where (6.1) is an equation of mean curvature type, we have no speciﬁc information about the mean curvature of Sf and, in particular, cannot expect the mean curvature of Sf to be bounded. Therefore, we cannot use exactly the same techniques used in [25] and [23] to obtain the regularity of X(u, v) on ∂00B

51 which is required to establish the existence of the radial limits of f at O. An alternative is to

reparametrize Sf in characteristic coordinates corresponding to equation (6.1) and attempt to use the results of section 3.2 to obtain the necessary regularity.

Let ω be the quasiconformal homeomorphism from B to B which maps ∂0B to ∂0B

00 00 and ∂ B to ∂ B such that X ◦ ω is a characteristic coordinate parametrization of Sf . (Here the results of chapters 4 and 5 are used to establish the existence of the parametrizations and Theorem 5 and properties of quasiconformal maps are used to show that ω is a homeomorphism from ∂B to ∂B.) Now deﬁne Z : B → S by

Z(σ, ρ) = X(ω(σ, ρ)), (σ, ρ) ∈ B,

3 where S is the closure in IR of Sf . Now the results of section 3.2 hold for Z; and we obtain the following result.

Theorem 8. Suppose (6.1) is an equation of mean curvature type whose normal form (3.15)

-(3.17) in characteristic coordinates (σ, ρ) satisﬁes |∆x| ≤ C|Dx|2 for some constant C; that is

2 2 2 2 2 2 |xσσ + xρρ| ≤ C(|xσ| + |xρ| + |yσ| + |yρ| + |zσ| + |zρ| )

2 2 2 2 2 2 |yσσ + yρρ| ≤ C(|xσ| + |xρ| + |yσ| + |yρ| + |zσ| + |zρ| )

2 2 2 2 2 2 |zσσ + zρρ| ≤ C(|xσ| + |xρ| + |yσ| + |yρ| + |zσ| + |zρ| ).

Then the radial limits Rf(θ) of f at 0 exist for almost all θ ∈ [−α, α].

Conjecture 1. Z ∈ C1,δ(E ∪ ∂00B : IR3) and Rf(θ) exists for almost every θ ∈ (−α, α).

52 CHAPTER 7

CONCLUSION

The motivating force of this thesis was to create some of the infrastructure required

to investigate the behavior at corners of solutions of boundary value problems for equation of mean curvature type. Further research can utilize the connections that were established between isothermal and harmonic coordinates as well as the relationship between isothermal

and characteristic coordinates. We showed that for a two-dimensional manifold, isothermal coordinates are harmonic, though this does not hold in higher dimensions unless the manifold is conformally ﬂat.

Next, we examined characteristic coordinates which arise in partial diﬀerential equa-

tions and focused on elliptic type. We derived the normal form for an elliptic partial differential equation and the corresponding Beltrami equation. Dr. Lancaster calculated and

simpliﬁed the Laplacian of the components of the position vector x with respect to the vari-

ables ρ and σ and plans to use this for a future paper. In section 3.3, given a function f we found the local representation for the metric in characteristic coordinates with respect to Cartesian coordinates by solving a system of equations. This representation of the metric in characteristic coordinates allowed us to reach the conclusion that if the graph of the function was minimal, then the characteristic coordinates are isothermal. We then used the local representation of the metric and an example to show that if the surface is not minimal then the characteristic coordinates are not a harmonic coordinate system.

In chapter 4, we introduced equations of minimal surface type and equations of mean curvature type. We then showed in the proof of Theorem 4 that the form for the two dimensional equation of mean curvature type given by Simon in [29] is equivalent to the structure conditions as given by Finn in [14]. We begin chapter 5 by introducing a structure class, Riemann surfaces and the Uniformization theorem. We use these ideas to show the global existence of characteristic and isothermal coordinates on a surface, based on a footnote

53 in [10]. Following the existence proof, we then deﬁne quaisconformal maps and note that if a mapping is quasiconformal, then its inverse is also quasiconformal. That allowed us to prove that given a domain parameterized by characteristic coordinates and a domain parameterized by isothermal coordinates, the composition map between them is quasiconformal.

The application section shows how the results of this paper can be utilized to further

Dr. Lancaster’s studies of the behavior of the radial limits when examining equations of mean curvature type. We were able to use the existence of isothermal and characteristic parameterizations and then the properties of quasiconformal maps to gain the result of Theorem 8.

54 REFERENCES

55 LIST OF REFERENCES

[1] Lars Ahlfors and Leo Sario. Riemann Surfaces. Princeton University Press, 1960.

[2] M Berger. A Panoramic View of Riemannian Geometry. Springer, 2003.

[3] L Bers, F John, and M Schechter. Partial Diﬀerential Equations. Number v. 3. American Mathematical Society, 1964.

[4] L. Bers, I. Kra, and B. Maskit. Selected Works of Lipman Bers: Papers on Complex Analysis. American Mathematical Society, 1998.

[5] K.S. Bhamra. Partial Diﬀerential Equations: An Introductory Treatment with Applica- tions. Prentice-Hall Of India Pvt. Ltd., 2010.

[6] WM Boothby. An Introduction to Diﬀerentiable Manifolds and Riemannian Geometry. Number nos. 1-2. Academic Press, 1975.

[7] Shiing-Shen Chern. An elementary proof of the existence of isothermal parameters on a surface. Proceedings of the American Mathematical Society, 6(5):771–782, Oct 1955.

[8] B Chow and D Knopf. The Ricci Flow: An Introduction. American Mathematical Society, 2004.

[9] R Courant. Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Springer, 1977.

[10] Richard Courant and D Hilbert. Methods of Mathematical Physics. John Wiley & Sons, Incorporated, 1989.

[11] Dennis DeTurck and Jerry Kazdan. Some regularity theorems in riemannian geometry. Annales Scientifiques de l’Ecole´ Normale Supérieure. QuatrièmeSérie, 14(3):249–260, 1981.

[12] Manfredo Do Carmo. Diﬀerential Geometry of Curves and Surfaces. Prentice-Hall, Inc, 1976.

[13] MP do Carmo. Riemannian geometry:. Birkh¨auser,1992.

[14] Robert Finn. On equations of minimal surface type. The Annals of Mathematics, 60(3):397–416, November 1954.

[15] J Foster and JD Nightingale. A Short Course in General Relativity. Springer, 2006.

56 LIST OF REFERENCES (continued)

[16] Theodore W. Gamelin. Complex Analysis. Undergraduate Texts in Mathematics. Birkh¨auser,2001.

[17] D Gilbarg and NS Trudinger. Elliptic Partial Diﬀerential Equations of Second Order. Springer, 2001.

[18] NK Ibragimov. Transformation Groups applied to Mathematical Physics. D. Reidel, 1985.

[19] T Iwaniec and G Martin. The Beltrami Equation. Number 893. American Mathematical Society, 2008.

[20] Thalia Jeﬀres. Lecture notes: Diﬀernetial geometry. 2009-2010.

[21] Z. Jin and K. Lancaster. Behavior of solutions for some dirichlet problems near reentrant corners. Indiana Uni. Math J., (46):827–862, 1997.

[22] Erwin Kreyszig. Diﬀerential Geometry, volume 11 of Mathematical Expositions. Uni- versity of Toronto Press, 1959.

[23] Kirk Lancaster. Boundary behavior of a nonparametric minimal surface in r3 at a nonconvex point. Analysis, 5(1-2):61–69, 1985.

[24] Kirk Lancaster and David Siegal. Behavior of a bounded non-parametric h-surface near a reentrant corner. Z. Anal. Anwendungen, 15(4):819–850, 1996.

[25] Kirk Lancaster and David Siegel. Existence and behavoior of the radial limits of a bounded capillary surface at a corner. Paciﬁc Journal of Mathematics, 176(1):165–194, 1996.

[26] F. Morgan. Riemannian Geometry: A Beginner’s Guide. A.K. Peters, 1998.

[27] R. Osserman. A Survey of Minimal Surfaces. Dover Publications, 2002.

[28] J Pleba´nskiand A Krasi´nski. An Introduction to General Relativity and Cosmology. Cambridge University Press, 2006.

[29] Leon Simon. Equations of mean curvature type in 2 independent variables. Paciﬁc Journal of Mathematics, 69(11):245, 1977.

57 APPENDIX

58 APPENDIX A

In this appendix we provide some of the global deﬁnitions and local derivations of

terms given throughout the thesis.

A.1 Connection and Covariant Derivative

Let X (M) be the set of all vector fields of class Ck on M. We defined a connection that is compatible with the metric in (1.7) to be the Levi Civita connection. Now we give the definition for an affine connection ∇. On a differentiable manifold M as a mapping we define an affine connection ∇ : X (M) × X (M) → X (M) which satisfies:

(i) ∇fX+gY Z = f∇X Z + g∇Y Z,

(ii) ∇X (Y + Z) = ∇X Y + ∇X Z,

(iii) ∇X (fY ) = f∇X Y +X(f)Y, in which X,Y,Z ∈ X (M) and f, g are real valued functions of class C∞ deﬁned on M.

Let M be a diﬀerentiable manifold with an aﬃne connection ∇. If γ is a curve and

V a vector ﬁeld along γ, then there exists a covariant derivative which is a vector ﬁeld

DV dt along γ with the following properties:

D DV DW (i) Linear over addition: dt (V + W ) = dt + dt

D Df DV (ii) Product rule: dt (fV ) = V dt + f dt ,

(iii) If V is the restriction to γ of a vector ﬁeld deﬁned on M, (i.e., there exists Y such that

DV V (t) = X(c(t))) then dt = ∇γ0(t)X.

In chapter 1, we found a local coordinate expression (1.8) for the covariant derivative.

To derive this local expression we ﬁrst choose a parameterization x : U ⊂ IRn → M where

59 APPENDIX A (continued)

Pn ∂ the curve γ(t) = (x1(t), ..., xn(t)). Writing the vector field v as v(t) = vi(t) |γ(t), we i=1 ∂xi DV find dt in local coordinates as given in (1.8) using the definition of covariant derivative as given above and [20]:

n DV X D ∂ = (v (t) ) [by (i)] dt dt i ∂x i=1 i n X dvi ∂ D ∂ = + v (t) [by (ii)] dt ∂x i dt ∂x i=1 i i n X dvi ∂ ∂ = + v (t)∇ 0 [by (iii)] dt ∂x i γ (t) ∂x i=1 i i n X dvi ∂ ∂ = + vi(t)∇Pn x0 (t) ∂ dt ∂x j=1 j ∂xj ∂x i=1 i i n " n # X dvi ∂ X 0 = + vixi(t)∇ ∂ ∂∂xi dt ∂x ∂xj i=1 i j=1 n " n # X dvi ∂ X dxi k ∂ = + vi Γij [by (1.9)]. dt ∂xi dt ∂xk i=1 j,k=1

A.2 Mean Curvature

The mean curvature H of a surface at the point under consideration is the arith- metic mean of the principal curvatures ki,

1 H = (κ + κ ). n 1 2

The maximum and minimum of the normal curvature at a given point on a surface are called the principal curvatures κ1, κ2. Note that the principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. [26] A surface with zero mean curvature is called a minimal surface.

Let w be a unit tangent vector of a regular surface M ⊂ IR3 at p . Then the normal curvature of M in the direction w is

κ(w) = S(w) · w

60 APPENDIX A (continued) where S is the shape operator. Consider a parameterization p = x(u, v) with v = ax(u, v) + bx(u, v). The normal curvature in the direction v is

ea2 + 2fab + gb2 κ(v) = Ea2 + 2F ab + Gb2 where E,F,G are the coefficients of the first fundamental form and e, f, g and are the coefficients of the second fundamental form. The second fundamental form can be written explicitly as

edu2 + 2fdudv + gdv2 where

2 X ∂ xi e = X i ∂u2 i 2 X ∂ xi f = X i ∂u∂v i 2 X ∂ xi g = X i ∂v2 i and Xi are the direction cosines of the surface normal.