Quasihyperbolic Distance, Pointed Gromov-Hausdorff Distance, and Bounded Uniform Convergence

Quasihyperbolic Distance, Pointed Gromov-Hausdorﬀ Distance, and Bounded Uniform Convergence

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Department of Mathematical Sciences College of Arts and Sciences University of Cincinnati, May 2019

Author: Abigail Richard Chair: Dr. David Herron Degrees: B.A. Mathematics, 2011, Committee: Dr. Michael Goldberg University of Indianapolis Dr. Nageswari Shanmugalingam M.S. Mathematics, 2013, Dr. Marie Snipes Miami University Dr. Gareth Speight Abstract

We present relationships between various types of convergence. In particular, we examine several types of convergence of sets including Hausdorff convergence, Gromov-Hausdorff convergence, etc. In studying these relationships, we also gain a better understanding of the necessary conditions for certain conformal metric distances to pointed Gromov- Hausdorff converge. We use pointed Gromov-Hausdorff convergence to develop approximations of the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances via spaces with only finitely many boundary points.

I would like to thank God, my family, Dr. Sourav Saha, Dr. David Herron, Dr. Marie Snipes, Dr. David Minda, my committee, Dr. Lakshmi Dinesh, Dr. Fred Ahrens, and Dr. Crystal Clough for all of their help and support while I have been a graduate student at UC. I would also like to thank and recognize the Taft Center for their ﬁnancial support.

iv Contents

Abstract ii

Acknowledgments iv

1 Introduction 1

2 Background 5 2.1 Preliminary Notation ...... 5 2.2 Conformal Metrics ...... 7 2.2.1 Quasihyperbolic Metric ...... 8 2.2.2 Ferrand Metric ...... 13 2.2.3 Kulkarni-Pinkall-Thurston Metric ...... 14 2.2.4 Hyperbolic Metric ...... 15 2.3 Convergence ...... 16 2.3.1 Hausdorff Distance ...... 16 2.3.2 Gromov-Hausdorff Distance ...... 17 2.3.3 Pointed Gromov-Hausdorff Distance ...... 19 2.4 Chordal and Inversive Distances ...... 22 2.4.1 Chordal Distance ...... 23 2.4.2 Inversive Distance ...... 24 2.4.3 Lipschitz Relationships ...... 26

3 Pointed Gromov-Hausdorﬀ Convergence of Conformal Metrics 31 3.1 Pointed Gromov-Hausdorﬀ Convergence ...... 32 3.2 Bounded Uniform Convergence ...... 36

4 Convergence of Sets 38 4.1 Hausdorﬀ Convergence Relationships ...... 38 4.2 Gromov-Hausdorﬀ Convergence ...... 42

v 4.3 Carath´eodory Convergence ...... 48

5 Applications 53 5.1 Quasihyperbolic Distance ...... 54 5.2 SNCF Distance ...... 57 5.3 Ferrand, Kulkarni-Pinkall-Thurston, and Hyperbolic Distances ...... 61

6 More General Results 67 6.1 An Alternative Approximation ...... 67 6.2 Gromov-Hausdorﬀ Convergence of General Complete Metric Spaces . . . . . 71

7 Quasihyperbolic Distance in Euclidean Domains 77 7.1 Quasihyperbolic Isometries ...... 77 7.2 Bounded Uniform Convergence of the Quasihyperbolic Metric in Euclidean Domains ...... 83

Bibliography 88

vi Chapter 1

Introduction

Quasihyperbolic geometry has been studied extensively since it was ﬁrst introduced by

Gehring and Palka in 1976, and since then, it has played an important role in metric space analysis. Additionally, it has proven useful in a wide range of areas such as quasiconformal mappings, function theory, the study of Banach spaces, and many other forms of analysis. Quasihyperbolic geometry is a non-Euclidean geometry that can be seen as a generalization of the well-known and highly used hyperbolic geometry. In fact, the quasihyperbolic and hyperbolic distances are equal in Euclidean half-spaces. However, even in some simple spaces, the hyperbolic metric can be hard to compute. In contrast, the quasihyperbolic metric is often easier to calculate than the hyperbolic metric. Fur- ther, although the standard hyperbolic metric is typically deﬁned and discussed only in the context of certain Euclidean domains, the quasihyperbolic metric is deﬁned in even

n more general metric spaces than R . Although the quasihyperbolic metric is sometimes easier to calculate than the hyperbolic metric, quasihyperbolic distance can indeed be challenging to compute in general spaces. Consequently, it is worthwhile to approximate quasihyperbolic distances in general spaces by quasihyperbolic distances in “simpler” spaces. In Chapter 5, we

n do this for proper domains in R via spaces that have only ﬁnitely many boundary

1 points. In addition to constructing such an approximation for quasihyperbolic distance, we also provide approximations for three other non-Euclidean distances: Ferrand distance, Kulkarni-Pinkall-Thurston distance, and hyperbolic distance.

In determining approximations of non-Euclidean distances, we are led to explore relationships between various forms of convergence, one of which is convergence of Hausdorff distance. Hausdorff distance is a way to measure how close two subsets of a metric space are to each other, by measuring the smallest neighborhoods of each set covering the other set. We examine Hausdorff distance in both the chordalization and inversion of metric

n n n spaces. In R , the chordalization is the one-point compactification Rˆ := R ∪ {∞}, n and the inversion of R is determined via reflection through the unit sphere. Thus the n n n inversion of R is the space {∞} ∪ (R \{0}), as this is the image of R after reflection through the unit sphere. We determine various relationships between Hausdorff convergence in the chordalization and inversion of metric spaces.

Although Hausdorﬀ distance allows us to measure the similarity of two sets within one metric space, it does not permit us to compare two separate metric spaces. Gromov-

Hausdorff distance is a tool that allows us to do this, by measuring the Hausdorff distance between isometric embeddings of two metric spaces into one single space. We demonstrate relationships between Gromov-Hausdorff convergence and Hausdorff convergence in Chapter 4. For instance, we establish the following relationship between Gromov-

n Hausdorﬀ distance and Hausdorﬀ distance in Rˆ .

n Theorem 1.0.1. Suppose Ai,A (R are non-empty closed sets so that (Ai) converges to A with respect to Gromov-Hausdorﬀ distance. Then there are Euclidean isometries

n n fi : R → R so that (fi(Ai)) converges to A with respect to Hausdorﬀ distance in the n chordalization Rˆ .

We also study the notion of pointed Gromov-Hausdorff distance, which is a helpful tool for investigating how “close” two metric spaces are near specific points. In this context, we discuss the pointed Gromov-Hausdorff distance between pointed metric spaces. A

2 pointed metric space is a metric space (X, d) along with a point x ∈ X. We denote such a pointed space as (X, x, d). In Chapter 3, we prove a relationship between convergence of metrics in bounded sets and pointed Gromov-Hausdorff convergence. In Chapter 5, we use this relationship to demonstrate connections between pointed Gromov-Hausdorff distance and Hausdorff distance. For instance, in Theorem 5.3.6, we prove the portion of Theorem 1.0.2 below regarding the Ferrand and Kulkarni-Pinkall-Thurston distances.

We note that the calculation of Hausdorﬀ distance depends on the metric space distance.

Hence we state the following results in terms of Euclidean distance. However, in Chapters

5 and 6, we demonstrate generalizations of these results for the quasihyperbolic metric in non-Euclidean metric spaces as well.

n Theorem 1.0.2. Suppose Ai,A (R are non-empty and closed so that (Ai) converges n n to A with respect to Hausdorﬀ distance in the chordalization Rˆ . For each x ∈ R \ A, n n there exists N ∈ N so that whenever i ≥ N, x ∈ R \ Ai, and (R \ Ai, x)i≥N pointed n Gromov-Hausdorﬀ converges to (R \ A, x) with respect to the quasihyperbolic, Ferrand, and Kulkarni-Pinkall-Thurston distances.

The following corollary is a combination of our results from Corollary 5.3.7 and Corol- lary 5.1.2. In Chapter 5, we use Theorem 1.0.2 and Theorem 1.0.1 to prove the next result.

n Corollary 1.0.3. Suppose Ai,A (R are non-empty closed sets so that (Ai) converges n to A with respect to Gromov-Hausdorﬀ distance. Then for each x ∈ R \ A, there n n exists a sequence (xi) in R so that (R \ Ai, xi) pointed Gromov-Hausdorﬀ converges to n (R \ A, x) with respect to the quasihyperbolic, Ferrand, and Kulkarni-Pinkall-Thurston distances.

While investigating the chordalization of metric spaces, we also prove the following result, which is a special case of our Proposition 5.1.4 combined with Theorem 1.0.2.

3 n Theorem 1.0.4. Suppose A is a closed set in R . Then there exists a sequence of finite sets (Ai) which converge to A with respect to Hausdorff distance in the chordalization ˆ n n R . Further (Ai) can be chosen so that for each i, Ai ⊂ A and for each x ∈ R \ A, the n n sequence (R \ Ai, x) pointed Gromov-Hausdorff converges to (R \ A, x) with respect to the quasihyperbolic, Ferrand, and Kulkarni-Pinkall-Thurston distances.

In Chapter 5, we prove a result for the hyperbolic metric similar to that of Theo- rem 1.0.2, and we also generalize Theorem 1.0.2 to the quasihyperbolic metric in the context of complete length spaces. Theorem 1.0.4 provides an approximation of the quasihyperbolic metric via ‘nice’ spaces with only finitely many boundary points, while The- orem 1.0.2 establishes a relationship between Hausdorff and pointed Gromov-Hausdorff convergence. By examining properties of quasihyperbolic isometries, we develop a further understanding of pointed Gromov-Hausdorff convergence of quasihyperbolic metric spaces in Chapter 7. In Chapter 6, we show that Theorem 1.0.4 does not necessarily hold in spaces which are not length spaces. As an additional result in Chapter 6, we provide an alternative construction for a sequence of spaces with finitely many boundary points, and prove that this construction still yields pointed Gromov-Hausdorff convergence of the quasihyperbolic metric.

4 Chapter 2

Background

In this chapter, we provide necessary background and notation.

2.1 Preliminary Notation

Throughout this paper, (X, d) denotes a metric space. For each x ∈ X and ε > 0,

Bd(x, ε) and Bd[x, ε] denote the open and closed balls respectively centered at x of radius

ε. Whenever the metric d is clear in context, we denote Bd(x, ε) and Bd[x, ε] simply as B(x, ε) and B[x, ε] respectively. Also, whenever A, B ⊂ X, the distance between A and B is deﬁned to be distd(A, B) := inf{d(a, b) a ∈ A and b ∈ B}. Additionally, the

ε-neighborhood of A is Nd(A, ε) := {z ∈ X distd(z, A) < ε}, and the diameter of A is diamd(A) := sup{d(y, z) y, z ∈ A}.

A path in X is a continuous map γ from an interval I ⊂ R to X. The interval I need not necessarily be compact. However, if r ≤ s and I = [r, s], we deﬁne the length of γ to be

( ) n X `(γ) = `d(γ) := sup d(γ(xi), γ(xi+1)) r = x1 ≤ x2 ≤ ... ≤ xn+1 = s, n ∈ N . i=1

5 If γ(r) = a and γ(s) = b, we say that γ is a path from a to b and denote this by

γ : a y b. We refer to the set γ(I) as the trajectory of γ, and whenever p is a point in this trajectory, we say that p ∈ γ.

We say that X is rectifiably connected if whenever x, y ∈ X, there is a path of finite length from x to y in X. We use X to denote the metric completion of X, and the metric boundary of X is ∂X := X \ X. The space X is said to be locally complete if each point of X belongs to the interior of a complete subspace, and X is proper if every closed and bounded set in X is compact. Whenever W ⊂ X, we let W ◦ denote the topological interior of W, and we say that the metric space X is locally compact if whenever x ∈ X and U is an open set containing x, there is a compact set V ⊂ U so that x ∈ V ◦. Also whenever Ω ⊂ X, the complement of Ω is defined to be Ωc := X \ Ω.

We say that (X, d) is isometric to another metric space (Y, d0) if there exists a surjective map f : X → Y so that whenever x and x0 are points in X, d(x, x0) = d0(f(x), f(x0)).

Here, the map f is referred to as an isometry, and we use the notation X ≡ Y to denote that (X, d) is isometric to (Y, d0).

n n n Euclidean space of dimension n is represented as R , and Rˆ := R ∪ {∞} is the n one point extension/compactiﬁcation of R . We use χ to denote the Euclidean chordal n distance on Rˆ given by

 2|x − y| n  if x, y ∈ R ,  p1+|x|2p1+|y|2 χ(x, y) := 2 n  if x ∈ R , and y = ∞.  p1+|x|2

n Here, |.| denotes the standard Euclidean norm in R . We note that χ is an actual distance function satisfying the triangle inequality. Let R > 0, and let 0 denote the origin

n in R . Note that in B[0,R], χ is bi-Lipschitz equivalent to |.|. In particular whenever x, y ∈ B[0,R], 2 |x − y| ≤ χ(x, y) ≤ 2|x − y|. 1 + R2

6 n n A similarity is a bijective map f : R → R for which there exists r > 0 so that n n whenever x, y ∈ R , |f(x) − f(y)| = r|x − y|. If y ∈ R and y1, y2, ..., yn ∈ R so that th y = (y1, y2, ..., yn), we say that the i component of y is yi for each i ∈ {1, 2, ..., n}. We use < . > to denote a vector. In particular < y1, y2, ..., yn > is said to be a vector in

n n n R . If p ∈ R is a point and v = < v1, v2, ..., vn > is a vector in R , the line through p in direction v is the set {p + tv : t ∈ R}. In contrast, the ray through p in direction v is the set {p + tv : t ∈ R, t ≥ 0}, and whenever s1, s2 ∈ R so that s1 ≤ s2, the set

{p+tv : t ∈ R, s1 ≤ t ≤ s2} is said to be a line segment from p+s1v to p+s2v. Whenever n −→ y, z ∈ R , we let y z be the ray beginning at y and going through z. If L is either a n line or ray containing the points a and b in R ,[a, b] denotes the line segment along L from a to b (where a and b are included along the segment). Also, (a, b) := [a, b] \{a, b},

(a, b] := [a, b] \{a}, and [a, b) := [a, b] \{b}.

We use the notation w ∧ x := min{w, x} for the minimum of w, x ∈ R. Whenever n n Ω ⊂ R is non-empty, open, and connected, we say that Ω is a domain in R . Sometimes, the dimension of Euclidean space which we are discussing may not be clear in context.

n n n As a consequence whenever x ∈ R , we use B (x, R) and B [x, R] to clearly represent n open and closed balls respectively in n-dimensional Euclidean space. Also, we use S to n+1 n n+1 represent the unit sphere in R . That is, S = {z ∈ R : |z| = 1}.

2.2 Conformal Metrics

The quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic metrics are all examples of conformal metrics. Here, we provide a formal deﬁnition of a conformal metric, and then we discuss the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances.

Whenever we discuss conformal metrics, it will be in the context that X is rectifiably connected. A conformal metric on a rectifiably connected space X is of the form ρ ds where ρ is a positive continuous function defined on X. We note that conformal metrics

7 additionally are deﬁned to satisfy particular inequalities, though we will not need to make use of this. The inequalities which conformal metrics are deﬁned to satisfy include a Harnack type inequality, an Ahlfors upper Q-regular inequality, and a Koebe inequality

(See pages 165-166 of [H3] for further information regarding these inequalities). When- ever a, b ∈ R with a ≤ b and γ :[a, b] → X is a path, we deﬁne the ρ-length of γ to be R b R the Lebesgue-Stieltjes integral a (ρ ◦ γ) ds =: γ ρ ds =: `ρ(γ), where s :[a, b] → [0, ∞) is given by s(t) := ` γ [a,t] . We deﬁne the length distance dρ on X by

dρ(x, y) := inf{`ρ(γ) γ : x y y is a rectiﬁable path in X}.

We often denote the open and closed balls with respect to dρ as Bρ(x, r) and Bρ[x, r] respectively for any r > 0. Also we say that a path γ : x y y is a ρ-geodesic if the distance dρ(x, y) = `ρ(γ). However, we note that ρ-geodesics need not necessarily exist and need not be unique. When a ρ-geodesic exists between every two points in X, we say that X is a geodesic space, and we say that X is uniquely geodesic if there is a unique ρ-geodesic between each two points in X. The distance which results when ρ = 1 is called the intrinsic length distance on X and is denoted lX . When the space X is clear in context, we use l to denote lX . The metric space (X, d) is said to be a length space if d = l. When X is a length space, we can utilize the widely known Hopf-Rinow

Theorem (see for instance [BBI, Theorem 2.5.28]). Below, we state this theorem.

Fact 2.2.1. Every locally compact, complete length space is proper and geodesic.

2.2.1 Quasihyperbolic Metric

We now discuss some examples of conformal metrics. The quasihyperbolic metric is one example of a conformal metric.

Definition 2.2.2. A metric space Ω is said to be a quasihyperbolic metric space if it is rectifiably connected, non-complete, and locally complete. The quasihyperbolic metric on such a space Ω is given by (1/δ)ds where δ(x) = δΩ(x) is defined to be dist(x, ∂Ω),

8 where dist(x, ∂Ω) is the distance from x to ∂Ω in Ω. We often denote quasihyperbolic distance in Ω by the notation k = kΩ := d1/δ.

We note that whenever x is a point in a quasihyperbolic metric space Ω, δ(x) > 0 since Ω is locally complete. In order to ensure that (Ω, k) is a complete metric space, we also often suppose that the identity map from (Ω, l) to (Ω, d) is a homeomorphism, and then in this setting the identity map from (Ω, k) to (Ω, d) is a homeomorphism (The fact that (Ω, l) is homeomorphic to (Ω, k) is demonstrated in Lemma A.4 on pg. 92 of

[BHK]. Also see [BHK, Prop. 2.8] for a proof that (Ω, k) is complete when the identity map from (Ω, k) to (Ω, d) is a homeomorphism). We note however that if Ω is a general quasihyperbolic metric space, it is not necessarily the case that the identity map from

(Ω, d) to (Ω, l) is a homeomorphism. Suppose Ω0 is also a quasihyperbolic metric space, and k0 is the quasihyperbolic distance with respect to Ω0. We say that a bijective map f :Ω → Ω0 is a quasihyperbolic isometry provided f : (Ω, k) → (Ω0, k0) is an isometry.

The following fact from [V1, Lemma 2.17] will be helpful. Here, we remind the reader

n that Ω ⊂ R is a domain if it is non-empty, open, and connected.

n Fact 2.2.3. Suppose Ω (R is a domain, a ∈ Ω, and ζ ∈ ∂Ω so that δ(a) = |a − ζ|. Let b be a point along the line segment from a to ζ so that a 6= ζ 6= b. Then the line segment from a to b is a geodesic with respect to the quasihyperbolic distance k in Ω, and

|a − ζ| k(a, b) = log . |b − ζ|

The following remark regarding quasihyperbolic isometries of a half-plane is already well-known in literature on the quasihyperbolic metric. Here, we follow the deﬁnition of

n n Möbiustransformations given in [B, Definition 3.1.1] and say that a map f : Rˆ → Rˆ is a Möbiustransformation if it is a finite composition of inversions through spheres and reflections through planes.

9 0 2 2 Remark 2.2.4. Suppose Ω is the upper half-plane in R . Suppose Ω (R is a domain and f :Ω → Ω0 is a quasihyperbolic isometry. Then f is a M¨obiustransformation.

Proof. Let k and k0 be quasihyperbolic distance in Ω and Ω0 respectively. Let a ∈ Ω, and let ζ ∈ ∂Ω so that δ(a) = |ζ − a|. Let [a, ζ) denote the line segment in Ω from a to ζ. Let b ∈ [a, ζ) so that a 6= b 6= ζ. By Fact 2.2.3, the line segment [a, b] from a |a − ζ| to b is a geodesic, and k(a, b) = log . Then f([a, b]) is a geodesic from f(a) to |b − ζ| f(b). In order to understand the geometry of geodesics in Ω0, we consider the hyperbolic metric (Note that the hyperbolic metric is deﬁned later in Section 2.2.4). Also note that the quasihyperbolic metric in Ω0 is equivalent to the hyperbolic metric in Ω0 (See for instance Section 5.3.2 of [BBI]). With respect to the hyperbolic metric, it is well known that the geodesics in the upper half-plane are vertical rays starting from the x-axis and semi-circles that meet the x-axis orthogonally (See Section 7.3 from [B] or Section 5.3 from [BBI]). Then f([a, b]) is either part of a vertical line or circle that intersects the x-axis orthogonally. First suppose that f([a, b]) is part of a line intersecting the x-axis orthogonally, and let α be the point of intersection along the x-axis. By Fact 2.2.3,

0 |f(a) − α| k (f(a), f(b)) = log . |f(b) − α|

|a − ζ| |f(a) − α| |a − ζ| |f(b) − α| Then since k(a, b) = k0(f(a), f(b)), either = or = . |b − ζ| |f(b) − α| |b − ζ| |f(a) − α| Note that if α = 0 = ζ, if [a, ζ) ⊂ {z ∈ R : z > 0}, and if f([a, b]) ⊂ {z ∈ R : z > 0}, then a b either f(b) = f(a) or f(b) = f(a). We note that this case does not actually occur b a since Ω0 is the upper half-plane. However, we have shown that in this non-existent case, f is M¨obius on the segment [a, ζ). This though implies that in all of the cases which actually occur, f is a composition of a M¨obiustransformation, Euclidean rotations, and

Euclidean translations on the segment [a, ζ). Note that since f is a quasihyperbolic isometry of domains in Euclidean space, it is conformal (See [MO, Theorem 2.6]). Then since

10 f is M¨obiuson the segment [a, ζ), f is M¨obiusin all of Ω (See the Principle of Analytic

Continuation (See Corollary 1.6 on page 307 of Chapter VIII of [P]).

Now suppose that f([a, b]) is part of a circle C that intersects the x-axis orthogonally, and let β be one of the points of intersection along the x-axis. Note that any Euclidean translation g is a quasihyperbolic isometry from Ω0 to g(Ω0). Thus since translations are quasihyperbolic isometries, we may assume that β is the origin.

2 Let J be the map on Rˆ given by

 x  2 if 0 6= x 6= ∞,  |x| ∗  J(x) := x := ∞ if x = 0,    0 if x = ∞.

Note that J then sends C to a line L, and since J is a conformal map, L intersects the x-axis orthogonally at some point ω. Note that J(Ω0) = Ω0, and so L is a geodesic in J(Ω0). In particular J(f([a, b])) is a geodesic, and by Fact 2.2.3,

0 |J(f(a)) − ω| k (J(f(a),J(f(b))) = log . |J(f(b)) − ω|

Note that it is well-known that J is an isometry of the hyperbolic metric on Ω0 (See for instance Section 5.3.2 of [BBI]). Hence as the quasihyperbolic metric is equivalent to the hyperbolic metric in Ω0,J is a quasihyperbolic isometry on Ω0, and

k(a, b) = k0(f(a), f(b)) = k0(J(f(a)),J(f(b))).

Hence

11 |a − ζ| |J(f(a)) − ω| |a − ζ| |J(f(b)) − ω| and so either = or = . Then J ◦ f is M¨obius |b − ζ| |J(f(b)) − ω| |b − ζ| |J(f(a)) − ω| along the segment [a, ζ), and so J ◦ f is M¨obiusin all of Ω (again by [P, Corollary 1.6]).

Then f must also be M¨obiusin Ω.

We would like to discuss quasihyperbolic metric spaces in the setting where they are non-empty, open, connected subspaces of a metric space (X, d). This setting occurs if

X is a quasihyperbolic super space.

Deﬁnition 2.2.5. A metric space (X, d) is called a quasihyperbolic super space if it is a complete rectiﬁably connected metric space for which the identity map from (X, l) to

(X, d) is a homeomorphism.

When (X, d) is a quasihyperbolic super space, each non-empty, open, connected subspace Ω ( X is a quasihyperbolic metric space, and the identity map from (Ω, l) to (Ω, d) is a homeomorphism. In the context that X is a quasihyperbolic super space, we will often let Ω be the complement of some closed, non-empty A ( X. Thus in this setting, Ω is an open set, but not necessarily connected. Hence Ω is not necessarily a quasihyperbolic metric space. Thus if a, b ∈ Ω, the quasihyperbolic distance kΩ(a, b) is ﬁnite only if a and b both lie in the same connected component of Ω. If instead a and b do not lie in the same component, then we let kΩ(a, b) := ∞. Also in this setting, the metric boundary of Ω is isometrically equivalent to the topological boundary of Ω. Then for each x ∈ Ω, δΩ(x) is the distance from x to the topological boundary bd(Ω), and so δ(x) = δΩ(x) ≥ dist(x, A). If X is further assumed to be a length space, then δ(x) = dist(x, A). However, equality does not necessarily hold in general. Let δ0 :Ω → [0, +∞) be either δ or distance to A.

Regardless of whether or not X is a length space, the following inequalities hold; these were ﬁrst established by Gehring and Palka [GP, 2.1]. If a and b are points in the same

12 connected component of Ω, then

! d(a, b) δ0(a) d 0 (a, b) ≥ log 1 + ≥ log . (1) 1/δ 0 0 0 min{δ (a), δ (b)} δ (b)

By the preceding inequalities, it is easy to see that whenever a ∈ Ω, R > 0, and x ∈ B [a, R], the following hold: d1/δ0

d(x, a) ≤ δ(a)(eR − 1) and δ(x) ≥ e−Rδ(a). (2)

2.2.2 Ferrand Metric

The Ferrand metric is another example of a conformal metric which is deﬁned on any

n domain Ω ⊂ Rˆ whose complement has at least two distinct points. The Ferrand metric was ﬁrst introduced by Jacqueline Ferrand in 1988 in [F], and since then it has been shown to have close connections to the hyperbolic and quasihyperbolic metrics (see for

n example [HJ], [HIM], [HMM2], and [H4]). For a quasihyperbolic metric space Ω in Rˆ , ϕ ds is often used in the literature to denote the Ferrand metric on Ω. We follow this

n convention so that for each x ∈ Ω ∩ R ,

|a − b| ϕ(x) = ϕ (x) := sup . Ω |x − a||x − b| a,b∈Rˆn\Ω

In [HJ] and [HIM, Prop. 4.4], ϕ is described equivalently via measuring the diameter

n of a set. To describe this set, we let J be the map on Rˆ given by

 x  2 if 0 6= x 6= ∞,  |x| ∗  J(x) := x := ∞ if x = 0,    0 if x = ∞.

13 n Geometrically, J is reflection in the unit sphere. Also for each x, y ∈ R , we define ∗ Jy(x) := J(x − y) = (x − y) . Here, Jy is obtained by first performing a Euclidean translation sending y to the origin, and then reflecting about the unit sphere. It is

n shown in [HJ] and [HIM, Prop. 4.4] that for each x ∈ Ω∩R , ϕ(x) can be determined by n calculating the Euclidean diameters of particular sets. In particular for each x ∈ Ω ∩R ,

c ϕ(x) = diam[Jx(Ω )] = diam[Jx(∂Ω)].

We also use the following remark regarding the inversion J from Section 3.1 of [B].

n Remark 2.2.6. Suppose x, y ∈ R . Then |x − y| |J(x) − J(y)| = . |x||y|

2.2.3 Kulkarni-Pinkall-Thurston Metric

The Kulkarni-Pinkall-Thurston metric is a conformal metric that is closely tied to the

Ferrand metric. It was first used by Kulkarni and Pinkall in [KP] as a canonical metric for Möbiusstructure on n-dimensional manifolds. Like the Ferrand metric, it is defined

n on any domain in Rˆ whose complement contains at least two distinct points. The Kulkarni-Pinkall-Thurston metric is also M¨obiusinvariant and bi-Lipschitz equivalent to the Ferrand metric (as well as bi-Lipschitz equivalent to the quasihyperbolic metric on

n domains in R ); see [HIM, Cor. 4.6]. In [HIM, Prop. 4.4], it is shown that the Kulkarni- Pinkall-Thurston metric can be deﬁned in a very similar manner to the Ferrand metric.

Whereas the Ferrand metric can be deﬁned by measuring the diameters of certain sets, the Kulkarni-Pinkall-Thurston metric can be deﬁned by measuring the circumdiameters

n of those same sets. It is well-known that whenever A is a non-empty bounded set in R having at least two distinct points, there is a unique smallest closed Euclidean ball C that contains A (See for instance [BW] or [EH]). In this case, the circumdiameter of A is the diameter of C and is denoted cdiam(A). In discussing the Kulkarni-Pinkall-Thurston metric, we will use the deﬁnition given in Proposition 4.4 of [HIM] using circumdiameters.

14 n n That is, whenever Ω is a quasihyperbolic domain in Rˆ and x ∈ Ω ∩ R , the Kulkarni- Pinkall-Thurston metric is given by

c µ(x) = µΩ(x) := cdiam[Jx(Ω )] = cdiam[Jx(∂Ω)].

n Here, Jx is as in the previous section. Also note that when x ∈ Ω ∩ R , there exists c r > 0 so that the Euclidean ball B(x, r) ⊂ Ω. Because of this fact, Jx(Ω ) is necessarily ˆ n bounded. We also point out that Jx(x) = ∞. Thus as long as R \ Ω contains at least c n two points and x ∈ Ω, Jx(Ω ) also contains at least two points in R .

2.2.4 Hyperbolic Metric

The hyperbolic metric is yet another example of a conformal metric. It is deﬁned on any domain Ω of the extended complex plane Cˆ whose complement contains at least three points; here a domain again refers to a non-empty, open, connected set. Such an Ω is called a hyperbolic region. We use λΩ ds to denote the hyperbolic metric on a hyperbolic region Ω. When the space Ω is clear in context, we denote the hyperbolic metric on Ω simply as λ ds. On the unit disk D, the hyperbolic metric is given by

2|dz| λ (z)|dz| = . D 1−|z|2

The concept of pull-back metrics can be used to generalize the hyperbolic metric on

D to any hyperbolic region in the complex plane. If Ω1 and Ω2 are hyperbolic regions, f :Ω1 → Ω2 is a locally injective holomorphic function, and ρ(z)|dz| is a conformal

∗ metric on Ω2, then the pull-back of ρ(z)|dz| by f is the conformal metric σ ds = f [ρ ds] on Ω1 where σ(z)|dz| = f ∗[ρ(z)|dz|] := ρ(f(z))|f 0(z)||dz|.

15 For a hyperbolic region Ω, the hyperbolic metric λΩ ds is the unique metric on Ω so that for any holomorphic universal covering projection f : D → Ω,

∗ f [λΩ(z)|dz|] = λD(z)|dz|.

Here, we remind the reader of the basic topological concept that a covering projection f : D → Ω is a universal covering projection since D is simply connected. Also we remind the reader that the hyperbolic metric is equal to the quasihyperbolic metric in half-spaces of C (See for instance Section 5.3.2 of [BBI]).

2.3 Convergence

To find approximations of conformal metrics, we investigate three main types of convergence: Hausdorff convergence, Gromov-Hausdorff convergence, and pointed Gromov-

Hausdorﬀ convergence.

2.3.1 Hausdorﬀ Distance

We use Hausdorﬀ distance to measure how close two subsets of a metric space X are to each other. The Hausdorﬀ distance between A, B ⊂ X is

X dH (A, B) := inf{ε > 0 B ⊂ Nd(A, ε) and A ⊂ Nd(B, ε)}.

Here, Nd(A, ε) and Nd(B, ε) are as defined in Section 2.1. When the metric space X is clear in context, we just denote the Hausdorff distance between A and B as dH (A, B). Sometimes, we will be considering various distances on the space X, and at these times it will be helpful to have notation to distinguish these different distances. Thus if d˜ is ˜ a distance function on X, we use dH (A, B) to denote the Hausdorff distance between A and B with respect to the distance d.˜ The following fact appears in [H2] listing various equivalent definitions for Hausdorff distance.

16 Fact 2.3.1. The following quantities are equal for non-empty subsets A and B of X : a) dH (A, B), b) inf{ε > 0 ∀ a ∈ A, distd(a, B) < ε and ∀ b ∈ B, distd(b, A) < ε}, c) max{[supa∈A distd(a, B)], [supb∈B distd(b, A)]}, d) supz∈X |distd(z, A) − distd(z, B)|.

We say that a sequence of sets (Ai) in X Hausdorff converge to a set A ⊂ X if dH (Ai,A) → 0. From the definition of Hausdorff distance, it is easy to see that dH (Ai,A) → 0 iff the distances distd(x, Ai) converge uniformly to distd(x, A) in X

(i.e. for each ε > 0, there exists N ∈ N so that whenever i ≥ N and x ∈ X,

|distd(x, A) − distd(x, Ai)| < ε).

2.3.2 Gromov-Hausdorﬀ Distance

Although Hausdorff distance allows us to measure the similarity of two sets within one metric space, it does not permit us to compare two metric spaces. Gromov-Hausdorff distance is a tool that allows us to do this. Formally, the Gromov-Hausdorff distance between non-empty metric spaces X and Y is

Z 0 0 0 0 0 0 dGH (X,Y ) := inf{dH (X ,Y ) ∃ Z ⊃ X ,Y so that X ≡ X and Y ≡ Y }.

Here we remind the reader that the notation ≡ is used to represent an isometric relationship (See Section 2.1). An equivalent deﬁnition for Gromov-Hausdorﬀ distance can be derived by using admissible distance functions. In this context given two sets X and Y, the disjoint union of X and Y is X t Y := (X × {1}) ∪ (Y × {2}). In the case that (X, d) and (Y, d0) are metric spaces, we say that a distance function d˜ on X t Y is admissible if whenever x, x0 ∈ X and y, y0 ∈ Y,

d˜((x, 1), (x0, 1)) = d(x, x0) and d˜((y, 2), (y0, 2)) = d0(y, y0).

˜ If d is an admissible distance and t > 0 so that X ⊂ Nd˜(Y, t) and Y ⊂ Nd˜(X, t), then d˜ is said to be t-admissible on X t Y. Note this is equivalent to the Hausdorﬀ distance

17 between (X × {1}) and (Y × {2}) being no more than t. Using this concept of admissible distance, the following fact appears in [H2].

Fact 2.3.2. For non-empty metric spaces (X, d) and (Y, d0), the following quantities are equal: a) dGH (X,Y ), b) inf{t > 0 ∃ a t-admissible distance on X t Y }, ˜XtY ˜ c) inf{dH (X,Y ) d is an admissible distance on X t Y }, and 0 0 0 0 Z 0 0 d) inf{t > 0 ∃ Z ⊃ X ,Y with X ≡ X ,Y ≡ Y , and dH (X ,Y ) < t}.

There is a relationship between Gromov-Hausdorff distance and certain maps known as ε-rough isometries. We next define ε-rough isometries, and then we present several known connections between ε-rough isometries and Gromov-Hausdorff distance.

Deﬁnition 2.3.3. Suppose A ⊂ X, B ⊂ Y , and ε > 0. A map f : A → Y is said to be an ε-rough isometry from A to B if the following are satisﬁed:

1) B ⊂ Nd0 (f(A), ε), and 2) whenever z and z0 are points in A, d(z, z0) − ε < d0(f(z), f(z0)) < d(z, z0) + ε.

The following fact demonstrates the relationship between Gromov-Hausdorﬀ distance and ε-rough isometries. It appears as part of Corollary 7.3.28 in [BBI].

Fact 2.3.4. Suppose A ⊂ X is non-empty and B ⊂ Y is non-empty. If there exists

ε > 0 so that the Gromov-Hausdorﬀ distance between A and B is less than ε, then there exists a 2ε-rough isometry f : A → B.

We make the following observation about the way that the 2ε-rough isometry can be chosen in the preceding fact.

Observation 2.3.5. Suppose (X, d) and (Y, d0) are metric spaces with ∅= 6 A ⊂ X and

∅= 6 B ⊂ Y. Also suppose ε > 0 and d˜ is an ε-admissible distance on A t B. Then there exists a 2ε-rough isometry f : A → B so that whenever a ∈ A, d˜(a, f(a)) < ε.

18 Fact 2.3.4 can also be used to make the following observation.

Observation 2.3.6. Suppose (Xi, di) is a sequence of metric spaces which Gromov-

Hausdorﬀ converges to a metric space (X, d). Then diamdi (Xi) → diamd(X).

We use the following fact concerning ε-rough isometries in chapter 4 to obtain relationships between Gromov-Hausdorﬀ convergence and other forms of convergence. This fact

n n appears as Theorem 2.2 in [ATV]. For A ⊂ R and φ : A → R , kφkA := supx∈A |φ(x)|.

n n Fact 2.3.7. Let A ⊂ R be compact. Suppose f : A → B ⊂ R is an εdiam(A)-rough n n isometry with ε ≤ 1. Then there exists an isometry S : R → R so that √ kS − fkA ≤ cn εdiam(A), where cn depends only on n.

√ √ Note that if kS − fkA ≤ cn εdiam(A), then dH (S(A), f(A)) ≤ cn εdiam(A).

2.3.3 Pointed Gromov-Hausdorﬀ Distance

Gromov-Hausdorﬀ distance provides an excellent measure of how “similar,” or “close,” two compact metric spaces are to each other. However, the following example illustrates that Gromov-Hausdorﬀ convergence is not suited to describe convergence of unbounded metric spaces at a particular point.

n+1 n Example 2.3.8. Let (Si)i∈N be a sequence of spheres in R , each tangent to R ×{0} at the origin. Assume that as i → ∞, the radius of Si approaches inﬁnity. Let δi, δ denote

n n+1 Euclidean distance to Si, (R ×{0}) respectively in R . Note that if you were to stand at the origin, it would seem that as i → ∞, the sequence of spheres (Si) approaches

n n+1 R × {0}. Additionally, δi → δ locally uniformly in R . However, in actuality, for each n i ∈ N, the Gromov-Hausdorﬀ distance between Si and R × {0} is ∞.

19 Because Gromov-Hausdorﬀ distance does not provide a good description of the convergence described in the preceding example, we turn our attention to pointed Gromov-

Hausdorff distance, and we see that the spaces in the preceding example do indeed pointed Gromov-Hausdorff converge. To define pointed Gromov-Hausdorff distance, we use isometries and admissible distances in the context of pointed metric spaces. A pointed metric space is simply a metric space taken with one of its points. For instance, if x is a point in the metric space (X, d), then (X, x, d) is a pointed metric space. When the metric d is clear in context, we refer to this pointed space as (X, x). We extend the definition of isometry to pointed metric spaces. Pointed metric spaces (X, x, d) and

(Y, y, d0) are isometric if there exists an isometry f : X → Y so that f(x) = y. Here, the map f is said to be an isometry of pointed metric spaces, and we denote the fact that

(X, x, d) is isometric to (Y, y, d0) by the notation (X, x, d) ≡ (Y, y, d0). Additionally, we extend the deﬁnition of ε-rough isometries to pointed spaces. If a ∈ A ⊂ X, b ∈ B ⊂ Y, and ε > 0, then a map f : A → Y is said to be an ε-rough isometry from (A, a) to (B, b) when the following two conditions are satisﬁed:

1) d0(f(a), b) < ε, and

2) f is an ε-rough isometry from A to B as in Deﬁnition 2.3.3.

We also extend the deﬁnition of admissible distances to the context of pointed metric spaces. For t > 0, d˜is said to be a (t, x, y)-admissible distance if d˜: XtY ×XtY → [0, ∞] is an admissible distance on X t Y and

˜ −1 −1 d(x, y) < t, Bd˜[x, t ] ⊂ Nd˜(Y, t), and Bd˜[y, t ] ⊂ Nd˜(X, t).

Using this extension of admissible distances to pointed spaces, we can extend the definition of Gromov-Hausdorff distance to that of pointed Gromov-Hausdorff distance for pointed spaces. We say that the pointed Gromov-Hausdorff distance between (X, x) and (Y, y) is ( ) 1 d ((X, x), (Y, y)) := min , d˜ ((X, x), (Y, y)) GH∗ 2 GH∗

20 where

˜ ˜ dGH∗ ((X, x), (Y, y)) := inf{t > 0 ∃ a (t, x, y)-admissible distance d on X t Y }.

A sequence of pointed metric spaces (Xi, xi) is said to pointed Gromov-Hausdorﬀ converge to (X, x) if dGH∗ ((Xi, xi), (X, x)) → 0. As the following result from [H2] illustrates, pointed Gromov-Hausdorﬀ convergence is a good tool for describing the convergence from the previous example.

n Example 2.3.9. Let S be a sphere of radius R ≥ 1 which is tangent to R × {0} at the n+1 n 1/3 origin in R . Then dGH∗ ((S, 0), (R , 0)) ≤ 1/R , and so the sequence of spheres (Si) n from Example 2.3.8 pointed Gromov-Hausdorﬀ converge to (R , 0).

We can also describe pointed Gromov-Hausdorﬀ convergence in terms of ε-rough isometries. The following fact from [H2] indicates how ε-rough isometries can be used to describe pointed Gromov-Hausdorﬀ convergence.

Fact 2.3.10. The following are equivalent for a sequence of pointed metric spaces

(Xi, xi): a) dGH∗ ((Xi, xi), (X, x)) → 0. b) For all R > 0 there exist Ri > R, εi > 0, and maps fi : B[xi,Ri] → X so that

Ri → R, εi → 0, and fi are εi-rough isometries from (B[xi,Ri], xi) to (B[x, R], x). Here, it is possible to choose fi so that fi(xi) = x for each i. c) For all R > ε > 0, ∃ N ∈ N so that whenever i ≥ N, there is an ε-rough isometry fi from (B[xi,R], xi) to (B[x, R − ε], x). Here whenever i ≥ N, it is possible to choose fi so that fi(xi) = x.

Additionally, the next fact from [H2] also shows how ε-rough isometries are related to pointed Gromov-Hausdorﬀ convergence.

21 Fact 2.3.11. Let (X, x) and (Y, y) be pointed metric spaces.

−1 a) If dGH∗ ((X, a), (Y, b)) < t < (1/2), then there exists a map f : B[x, t ] → Y which is a 2t-rough isometry from (B[x, t−1], x) ⊂ (X, x) to (B[y, t−1 − 2t], y) ⊂ (Y, y). Here, f can be chosen so that f(x) = y. b) Let R > ε > 0, and suppose f : B[x, R] → Y is an ε-rough isometry from (B[x, R], x) to (B[y, R − ε], y). Then dGH∗ ((X, x), (Y, y)) < max{2ε, 1/(R − ε)}.

We also use the following fact from [H2] concerning pointed proper metric spaces.

Fact 2.3.12. Let GH∗ denote the collection of all isometry classes of pointed proper metric spaces. Then (GH∗, dGH∗ ) is a metric space.

2.4 Chordal and Inversive Distances

The starting point for this thesis is the following fact, and we focus on generalizing and improving this result.

n n n Starting Point: Let Ω, Ωi (R be domains. Suppose dH (R \ Ωi, R \ Ω) → 0 with n respect to Euclidean distance in R . Then the quasihyperbolic distances in Ωi converge locally uniformly in Ω to the quasihyperbolic distance in Ω.

Note that in the above scenario, the Euclidean distances dist(x, ∂Ωi) converge uni-

n formly in R to dist(x, ∂Ω), and this uniform convergence can be used to prove the validity of the preceding fact. However, Hausdorﬀ convergence is a strong condition, and so we would like to explore possible weaker hypotheses. One possible weaker hypothesis is that of Hausdorﬀ convergence with respect to chordal distance. Chordal distance is very closely related to inversive distance, and we explore this relationship further in Section 2.4.3.

22 2.4.1 Chordal Distance

Let (X, d) be any metric space. Chordal distance is deﬁned on X,ˆ where Xˆ is deﬁned as in [BHX]. In the case that X is bounded, Xˆ := X in [BHX], and in the scenario that

X is unbounded, Xˆ denotes X ∪ {∞}. Fix a base point o ∈ X. Before defining chordal distance, we first define a map c : X × X → [0, +∞) which we will use in our definition of chordal distance. Whenever x, y ∈ X,

d(x, y) c(x, y) := . [1 + d(x, o)][1 + d(y, o)]

In the case that X is unbounded, we extend c to Xˆ × Xˆ so that for each x ∈ X,

1 c(x, ∞) := c(∞, x) := and c(∞, ∞) := 0. 1 + d(x, o)

In general, c may not satisfy the triangle inequality. Thus to ensure that chordal distance satisﬁes the triangle inequality, we deﬁne the metric chordal distance between x and y to be

( ) n ˆ X ˆ d(x, y) := inf c(xj, xj−1) x = x0, ..., xn = y ∈ X . j=1

With respect to the distance d,ˆ a set U ⊂ Xˆ is open iﬀ either U is an open set in X or Xˆ \ U is a bounded closed set in X. We note that if the original metric space X is non-complete, Xˆ may not be complete as well. We also note that even if X is bounded, d may not be equal to d.ˆ However, if X is bounded, it is true that the chordal distance in X = Xˆ is bi-Lipschitz equivalent to d (See parts (a) and (d) of Remark 2.4.1 for further information). Additionally, we remark that the metric chordal distance may not

n be equal to the Euclidean chordal distance on Rˆ . However, they are bi-Lipschitz. The definition of metric chordal distance is most certainly “inspired” by that of Euclidean chordal distance. However, in defining metric chordal distance, efforts were taken in

23 [BHX] to create a “simplified” version of Euclidean chordal distance. This was done with the idea that the metric chordal distance can be extended to very complex general metric spaces, and due to the possible complexity of the underlying metric space, it is beneficial to define the metric chordal distance in a very simple manner. We use both

n the metric chordal distance and Euclidean chordal distance on Rˆ in Chapters 4-5, and thus it is important to introduce both of them here.

To represent Hausdorff distance between sets A, B ⊂ Xˆ with respect to chordal dis- ˆ tance, we use the notation dH (A, B). We call this the chordal Hausdorff distance between ˆ A and B. We also use the notation dGH (A, B) to refer to the Gromov-Hausdorff distance between A and B with respect to d,ˆ and we refer to this as the chordal Gromov-Hausdorff distance between A and B.

Chordal distance satisﬁes various useful inequalities appearing throughout the literature (see for example [BHX]). We give some of these inequalities here. First, for all x, y ∈ X, 1 1 1 c(x, y) ≤ dˆ(x, y) ≤ c(x, y) ≤ + . 4 1 + d(x, o) 1 + d(y, o)

Secondly, whenever 0 < r ≤ R, r < d(x, o) < R, and r < d(y, o) < R,

d(x, y) d(x, y) ≤ dˆ(x, y) ≤ . 4(1 + R)2 (1 + r)2

2.4.2 Inversive Distance

Inversive distance is defined similarly to chordal distance. The base point o ∈ X is as in the preceding section. As with chordal distance, we first define a quantity that will be used in the definition of inversive distance. Here for any x, y ∈ X \{o},

d(x, y) i(x, y) := . d(x, o)d(y, o)

24 Let X∗ := X \{o} and Xˆ ∗ := Xˆ \{o}. In the case that X is unbounded, we extend i to Xˆ ∗ × Xˆ ∗ so that for each x ∈ X,

1 i(x, ∞) := i(∞, x) := and i(∞, ∞) := 0. d(x, o)

As with c in chordal distance, in general i may not satisfy the triangle inequality, and so we deﬁne the inversive distance d∗ on Xˆ ∗ by

( ) n ∗ X ˆ ∗ d (x, y) := inf i(xj, xj−1) x = x0, ..., xn = y ∈ X . j=1

∗ Note that if (xj) is a sequence in X so that d(xj, o) → ∞, then d (xj, ∞) → 0. Also note that if X is non-complete, Xˆ ∗ may not be complete. However if (X, d) is a complete metric space, then (Xˆ ∗, d∗) is indeed also complete.

∗ Again similar to chordal Hausdorﬀ distance, we use dH (A, B) to represent Hausdorﬀ distance between two sets A, B ⊂ Xˆ ∗ with respect to inversive distance. We say that

∗ ∗ dH (A, B) is the inversive Hausdorﬀ distance between A and B. Also, dGH (A, B) repre- sents the Gromov-Hausdorﬀ distance between A and B with respect to d∗, and we say

∗ that dGH (A, B) is the inversive Gromov-Hausdorﬀ distance between A and B. Also like chordal distance, inversive distance satisﬁes multiple inequalities which ap- pear throughout the literature (see for instance [BHX]). For any x, y ∈ X \{o},

1 1 1 i(x, y) ≤ d∗(x, y) ≤ i(x, y) ≤ + . 4 d(x, o) d(y, o)

Also whenever 0 < r ≤ R, r < d(x, o) < R, and r < d(y, o) < R,

d(x, y) d(x, y) ≤ d∗(x, y) ≤ . 4R2 r2

The multiple similarities between inversive and chordal distance lead to interesting relationships in terms of Hausdorﬀ convergence with respect to these distances, and we explore this idea further in the next section.

25 2.4.3 Lipschitz Relationships

In this section, we compare Hausdorﬀ convergence in (X, d) to Hausdorﬀ convergence in both the chordalization and inversion of X. In this comparison, we use several Lipschitz relationships between chordal and inversive distance. The following relationships easily follow from the previously mentioned inequalities for chordal and inversive distance.

Remark 2.4.1. The following Lipschitz relationships hold:

(a) dˆ(x, y) ≤ c(x, y) ≤ d(x, y) for all x, y ∈ X.

(b) Whenever x, y ∈ X \{o}, dˆ(x, y) ≤ 4d∗(x, y). !2 1 + m (c) Let m > 0, and suppose x, y ∈ X \ B [o, m]. Then d∗(x, y) ≤ 4 dˆ(x, y). d m 2 ˆ (d) Let R > 0, and suppose x, y ∈ Bd[o, R]. Then 4(1 + R) d(x, y) ≥ d(x, y).

Let Ai,A ⊂ X be non-empty. As a direct consequence of the preceding Lipschitz relationships, we have the following relationships among diﬀerent types of Hausdorﬀ convergence.

Remark 2.4.2. The following implications always hold: ∗ ˆ (a) dH (Ai \{o},A \{o}) → 0 implies dH (Ai \{o},A \{o}) → 0 ˆ (b) dH (Ai,A) → 0 implies dH (Ai,A) → 0

We note that in part (a) of the preceding remark, we had to remove o from the sets Ai and A because the metric space (X∗, d∗) does not contain the point o. Such situations will continue to arise, and so it is helpful for us to deﬁne

∗ ∗ Ai := Ai \{o} and A := A \{o}.

When A is bounded, any of the three types of Hausdorﬀ convergence will automatically imply Hausdorﬀ convergence in (X, d). To prove this, we begin by demonstrating the following remark.

26 ˆ Remark 2.4.3. Suppose Ai,A ⊂ X so that dH (Ai,A) → 0. If A is bounded, there exists

S > 0 and N ∈ N so that Ai ⊂ Bd[o, S] for all i ≥ N.

Proof. The case that X is bounded is trivial. So suppose that X is unbounded, and suppose that A is bounded. Then α := distdˆ(∞,A) > 0. Then the chordal distance between ∞ and Ndˆ(A, α/2) is no less than α/2. Then we can let S > 0 so that ˆ Ndˆ(A, α/2) ⊂ Bd(o, S). Let N ∈ N so that whenever i ≥ N, dH (Ai,A) < α/4. Then whenever i ≥ N,Ai ⊂ Bd(o, S).

ˆ Note that if A is bounded and dH (Ai,A) → 0, then dH (Ai,A) → 0 by Remark 2.4.3 and Remark 2.4.1. We then have the following relationships among forms of Hausdorﬀ convergence.

Remark 2.4.4. Suppose that A is bounded with respect to the metric d. The following implications then hold: ˆ a) dH (Ai,A) → 0 implies dH (Ai,A) → 0

∗ ∗ ∗ ∗ ∗ b) dH (Ai ,A ) → 0 implies dH (Ai ,A ) → 0

We let δi, δ : X → [0, ∞) be given by δi(x) := distd(x, Ai) and δ(x) := distd(x, A) ˆ ˆ ˆ ˆ for each x ∈ X. We also let δi, δ : X → [0, ∞) be given by δi(x) := distdˆ(x, Ai) and ˆ ˆ δ(x) := distdˆ(x, A) for each x ∈ X. Remember from Section 2.3.1, dH (Ai,A) → 0 if and only if δi → δ uniformly in X. Thus the preceding remark indicates that when A ˆ is bounded and dH (Ai,A) → 0, δi → δ uniformly in X. We explore the implications of chordal Hausdorﬀ convergence further in chapter 4.

n n ∗ ∗ Note that it is known that standard Euclidean space (R , |.|) is isometric to ((Rˆ ) , d ) n n ∗ via the mapping J : R → (Rˆ ) (see for instance [BHX, Section 3.A]). Here J is as in Section 2.2.2. We now consider an example in Euclidean space demonstrating that if A ˆ ∗ is unbounded with respect to d, then it is possible for dH (Ai,A) → 0 and dH (Ai,A) → 0 while dH (Ai,A) 6→ 0. In this example, we choose 0 to act as the base point o.

27 Example 2.4.5. Consider the space R with the standard Euclidean distance. Let A be ˆ the entire space R, and for each i, let Ai be the interval [−i, i]. Then dH (Ai,A) → 0 and ∗ ∗ ∗ ∗ ∗ dH (Ai ,A ) → 0, but for each i, dH (Ai,A) = ∞ = dH (Ai ,A ).

∗ ∗ Proof. By the way Euclidean distance is deﬁned, dH (Ai,A) = ∞ = dH (Ai ,A ) for each ˆ ∗ ∗ i ∈ N. For each i, let Bi := (−∞, −1/i]∪[1/i, ∞). Since (R, |.|) is isometric to (R , d ) via ∗ ∗ ∗ J, dH (Ai ,A ) = dH (Bi,A). By the way Euclidean distance is deﬁned, dH (Bi,A) → 0, ∗ ∗ ∗ ˆ and so dH (Ai ,A ) → 0. Then dH (Ai,A) → 0 by Remark 2.4.2.

We additionally provide a second example demonstrating that it is possible to have ˆ ∗ dH (Ai,A) → 0 and dH (Ai,A) → 0 while dH (Ai,A) 6→ 0. In this example, we let the origin 0 act as the base point o.

n+1 Example 2.4.6. Consider the space R with the standard Euclidean distance. Let n n+1 n A := R × {0}. For each i, let Ai := Si be a sphere in R which is tangent to R × {0} at the origin. As in Example 2.3.8, we choose Si so that as i → ∞, the radius of the ˆ ∗ ∗ ∗ sphere Si converges to ∞. Then dH (Ai,A) → 0 and dH (Ai ,A ) → 0. However, for each ∗ ∗ i ∈ N, we have dH (Ai,A) = ∞ = dH (Ai ,A ).

∗ ∗ Proof. By the way Euclidean distance is deﬁned, dH (Ai,A) = ∞ = dH (Ai ,A ) for each i ∈ N. Note that J(Si) is a half-space containing the points in the intersection of Si and the unit sphere. Let Ci be this intersection of Si with the unit sphere, and let C be

n the intersection of the unit sphere and R × {0}. Since the radius of Si converges to ∞ as i → ∞, dH (Ci,C) → 0. This means that the half-spaces J(Si) Hausdorﬀ converge

n ∗ ∗ ∗ to (R × {0}) with respect to Euclidean distance. Then since dH (Ai ,A ) is equal to n ∗ ∗ ∗ ˆ dH (J(Si), R ×{0}), we have dH (Ai ,A ) → 0. Then dH (Ai,A) → 0 by Remark 2.4.2.

When we consider the case where A is bounded away from o, each type of Hausdorﬀ convergence implies inversive Hausdorﬀ convergence. This directly follows from part (b) of Remark 2.4.2 and part (c) of Remark 2.4.1.

28 Remark 2.4.7. Suppose r > 0 so that A ⊂ X \ Bd(o, r). Then the following hold:

∗ a) dH (Ai,A) → 0 implies dH (Ai,A) → 0 ˆ ∗ b) dH (Ai,A) → 0 implies dH (Ai,A) → 0

As in the preceding remark, suppose r > 0 so that A ⊂ X \ Bd(o, r). Note that if ˆ either dH (Ai,A) → 0 or dH (Ai,A) → 0, then Ai ⊂ X \ Bd(o, r/2) for all suﬃciently

∗ large i. Thus for all but at most ﬁnitely many i, the quantity dH (Ai,A) is deﬁned.

If A is not bounded away from o, then it is possible to have dH (Ai,A) → 0 and ˆ ∗ ∗ ∗ dH (Ai,A) → 0 while dH (Ai ,A ) 6→ 0. As in the preceding example, we let 0 act as the base point o.

∗ Example 2.4.8. Consider the Euclidean space R, and let A := R = R \{0}. For each ˆ i, let Ai := (−∞, 1/i] ∪ [1/i, ∞). Then dH (Ai,A) → 0 and dH (Ai,A) → 0. However for

∗ each i ∈ N, dH (Ai,A) = ∞.

ˆ Proof. By the way Euclidean distance is deﬁned, dH (Ai,A) → 0, and so dH (Ai,A) → 0 by Remark 2.4.2. For each i ∈ N, let Bi := [−i, 0) ∪ (0, i]. Since J is an isometry from ˆ ∗ ∗ ∗ standard Euclidean space (R, |.|) to (R , d ), dH (Ai,A) = dH (Bi,A) = ∞ for each i.

The following chart summarizes the relationships between the different types of Haus- dorff convergence that we have discussed. In this chart, diam∗(A) denotes the diameter of A with respect to d∗. Also, black arrows denote implications which always hold, blue arrows denote implications which hold when diamd(A) < ∞, and red arrows denote im- ∗ plications which hold when diam (A) < ∞. Note that in the chart, it is assumed that Ai and A do not contain the point o where necessary to properly define inversive distance.

Figure 2.1: Hausdorﬀ Convergence

30 Chapter 3

Pointed Gromov-Hausdorﬀ Convergence of Conformal Metrics

In this chapter, we introduce the concept of bounded uniform convergence, and we prove that there is a relationship between pointed Gromov-Hausdorﬀ convergence and bounded uniform convergence. We later use these results in Chapter 5 to provide approximations of the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances.

Throughout this chapter (X, d) denotes a non-empty metric space, and o ∈ X is a ﬁxed base point. Let Ai,A ( X be non-empty and closed. Let Ω := X\A, and let Ωi := X\Ai.

Also let δ, δi represent distance to A, Ai respectively. For each i, let ρi ds be a conformal metric on Ωi. We say that the sequence (ρi ds) converges boundedly uniformly in Ω to ρ ds if ρ ds is a conformal metric on Ω and whenever x ∈ Ω and r > 0,

ρ B [x, r] ⊂ Ω for all suﬃciently large i and i → 1 uniformly in B [x, r]. ρ i ρ ρ

Note that by deﬁnition if Ω is bounded with respect to the distance dρ, then bounded ρ uniform convergence of (ρ ds) to ρ ds in Ω automatically implies that i → 1 uniformly i ρ in Ω. Also, we say that a metric ρ ds on Ω is complete if (Ω, dρ) is a complete metric space.

31 Let fi, f be functions deﬁned on Ωi, Ω respectively. We say that a sequence of functions

(fi) converges boundedly uniformly in Ω to a function f if whenever B is a bounded set in Ω,

B ⊂ Ωi for all suﬃciently large i and fi → f uniformly in B.

Note that in the above deﬁnition, it is necessary for B to be contained in the domains of f and fi for all suﬃciently large i.

3.1 Pointed Gromov-Hausdorﬀ Convergence

Throughout this section, ρ ds and ρi ds denote conformal metrics on Ω and Ωi respectively. We aim to determine a relationship between pointed Gromov-Hausdorﬀ convergence and bounded uniform convergence of complete conformal metrics. In order to do this, we ﬁrst prove the following lemma.

Lemma 3.1.1. Suppose X is rectiﬁably connected and locally rectiﬁably connected. Let a ∈ Ω, R > 0, and 0 < ε < 1. Suppose (ρi ds) is a sequence of complete conformal metrics converging boundedly uniformly in (Ω, dρ) to a complete conformal metric ρ ds.

Then there exists N ∈ N so that

i ≥ N and b, c ∈ Bρi [a, R] =⇒ Bρi [a, R] ⊂ Ω and |dρ(b, c) − dρi (b, c)| < ε.

Proof. We will ﬁrst show that for all suﬃciently large i, Bρi (a, R) ⊂ Bρ[a, (1 + ε)R] and

Bρ(a, R) ⊂ Bρi [a, (1 + ε)R]. The idea for the proof of this ball containment is due to [BM], where a similar result is proved speciﬁcally for the hyperbolic metric.

0 By the bounded uniform convergence of (ρi ds) to ρ ds, we can let M ∈ N so that 0 whenever i ≥ M and x ∈ Bρ[a, (1 + ε)R],

ρ(x) 1 − ε < < 1 + ε. ρi(x)

32 0 Let i ≥ M , and suppose b ∈ Ωi so that b∈ / Bρ[a, (1+ε)R]. We show that b∈ / Bρi (a, R).

Let γ : a y b be a rectiﬁable path in Ωi, and let c be the ﬁrst point that you reach while traversing γ from a to b for which dρ(a, c) = R(1 + ε). In this context, let γ[a, c] denote this traversed portion of γ from a to c. Note that since (Ω, dρ) is complete and Ω is an open set in X, such a c exists. Then

Z Z Z (1 + ε) ρi ds ≥ (1 + ε) ρi ds ≥ ρ ds ≥ dρ(a, c) = (1 + ε)R. γ γ[a,c] γ[a,c]

Then dρi (a, b) ≥ R by taking an inﬁmum over all such γ in the above argument. So 0 b∈ / Bρi (a, R). Thus Bρi (a, R) ⊂ Bρ[a, (1 + ε)R]. Note that then there exists P ∈ N so 0 that whenever i ≥ P ,Bρi [a, R] ⊂ Bρ[a, R + ε]. Note also that in a very similar manner, it can be shown that there exists T ∈ N so that whenever i ≥ T,Bρ[a, R] ⊂ Bρi [a, R+ε]. 0 0 Let α, β > 0 be so that 2α(R + α) + β + αβ < ε. Let N ∈ N so that whenever i ≥ N ,

Bρi [a, R] ⊂ Bρ[a, R + α] ⊂ Ω. By the bounded uniform convergence of (ρi ds) to (ρ ds), we let N ∈ N so that whenever i ≥ N and x ∈ Bρ[a, 3(R + α) + β] ⊂ Ω, we have x ∈ Ωi and ρ (x) 1 − α ≤ i ≤ 1 + α. ρ(x)

Let b, c ∈ Bρ[a, R + α] and γ : b y c a rectiﬁable path in Ω so that `ρ(γ) is no more than dρ(b, c) + β. Then γ ⊂ Bρ[a, 3(R + α) + β]. Then whenever i ≥ N,

Z Z

dρi (b, c) ≤ ρi ds ≤ (1 + α)ρ ds ≤ (1 + α)(dρ(b, c) + β). γ γ

Let P ∈ N so that for all i ≥ P,Bρi [a, 3R + β] ⊂ Bρ[a, 3R + β + α]. Let M ∈ N so that whenever i ≥ M and x ∈ Bρ[a, 3R + β + α],

ρ(x) 1 − α < < 1 + α. ρi(x)

33 Let i ≥ max{M,P } and u, v ∈ Bρi [a, R]. Let γi : u y v be a rectiﬁable path in Ωi so that dρi (u, v) + β ≥ `ρi (γi). Then

γi ⊂ Bρi [a, 3R + β] ⊂ Bρ[a, 3R + β + α] ⊂ Ω.

Then Z Z

dρ(u, v) ≤ ρ ds ≤ (1 + α) ρi ds ≤ (1 + α)(dρi (u, v) + β). γi γi

0 Then whenever i ≥ max{N,N ,M,P } and x, y ∈ Bρi [a, R],

dρi (x, y) ≤ (1 + α)(dρ(x, y) + β) ≤ dρ(x, y) + α(dρ(x, a) + dρ(a, y)) + β + αβ

≤ dρ(x, y) + 2α(R + α) + β + αβ < dρ(x, y) + ε, and also

dρ(x, y) ≤ (1 + α)(dρi (x, y) + β) ≤ dρi (x, y) + α(dρi (x, a) + dρi (a, y)) + β + αβ

≤ dρi (x, y) + 2αR + β + αβ < dρi (x, y) + ε.

0 Then whenever i ≥ max{N,N ,M,P } and x, y ∈ Bρi [a, R],

|dρi (x, y) − dρ(x, y)| < ε.

Using the previous lemma, we now prove the desired pointed Gromov-Hausdorﬀ convergence. Here, ρ ds is considered to be a complete metric on Ω when (Ω, dρ) is complete.

Similarly, ρi ds is a complete metric on Ωi when (Ωi, dρi ) is complete.

Theorem 3.1.2. Suppose X is rectiﬁably connected and locally rectiﬁably connected. Let a ∈ Ω and ai ∈ Ωi so that ai → a. Suppose (ρi ds) is a sequence of complete conformal metrics converging boundedly uniformly in (Ω, dρ) to a complete conformal metric ρ ds.

Then dGH∗ ((Ωi, ai, dρi ), (Ω, a, dρ)) → 0.

Proof. Let R > ε > 0. We use Fact 2.3.10 to attain pointed Gromov-Hausdorﬀ convergence by showing that the identity map is an ε-rough isometry from (Bρi [ai,R], ai) to

34 (Bρ[a, R − ε], a) for suﬃciently large i. By the proof of Lemma 3.1.1, we let N1 ∈ N so that whenever i ≥ N1,Bρi [ai,R] ⊂ Bρi [a, R + (ε/2)] and dρi (ai, a) < ε/2.

By Lemma 3.1.1, we can let N2 ∈ N be so that N2 ≥ N1 and whenever i ≥ N2 and b, c ∈ Bρi [a, R+(ε/2)], the space Ω contains Bρi [a, R+(ε/2)] and |dρ(b, c)−dρi (b, c)| < ε.

Whenever i ≥ N2, let fi : Bρi [ai,R] → Ω be given by fi(x) = x. Let N3 ∈ N so that

N3 ≥ N2 and whenever i ≥ N3, dρ(ai, a) < ε. Then whenever i ≥ N3, dρ(fi(ai), a) < ε.

ε Also whenever i ≥ N1 and z ∈ Bρi [a, R − 2 ], ε ε d (a , z) ≤ d (a , a) + d (a, z) < + R − = R. ρi i ρi i ρi 2 2 ε Then whenever i ≥ N1,Bρi [a, R− 2 ] ⊂ Bρi [ai,R]. By Lemma 3.1.1, we can let N4 ∈ N ε be so that N4 ≥ N3 and whenever i ≥ N4,Bρ[a, R − ε] ⊂ Bρi [a, R − 2 ]. Then whenever i ≥ N4,Bρ[a, R − ε] ⊂ Bρi [ai,R]. Then whenever i ≥ N4, fi(Bρi [ai,R]) = Bρi [ai,R] is most certainly an ε-net for Bρ[a, R − ε].

We have shown that whenever i ≥ N4, fi is an ε-rough isometry from (Bρi [ai,R], ai) to (Bρ[a, R − ε], a). Then (Ωi, ai, dρi ) pointed Gromov-Hausdorﬀ converges to (Ω, a, dρ) as i → ∞.

We next provide an example to demonstrate that in the previous theorem, pointed

Gromov-Hausdorﬀ convergence cannot be replaced with either Hausdorﬀ convergence or

Gromov-Hausdorﬀ convergence.

2 2 Example 3.1.3. Let Ω := {(x, y): x + y < 1 and x, y ∈ R}. For each i ∈ N, let

2 2 Ωi := {(x, y):(x + 1/i) + y < 1 and x, y ∈ R} ∪ {(x, y): x < −1 and x, y ∈ R}.

Let δi, δ denote Euclidean distance to ∂Ωi, ∂Ω respectively, and let ki, k denote quasihyperbolic distance in Ωi, Ω respectively. Note that δi → δ boundedly uniformly in ˆ (Ω, k). However dH (∂Ωi, ∂Ω) 6→ 0, dH (∂Ωi, ∂Ω) 6→ 0, and dGH (∂Ωi, ∂Ω) 6→ 0.

35 Theorem 3.1.2 relies on the concept of bounded uniform convergence. In order to employ this to approximate a given conformal metric, we need to understand situations where bounded uniform convergence of conformal metrics occur. An exploration of this follows in the next section. In the next chapter, we investigate various relationships between other forms of convergence outside of pointed Gromov-Hausdorﬀ convergence.

3.2 Bounded Uniform Convergence

Throughout this section, (X, d) is assumed to be a quasihyperbolic super space. We look at an example of bounded uniform convergence of conformal metrics. Remember that

A, Ai ( X are non-empty closed sets, Ω = X \ A, and Ωi = X \ Ai. Also recall that δ and δi are the distances to A and Ai respectively with respect to d. In the case that X ˆ is a quasihyperbolic super space, we show in Chapter 4 that dH (Ai,A) → 0 implies that

δi → δ boundedly uniformly in Ω. In this section, we begin by noting some observations that we use in proving bounded uniform convergence of δi to δ. We use the notation ˜ ˜ d, di to represent dδ−1 , d −1 respectively. We also use k, ki to denote quasihyperbolic δi distance in Ω, Ωi respectively (Remember that quasihyperbolic distance is determined by the distance to the metric boundary. See section 2.2.1 for further details). Note that if X is a length space, then d˜ = k and d˜i = ki. However, in general, d˜ and d˜i may not necessarily be quasihyperbolic distances.

Using the inequalities in (2) from section 2.2.1, it is not too hard to see the validity of the following claim.

Claim 3.2.1. Suppose δi → δ boundedly uniformly in (X, d). Then for each R > 0 and for each z ∈ Ω,Bd˜[z, R] ⊂ Ωi for all but ﬁnitely many i.

We provide a very brief idea of the proof of the preceding claim. If R > 0 and

−R z ∈ Ω, δ(x) ≥ e δ(z) for each x ∈ Bd˜[z, R]. Thus if δi → δ uniformly in the set

36 R Bd[z, δ(z)(e − 1)] ⊃ Bd˜[z, R], then for all suﬃciently large i,

−R x ∈ Bd˜[z, R] =⇒ δi(x) ≥ (1/2)e δ(z) > 0.

Hence Bd˜[z, R] ⊂ Ωi for all but ﬁnitely many i. Alternatively, the validity of the preceding claim can be shown through the proof of Lemma 3.1.1.

We demonstrate the following example of bounded uniform convergence of conformal metrics. We use this example in Chapter 4 during our study of chordal Hausdorﬀ convergence. Remember that d˜ and d˜i are not necessarily quasihyperbolic distances in this setting.

−1 −1 Example 3.2.2. Suppose that δi → δ boundedly uniformly in (X, d). Then δi → δ boundedly uniformly in (Ω, d˜).

Proof. Let a ∈ Ω, R > 0, and x ∈ Bd˜[a, R]. By the inequalities in (2) from section 2.2.1, R d(x, a) ≤ δ(a)(e − 1). Thus Bd˜[a, R] is a bounded set in X with respect to d, and so R R δi → δ uniformly in Bd˜[a, R]. Note also that δ(x) ≤ δ(a)(e − 1) + δ(a) = δ(a)e , and −R δ δ(x) ≥ e δ(a). Then → 1 uniformly in B ˜[a, R]. Because of this and Claim 3.2.1, δi d −1 −1 we then have that δi → δ boundedly uniformly in Ω.

In Chapter 4, we study the implications of chordal Hausdorff convergence. The previous example, combined with our upcoming study of chordal Hausdorff convergence, yields a relationship between chordal Hausdorff convergence and bounded uniform convergence of the quasihyperbolic metric. We then later use this relationship to provide an approximation for quasihyperbolic distance in Chapter 5.

37 Chapter 4

Convergence of Sets

In this chapter, we study relationships among various forms of convergence of sets. In the process, we learn that the concept of chordal Hausdorﬀ convergence can be used as an alternative hypothesis in the starting point for our work (See Section 2.4 for a discussion of the starting point.). In this chapter, (X, d) denotes a non-empty metric space with at least two distinct points, and o ∈ X is a ﬁxed base point. Let Ai,A ( X be non-empty. Let Ω := X \A, and let Ωi := X \Ai. Also let δ, δi represent the distances to A, Ai respectively.

4.1 Hausdorﬀ Convergence Relationships

We now investigate the implications of chordal Hausdorff convergence. In particular, we demonstrate a relationship between chordal Hausdorff convergence and bounded uniform convergence of (δi) to δ. In visualizing the following theorem, it may be helpful to keep some instances of chordal Hausdorff convergence in mind. We remind the reader that

Examples 2.4.5, 2.4.6, and 2.4.8 are all situations where we have chordal Hausdorﬀ convergence.

38 ˆ Theorem 4.1.1. Suppose dH (Ai,A) → 0. Then δi → δ boundedly uniformly in X.

ˆ Proof. If X is bounded, then dH (Ai,A) → 0 =⇒ dH (Ai,A) → 0 by Remark 2.4.4.

Recall that when dH (Ai,A) → 0, δi → δ uniformly in X (See the discussion at the end of Section 2.3.1). Thus we may suppose that X is unbounded.

Let C ⊂ X be bounded with respect to the distance d, and let ε > 0. Pick S > 0 so that C ⊂ Bd(o, S). We will show that [ Bd[z, δ(z) + ε] ⊂ Bd[o, S + s] z∈C where s := diamd(C) + distd(C,A) + 2ε. Let c ∈ C and a ∈ A so that

distd(C,A) + ε > d(c, a).

Then whenever z ∈ C,

δ(z) + ε ≤ d(z, a) + ε ≤ d(z, c) + d(c, a) + ε ≤ diamd(C) + distd(C,A) + 2ε. [ Then Bd[z, δ(z) + ε] ⊂ Bd[o, S + s]. Note that in a similar manner, we can show z∈C that for each i, there exists Ri > 0 so that [ Bd[z, δi(z) + ε] ⊂ Bd[o, Ri]. z∈C

The preceding statement indicates that for each i, there exists Ri > 0 so that for each z ∈ C, we have Bd[z, δi(z) + ε] ⊂ Bd[o, Ri]. That is, Ri is not at all dependent on z.

We now show that whenever z ∈ C, δi(z) ≤ δ(z) + 2ε for suﬃciently large i. Because

0 X is unbounded, β := distdˆ(∞,Bd[o, S + s]) > 0. Then there exists S > 0 so that 0 Ndˆ(Bd[o, S + s], β/4) ⊂ Bd(o, S ). For each z ∈ C and each i ∈ N, let az ∈ A so that ˆ ˆ az ∈ Bd[z, δ(z) + ε], and let azi ∈ Ai so that d(az, azi) < 2dH (Ai,A) =: εi. Note that for each z ∈ C, az ∈ Bd[z, δ(z) + ε] ⊂ Bd[o, S + s]. Let N ∈ N so that whenever i ≥ N, εi < β/4. Observe that N and εi are independent of z. Also we remark that for each i ≥ N and each z ∈ C, azi ∈ Ndˆ(Bd[o, S + s], β/4) since az ∈ Bd[o, S + s] and dˆ(az, azi) < εi < β/4. Then

39 0 {az, azi | i ≥ N and z ∈ C} ⊂ Bd(o, S ).

Then whenever i ≥ N and z ∈ C, d(a , a ) ε > dˆ(a , a ) ≥ z zi . i zi z 4(1 + S0)2 That is,

0 2 d(az, azi) < εi4(1 + S ) .

Then whenever z ∈ C and i ≥ N,

0 2 δi(z) ≤ d(z, azi) ≤ d(z, az) + d(az, azi) ≤ δ(z) + ε + 4εi(1 + S ) .

ˆ Since εi = 2dH (Ai,A) → 0, we can pick M ∈ N so that whenever i ≥ M,

0 2 4εi(1 + S ) < ε.

Then whenever z ∈ C and i ≥ max{M,N}, δi(z) ≤ δ(z) + 2ε. (∗) We now prove that there exists R > 0 so that

[ Bd[z, δi(z) + ε] ⊂ Bd(o, R). z∈C, i∈N

For each z ∈ C and each i, let bzi ∈ Ai so that bzi ∈ Bd[z, δi(z) + ε], and let czi ∈ A ˆ so that d(bzi, czi) < εi. Then whenever z ∈ C, i ≥ max{M,N}, and w ∈ Bd[z, δi(z) + ε],

d(o, w) ≤ d(o, z) + d(z, w) < S + δi(z) + ε

≤ S + δ(z) + 3ε by (*)

≤ S + diamd(C) + distd(C,A) + 4ε.

Then whenever i ≥ j := max{M,N}, we have that

[ 0 Bd[z, δi(z) + ε] ⊂ Bd(o, R ), z∈C 0 0 where R = S+diamd(C)+distd(C,A)+4ε. Then letting R > max{R ,R1,R2,R3, ..., Rj}, we have that [ Bd[z, δi(z) + ε] ⊂ Bd(o, R). z∈C, i∈N

40 To finish off the proof, we show that whenever z ∈ C, δ(z) ≤ δi(z) + 2ε for sufficiently large i. Because X is unbounded, α := distdˆ(∞,Bd[o, R]) > 0. Then there exists Q > 0 so that Ndˆ(Bd[o, R], α/4) ⊂ Bd(o, Q). Note that for each z ∈ C and each i ∈ N, we have bzi ∈ Bd[z, δi(z) + ε] ⊂ Bd(o, R). Let T ∈ N so that whenever i ≥ T, εi < α/4. Then whenever z ∈ C and i ≥ T, we have czi ∈ Ndˆ(Bd[o, R], α/4) since bzi ∈ Bd(o, R) and since dˆ(bzi, czi) < εi < α/4. So whenever z ∈ C and i ≥ T, bzi and czi are elements of

Bd(o, Q). Then whenever z ∈ C and i ≥ T, d(b , c ) ε > dˆ(b , c ) ≥ zi zi . i zi zi 4(1 + Q)2 2 That is, d(bzi, czi) < 4εi(1 + Q) . Then whenever z ∈ C and i ≥ T,

2 δ(z) ≤ d(z, czi) ≤ d(z, bzi) + d(bzi, czi) ≤ δi(z) + ε + 4εi(1 + Q) .

2 Pick P ∈ N so that whenever i ≥ P, 4εi(1+Q) < ε. Then whenever i ≥ max{M,N,P,T } and z ∈ C, δi(z) − 2ε ≤ δ(z) ≤ δi(z) + 2ε. Then δi → δ uniformly in C.

ˆ Let Aî, Aˆ denote the closure of Ai and A respectively in X.ˆ Note that dH (Ai,A) is ˆ ˆ equal to dH (Aî, Aˆ). Thus if dH (Aî, Aˆ) → 0, the preceding theorem tells us that δi → δ boundedly uniformly in X. Also observe that the proof of the preceding theorem still holds if A ⊂ Xˆ as long as A ∩ X 6= ∅ (i.e., in this situation, A could possibly contain the point ∞). In this case, we want A ∩ X 6= ∅ so that it makes sense to define and speak of δ.

By Remark 2.4.2, the following corollary directly follows from the preceding theorem.

Corollary 4.1.2. δi → δ boundedly uniformly in X if either of the following hold:

(a) dH (Ai,A) → 0, or

∗ (b) A, Ai ⊂ X \{o} and dH (Ai,A) → 0.

41 4.2 Gromov-Hausdorﬀ Convergence

We have studied various relationships between Hausdorff convergence, bounded uniform convergence, and pointed Gromov-Hausdorff convergence. Now we wish to investigate a possible relationship between Gromov-Hausdorff convergence and pointed Gromov-

Hausdorﬀ convergence. Does Gromov-Hausdorﬀ convergence imply pointed Gromov-

Hausdorﬀ convergence?

n In this section, we focus our study on Euclidean space, and we let A, Ai (R be closed and non-empty. We show that there is indeed a relationship in Euclidean space between Gromov-Hausdorff convergence and chordal Hausdorff convergence, and we will see in Chapter 5 that there is a relationship between chordal Hausdorff convergence and pointed Gromov-Hausdorff convergence with respect to various conformal metrics.

Combining the results of this section with those in Chapter 5, we determine a relationship between Gromov-Hausdorﬀ convergence and pointed Gromov-Hausdorﬀ convergence in

Euclidean space.

We use Fact 2.3.4 to help us determine a relationship between Gromov-Hausdorﬀ convergence and chordal Hausdorﬀ convergence in Euclidean space. We note that the following result easily follows from Fact 2.3.4:

Suppose dGH (Ai,A) → 0. Then there exists a sequence (εi) of positive numbers converging to 0 so that for each i, there exists an εi-rough isometry fi : Ai → A so that fi(Ai) Hausdorﬀ converges to A. Although the preceding result is trivial, we now discuss better and “less obvious”

n results in Euclidean space. We first present a relationship in Rˆ between Gromov- n Hausdorff convergence and Hausdorff convergence. Here, remember that A, Ai (R ˆ ˆ ˆ n are closed and non-empty. Also let A and Ai denote the closures of A and Ai in R respectively. We remind the reader that the Euclidean chordal distance χ was defined

χ in Section 2.1. We use dGH to represent Gromov-Hausdorﬀ distance with respect to χ, χ and we use dH to denote Hausdorﬀ distance with respect to χ.

42 χ ˆ ˆ Theorem 4.2.1. Suppose that dGH (Ai, A) → 0. Then there exist isometries ˆ n ˆ n χ ˆ ˆ fi : R → R so that dH (fi(Ai), A) → 0.

ˆ n n n+1 ˆ ˆ Proof. Note that R = S ⊂ R . Thus we will think of Ai and A as being subsets of n ˆ ˆ S with dGH (Ai, A) → 0, and in this context, we will show that there exist isometries n n ˆ ˆ Ti : S → S so that dH (Ti(Ai), A) → 0. By Fact 2.3.4, we know that for each i, we can let hi : Aî → Aˆ be a 4dGH (Aî, Aˆ)−rough isometry. Recall that 0 denotes the origin in ˆ ˆ n Euclidean space (see Section 2.1). Note that since Ai and A are subsets of S , neither ˆ ˆ n+1 n+1 Ai nor A contains the Euclidean origin 0 in R . Extend hi to 0 ∈ R by letting hi ˆ ˆ fix 0. Let N ∈ N so that whenever i ≥ N, 4dGH (Ai, A) ≤ 1. Then whenever i ≥ N, we n+1 n+1 know by Fact 2.3.7 that there exists an isometry Si : R → R so that s 4dGH (Aî, Aˆ) sup{|Si(x) − hi(x)| : x ∈ {0} ∪ Aî} ≤ cn diam(Aî ∪ {0}) diam(Aî ∪ {0}) q = 2cn dGH (Aî, Aˆ)diam(Aî ∪ {0}) q ≤ 2cn 2dGH (Aî, Aˆ),

n+1 n+1 where cn depends only on the dimension n. For each i ≥ N, let Ti : R → R be the n isometry given by Ti(x) = Si(x) − Si(0). Since Si is surjective, Si(S ) is the boundary of n n ˆ ˆ B(Si(0), 1). Then Ti(S ) = S . We will show that dH (Ti(Ai), A) → 0, and so the desired n ˆ ˆ isometries fi are then the restriction of Ti to S . For each i, let Mi := dGH (Ai, A). Let a ∈ A,ˆ and let i ≥ N. Because hi : Aˆi → Aˆ is a 4Mi-rough isometry, we will let ai ∈ Aˆi so that |hi(ai) − a| ≤ 4Mi. Then p |Si(ai) − a| ≤ |Si(ai) − hi(ai)| + |hi(ai) − a| ≤ 2cn 2Mi + 4Mi.

Then

|Ti(ai) − a| = |Si(ai) − Si(0) − a| ≤ |Si(ai) − a| + |Si(0)| p = |Si(ai) − a| + |hi(0) − Si(0)| ≤ 4cn 2Mi + 4Mi.

43 √ Note that 4cn 2Mi + 4Mi → 0 as i → ∞. Now let i ≥ N, and let bi ∈ Aˆi. Then hi(bi) ∈ Aˆ so that

|Ti(bi) − hi(bi)| = |Si(bi) − Si(0) − hi(bi)| ≤ |Si(bi) − hi(bi)| + |Si(0)| p = |Si(bi) − hi(bi)| + |Si(0) − hi(0)| ≤ 4cn 2Mi. √ Note that 4cn 2Mi → 0 as i → ∞. Then dH (Ti(Aˆi), Aˆ) → 0.

We now prove Theorem 1.0.1. We remind the reader that the statement of Theo- rem 1.0.1 is as follows:

n Suppose A, Ai (R are non-empty closed sets so that dGH (Ai,A) → 0. Then there n n ˆ exist isometries Si : R → R so that dH (Si(Ai),A) → 0.

Proof. First suppose that A contains only one point a. For each i, let ai ∈ Ai, and let

n n Si : R → R be given by Si(z) := z − ai + a. Then note that Si(Ai) contains {a} = A.

Since dGH (Ai,A) → 0, we know that diam(Ai) → diam(A) = 0 by Observation 2.3.6.

Then since diam(Si(Ai)) = diam(Ai), diam(Si(Ai)) → 0. Then dH (Si(Ai), {a}) → 0. ˆ That is, dH (Si(Ai),A) → 0, and so dH (Si(Ai),A) → 0 by Remark 2.4.2.

Now suppose that A contains at least two distinct points. Then note that Ai has at least two distinct points for all suﬃciently large i. Let (εi) be a sequence of positive numbers so that εi → 0 and so that for each i, dGH (Ai,A) < εi/2. Also let di be an

(εi/2)-admissible distance on Ai t A. Since translations are Euclidean isometries, there is no harm in assuming that 0 ∈ A.

Note that isometrically, Ai can be seen as both a subset of Ai t A and as a subset of n. Similarly C := A ∩ B [0, √1 ] can be seen isometrically as both a subset of R i i di 2 εi n Ai ⊂ (Ai t A) and as a subset of Ai ⊂ R . Thus we can think of Ci as being a bounded n n and closed subset of Ai ⊂ R . In particular, Ci can be seen as a compact subset of R . We will use this compact set in conjunction with Fact 2.3.7 to ﬁnd Euclidean isometries n n ˆ Si : R → R so that dH (Si(Ai),A) → 0.

44 By Fact 2.3.4, there is an εi−rough isometry fi : Ai → A. Note that by Observa- tion 2.3.5, fi can be chosen so that for each b ∈ Ai, di(fi(b), b) < εi/2. √ Let Di := diamdi (Ci) = diam|.|(Ci) ≤ 1/ εi. Let c 6= 0 be a ﬁxed point in A. Let √ N1 ∈ N be so that if i ≥ N1, 1/(2 εi) ≥ |c| + (εi/2) > (5/2)εi. Suppose i ≥ N1. In order to be able to use Fact 2.3.7, we show that εi/Di < 1. Since di is an εi/2-admissible distance on Ai t A, there exist wi, zi ∈ Ai so that di(0, wi) < εi/2 and di(zi, c) < εi/2. Note that since εi 1 1 di(0, wi) < ≤ √ − |c| < √ , 2 2 εi 2 εi we have that wi ∈ Ci. Also

εi 1 di(0, zi) ≤ di(0, c) + di(zi, c) < |c| + ≤ √ , 2 2 εi and so zi ∈ Ci. Then Di ≥ |zi − wi| = di(zi, wi) ≥ di(0, c) − di(c, zi) − di(0, wi) > |c| − εi.

Since |c| + (εi/2) > (5/2)εi, εi < |c|/2, and so |c| − εi > |c|/2. Then Di > |c|/2 > εi. So

εi/Di < 1.

n n Then by Fact 2.3.7 whenever i ≥ N1, there exists an isometry Si : R → R and a constant cn so that p 1/4 kSi − fikCi ≤ cn εiDi ≤ cnεi .

ˆ n Let α > 0, and let R > 0 so that diamdˆ(R \ B|.|[0,R]) < α. We will show that for ˆ suﬃciently large i, dH (Si(Ai),A) < α. We will start by showing that Si(Ai) and A are pretty “close” in B[0,R]. That is, we show that each point in A ∩ B[0,R] is “close” to a point in Si(Ai), and then we show that each point in Si(Ai) ∩ B[0,R] is “close” to a √ point in A. Let N2 ∈ N be so that N2 ≥ N1 and whenever i ≥ N2, 1/(2 εi)−(εi/2) ≥ R 1/4 and cnεi + εi < α. Let i ≥ N2, and let a be a point in B[0,R] ∩ A. Recall that di is an (εi/2)-admissible distance on Ai t A. Then for each i, we can let ai ∈ Ai so that di(ai, a) ≤ εi/2. We will show that Si(ai) is very “close” to a. Now 1 di(0, ai) ≤ di(0, a) + di(ai, a) ≤ √ . 2 εi

45 1/4 Then ai ∈ Ci, and so |Si(ai) − fi(ai)| ≤ cnεi . Also,

|fi(ai) − a| = di(fi(ai), a) ≤ di(fi(ai), ai) + di(ai, a) ≤ εi.

Now we will show that each point in Si(Ai) ∩ B[0,R] is “close” to a point in A. Pick √ 1/4 N3 ∈ N so that N3 ≥ N2 and whenever i ≥ N3, 1/(2 εi) − (3εi/2) − cnεi ≥ R. Let i ≥ N3. Suppose bi ∈ Ai so that Si(bi) ∈ B[0,R]. Let b0 ∈ Ai with di(b0, 0) ≤ εi/2. Since √ εi/2 ≤ 1/(2 εi), b0 ∈ Ci. So

1/4 |fi(b0) − Si(b0)| ≤ cnεi . (∗)

Also

|fi(b0)| = di(0, fi(b0)) ≤ di(0, b0) + di(b0, fi(b0)) ≤ εi. (∗∗)

Consequently,

di(bi, b0) = |bi − b0| = |Si(bi) − Si(b0)|

≤|Si(b0) − fi(b0)|+|fi(b0)|+|Si(bi)|

1/4 1 εi ≤ cnεi + εi + R ≤ √ − . 2 εi 2 So √ di(bi, 0) ≤ di(bi, b0) + di(b0, 0) ≤ 1/(2 εi).

ˆ 1/4 As a result, bi ∈ Ci. Thus d(Si(bi), fi(bi)) ≤ |Si(bi) − fi(bi)| ≤ cnεi < α, and so

Si(bi) is “close” to the point fi(bi) ∈ A. We now brieﬂy summarize the above results. For each ε > 0 and T 0 > 0, there exists

Q ∈ N so that whenever i ≥ Q, the following conditions hold: 0 (i) whenever a ∈ A ∩ B[0,T ], there exists ai ∈ Ai so that dˆ(a, Si(ai)) < ε, and

0 (ii) whenever ai ∈ Ai so that Si(ai) ∈ B[0,T ], there exists a ∈ A so that dˆ(a, Si(ai)) < ε.

46 We break the remaining portion of the proof into two cases. First suppose that A is bounded with respect to Euclidean distance. Let T > 0 so that A ⊂ B[0,T ]. We will

0 0 show that there exists T > 0 so that Si(Ai) ⊂ B[0,T ] for all but ﬁnitely many i. This will be of great beneﬁt as it will allow us to use result (ii) above. Let i ≥ N3. Now

1/4 |Si(b0)| ≤ |Si(b0) − fi(b0)| + |fi(b0)| ≤ cnεi + εi by (*) and (**).

Let z ∈ Ai. Then

1/4 |Si(z)| ≤ |Si(z) − Si(b0)| + |Si(b0)| = |z − b0| + |Si(b0)| ≤ |fi(z) − fi(b0)| + cnεi + 2εi 1/4 1/4 ≤ |fi(z)| + |fi(b0)| + cnεi + 2εi ≤ 2T + cnεi + 2εi.

1/4 Then Si(Ai) ⊂ B[0, 2T +cnεi +2εi]. Note that for all suﬃciently large i, the inequality 1/4 cnεi + 2εi < 1 holds. Then for suﬃciently large i, both A and Si(Ai) are contained in ˆ B[0, 2T + 1]. Then dH (Si(Ai),A) → 0 by results (i) and (ii) above. Now suppose that A is unbounded with respect to Euclidean distance. Since A is unbounded and since dGH (Ai,A) → 0, we let N4 ∈ N so that whenever i ≥ N4,Ai n is unbounded. Then whenever i ≥ N4,Si(Ai) ∩ (R \ B|.|[0,R]) 6= ∅. Also note that n A∩(R \B|.|[0,R]) 6= ∅. Remember that by the way we chose R, points in Si(Ai)\B|.|[0,R] and A \ B|.|[0,R] all lie within the chordal distance α of each other. By the preceding ˆ sentence and by (i)-(ii) above, dH (Si(Ai),A) → 0.

In Chapter 5, we use the preceding theorem to demonstrate a relationship between

Gromov-Hausdorff convergence and pointed Gromov-Hausdorff convergence for domains in Euclidean space. We then use this pointed Gromov-Hausdorff convergence to provide approximations of the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances.

47 4.3 Carath´eodory Convergence

Here we study Carathéodory convergence and compare it to chordal Hausdorff convergence. Throughout this section, X is a metric space, and (Ui) is a sequence of open sets in X. Following [BM], we define the core of (Ui) by

[ ◦ core(Ui) := (Ui ∩ Ui+1 ∩ Ui+2 ∩ ...) , i≥1 and we say that (Ui) Carath´eodory converges to its core if and only if each subsequence of (Ui) has the same core. There are indeed several relationships between chordal Hausdorﬀ convergence and

Carath´eodory convergence. We present one of these relationships below. Recall that when X is unbounded, Xˆ = X ∪ {∞}, and when X is bounded, Xˆ = X. The following theorem utilizes the idea of a closed subsequential limit. If (X, d) is a metric space and

(Ai) is a sequence of sets in X,ˆ we say that A ⊂ Xˆ is a closed subsequential limit of (Ai) ˆ if dH (Ai,A) → 0 and A is a closed set in X.ˆ

Theorem 4.3.1. Let (X, d) be a metric space. Let Ai,A ⊂ Xˆ be closed sets so that

Ωi := X \ Ai 6= ∅ and Ω := X \ A 6= ∅. Suppose (Ωi) Carath´eodory converges to core(Ωi) = Ω. Then A ∩ X and A ∪ (Xˆ \ X) are the only two sets in Xˆ that are possible closed subsequential limits of (Ai) with respect to chordal Hausdorﬀ convergence.

ˆ Proof. Suppose that (Aij ) is a subsequence of (Ai), and suppose B is a closed set in X so ˆ ◦ that dH (Aij ,B) → 0. Suppose z ∈ Ω, and let p ∈ N so that z ∈ (Ωp ∩ Ωp+1 ∩ ...) . There ˆ exists t > 0 so that Bdˆ(z, t) ⊂ (Ωp ∩ Ωp+1 ∩ ...). Then remembering that dH (Aij ,B) → 0, we know that z∈ / B. Thus Ω ⊂ X \ B. ˆ Now suppose that w ∈ X \B, and let r > 0 so Bdˆ(w, r) ⊂ X \B. Since dH (Aij ,B) → 0, we can let n ∈ N so that whenever j ≥ n, Bdˆ(w, r/2) ⊂ Ωij . Then we know that ◦ w ∈ (Ωin ∩ Ωin+1 ∩ ...) . Then w ∈ core(Ωij ) = Ω. Then X \ B ⊂ Ω. Since X \ B ⊂ Ω and Ω ⊂ X \ B, Ω = X \ B. Thus B is either A ∩ X or A ∪ (Xˆ \ X).

48 We remark that in the setting of the preceding theorem, any closed subsequential limit of (Ai) is equal to the closure of \ [ Ai n∈N i≥n in Xˆ by De Morgan’s Laws.

Also note that in the preceding theorem, it is possible for there to be subsequential limits of (Ai) other than the two mentioned. However, A ∩ X and A ∪ (Xˆ \ X) are the only possible subsequential limits which could be closed in X.ˆ Note though that if A is unbounded, then A ∩ X is not closed in X.ˆ Also note that we are not claiming the existence of a subsequential limit. In fact if X is not proper, there may potentially be no subsequential limits of (Ai) with respect to chordal Hausdorﬀ convergence. Below we give such an example.

Example 4.3.2. Let X = R, and let d be the distance function on X given by

d(x, y) := min{|x − y|, 1} whenever x, y ∈ R. For each i, let Ai := {0, 1, 2, ..., i}, and let Ωi := X \ Ai. Also let A := {0, 1, 2, 3, ...}, and let Ω := X \ A. Then (Ωi) Carath´eodory converges to core(Ωi) = Ω. However, (Ai) has no subsequential limit with respect to chordal Hausdorﬀ convergence.

In the case that X is proper, we can prove the following relationship between Carath´eodory convergence and chordal Hausdorﬀ convergence. It is important to note that in the next theorem, ∞ ∈ A ∩ Ai when X is unbounded.

Theorem 4.3.3. Let (X, d) be a proper metric space. Let Ωi, Ω ⊂ X be non-empty and ˆ open. Let Ai,A denote Xˆ \ Ωi, Xˆ \ Ω respectively. Then dH (Ai,A) → 0 if and only if

(Ωi) Carath´eodory converges to core(Ωi) = Ω.

ˆ Proof. (⇒). First suppose that dH (Ai,A) → 0. Let (Ωij ) be a subsequence of (Ωi).

Suppose x ∈ Ω. Then there exists r > 0 so that Bdˆ(x, r) ⊂ Ω. Then remembering that

49 ˆ dH (Ai,A) → 0, we will let n ∈ N so that whenever j ≥ n, Bdˆ(x, r/2) ⊂ Ωij . Then x ∈ core(Ωij ), and so Ω ⊂ core(Ωij ). ◦ Suppose y ∈ core(Ωij ), and let m ∈ N so that y ∈ (Ωim ∩ Ωim+1 ∩ ...) . Then we will let s > 0 so that Bdˆ(y, s) ⊂ (Ωim ∩ Ωim+1 ∩ ...). By the preceding sentence and because ˆ dH (Ai,A) → 0, y∈ / A. Then y ∈ Ω. Then core(Ωij ) ⊂ Ω. Thus core(Ωij ) = Ω, and as

(Ωij ) is an arbitrary subsequence of (Ωi), we know that (Ωi) Carath´eodory converges to core(Ωi) = Ω.

(⇐). Now suppose that (Ωi) Carath´eodory converges to core(Ωi) = Ω. Let (Ain ) be a subsequence of (Ai). Since X is proper, (Ain ) must subconverge to some closed set ˆ B in X with respect to chordal Hausdorﬀ distance. Say that (Aij ) is a subsequence of ˆ ˆ (Ain ) so that dH (Aij ,B) → 0. If X is bounded, A = A ∩ X = A ∪ (X \ X), and then by

Theorem 4.3.1, B = A. Suppose X is unbounded. Then ∞ ∈ Ai for each i, and so ∞ is contained in the closure of any chordal Hausdorﬀ limit of (Aij ). In particular since B is closed in X,ˆ ∞ ∈ B. Then B 6= A ∩ X, and so B = A ∪ (Xˆ \ X) by Theorem 4.3.1.

However since ∞ ∈ A, A = A ∪ (Xˆ \ X), and so B = A. Thus every subsequence of

(Ai) has a further subsequence which converges to A with respect to chordal Hausdorﬀ ˆ distance. Thus dH (Ai,A) → 0.

We note the following example of Carath´eodory convergence in the Banach space `2.

Here `2 is the standard space of square summable real sequences, and we denote the norm in `2 by k.k2.

Example 4.3.4. Let 0 := (0, 0, 0, 0, ...), let e1 := (1, 0, 0, 0, 0, ...), let e2 := (0, 1, 0, 0, 0, ...), let e3 := (0, 0, 1, 0, 0, 0, ...), and continue this pattern deﬁning ei for each i ∈ N. Let

V := `2 \{0}, and for each i, let Vi := `2 \{0 ∪ ei} . Let ki, k denote quasihyperbolic distance in Vi,V respectively, and let δi, δ be the distance to ∂Vi, ∂V respectively in `2.

Note that (Vi) Carath´eodory converges to core(Vi) = V. However, we later show that

(Vi, ki) does not pointed Gromov-Hausdorﬀ converge to (V, k).

50 We note that each space (Vi, ki) can be seen to be isometric to (V1, k1) via the mapping

th st which switches the i and 1 components of each element of `2. However, we also remark that each space (Vi, ki) is not isometric to (V, k). This can be seen simply by showing that (V1, k1) is not isometric to (V, k). From this, we then see that (Vi, ki) does not pointed Gromov-Hausdorﬀ converge to (V, k). In proving that (V1, k1) is not isometric to (V, k), we make use of the following observation.

Observation 4.3.5. If a, b ∈ V so that kak2 = kbk2, then k(a, b) ≤ π.

The validity of the above observation can easily be seen by noting that a, b, and 0 belong to an isometric embedding of C in `2. If a, b ∈ C \{0} so that |a| = |b|, it is not hard to see that the quasihyperbolic distance in C \{0} between a and b is no more than the quantity π. We now demonstrate that (V1, k1) is not isometric to (V, k).

Remark 4.3.6. The space (V1, k1) is not isometric to (V, k).

Proof. Suppose f : V1 → V is a bijective map. Suppose for contradiction that f is a quasihyperbolic isometry. Let g1 be the map from the interval (0, 1) ⊂ R to V1 given + + by g1(t) := te1. Let R := {t ∈ R | t > 0}, and let g2 : R → V1 be the map given by g2(t) := −te1. Let γ1, γ2 be the images of g1, g2 respectively in V1. Note that

k1 k dH (γ1, γ2) = ∞. Then dH (f(γ1), f(γ2)) = ∞.

We will show that f(γ1) is unbounded and that 0 is an accumulation point of f(γ1).

Let α0 be the image under g1 of the interval (0, 1/2] ⊂ R, and let α1 be the image under g1 of the interval [1/2, 1) ⊂ R. Note that neither α0 nor α1 is bounded with respect to quasihyperbolic distance in V1. Hence neither f(α0) nor f(α1) is bounded with respect to quasihyperbolic distance in V.

Let R ≥ r > 0. Let S[0, r, R] := {x ∈ V | r ≤ kxk2 ≤ R}. We will show that S[0, r, R] is bounded in V with respect to the distance k. Let a, b ∈ `2 so that kak2 = r and kbk2 = R. Let β be a rectiﬁable path in V from a to b. Then whenever x ∈ S[0, r, R], Z ds k(a, x) ≤ π + < ∞, β δ

51 where the quantity π comes directly from Observation 4.3.5. Hence neither f(α0) nor f(α1) is contained in S[0, r, R] for any R ≥ r > 0.

Consequently, if both f(α0) and f(α1) are bounded, then 0 is an accumulation point

k of both f(α0) and f(α1), and dH (f(α0), f(α1)) < ∞ by Observation 4.3.5. Also if there exists T > 0 so that f(α0) ∪ f(α1) ⊂ V \ Bk.k2 (0,T ), then both f(α0) and k f(α1) are unbounded, and dH (f(α0), f(α1)) < ∞ by Observation 4.3.5. However since

k1 k dH (α0, α1) = ∞, we have dH (f(α0), f(α1)) = ∞. Thus f(α0) ∪ f(α1) = f(γ1) is unbounded, and 0 is an accumulation point of f(γ1). In a similar manner, it can be shown that f(γ2) is unbounded and that 0 is an accumulation point of f(γ2).

We have shown that f(γ1) and f(γ2) are both unbounded, and 0 is an accumula-

k tion point of both f(γ1) and f(γ2). Then dH (f(γ1), f(γ2)) < ∞ by Observation 4.3.5. k However, recall that dH (f(γ1), f(γ2)) = ∞ if f is a quasihyperbolic isometry. Thus f absolutely cannot be a quasihyperbolic isometry.

As a direct result of the preceding remark, we make the following observation.

Observation 4.3.7. Although (Vi) Carath´eodory converges to core(Vi) = V, the quasihyperbolic spaces (Vi, ki) do not pointed Gromov-Hausdorﬀ converge to (V, k).

Thus far, we have studied relationships between various forms of convergence. In the next chapter, we will focus on using these relationships to approximate the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances.

52 Chapter 5

Applications

In this chapter, we use our previous results to provide approximations of various conformal metrics such as the quasihyperbolic metric, Ferrand metric, Kulkarni-Pinkall-

Thurston metric, and hyperbolic metric. In order to do this, we show that chordal

Hausdorff convergence implies bounded uniform convergence of the metrics, and this then implies pointed Gromov-Hausdorff convergence by Theorem 3.1.2. Thus we can es- timate the metrics by determining “nice” spaces which Hausdorff converge in the chordal sense. We therefore then focus on finding sequences of finite sets which yield chordal

Hausdorﬀ convergence.

As before, (X, d) is a metric space. A and Ai are closed, non-empty subsets of X so that A 6= X 6= Ai. We deﬁne the open sets Ω := X \ A and Ωi := X \ Ai. We also use δ and δi to represent distance to A and Ai respectively. Using this notation, we begin by discussing an approximation for quasihyperbolic distance, and then we extend our ideas to the Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances.

53 5.1 Quasihyperbolic Distance

We want to discuss the quasihyperbolic metric. However, even in the case that X is a quasihyperbolic super space, (1/δ) ds and (1/δi) ds do not necessarily represent the quasihyperbolic metrics on Ω and Ωi respectively. Most certainly in the case that X is a quasihyperbolic super space, the metric boundary ∂Ω is isometric to the topological boundary of Ω. Recall that the topological boundary of Ω is denoted bd(Ω) (see Sec- tion 2.2.1). Note that bd(Ω) ⊂ X \ Ω. Thus whenever x ∈ Ω, the distance from x to

X \ Ω is no more than the distance from x to the topological boundary of Ω. However, the distance from x to the topological boundary of Ω is not necessarily the same as the distance from x to X \Ω. In some cases, the distance from x to the topological boundary of Ω is greater than the distance from x to X \ Ω. However if X is a complete length space, then X is a quasihyperbolic super space and δ, δi are equal to the distances to the metric boundaries of Ω, Ωi respectively. When X is a complete length space, k := d1/δ and ki := d1/δi denote the quasihyperbolic distances in Ω and Ωi respectively. Remem- ber that A, Ai ( X are non-empty closed sets, and also recall that Ω and Ωi are the complements of A and Ai respectively. In this setting, we have already done much of the work in approximating quasihyperbolic distance, and the following theorem directly follows from our results.

Theorem 5.1.1. Suppose X is a complete length space. Let a ∈ Ω and ai ∈ Ωi so that ˆ ai → a. Suppose dH (Ai,A) → 0. Then dGH∗ ((Ωi, ai, ki), (Ω, a, k)) → 0.

Proof. This directly follows from Theorem 4.1.1, Example 3.2.2, and Theorem 3.1.2.

As the next corollary demonstrates, we can replace chordal Hausdorﬀ convergence with

Gromov-Hausdorﬀ convergence in the preceding theorem when X is Euclidean space.

54 n Corollary 5.1.2. Suppose A, Ai ( X = R are non-empty closed sets and a ∈ Ω. If dGH (Ai,A) → 0, then there exist ai ∈ Ωi so that dGH∗ ((Ωi, ai, ki), (Ω, a, k)) → 0.

n n Proof. Suppose dGH (Ai,A) → 0. By Theorem 1.0.1, there exist isometries Si : R → R ˆ so that dH (Si(Ai),A) → 0. Suppose a ∈ Ω. Then there exists N ∈ N so that whenever n n i ≥ N, a ∈ R \Si(Ai), and by Theorem 5.1.1, the sequence of spaces (R \Si(Ai), a)i≥N pointed Gromov-Hausdorﬀ converges to (Ω, a, k) with respect to the quasihyperbolic

n metric. However, with respect to the quasihyperbolic metric, each space R \ Si(Ai) is −1 isometric to (Ωi, ki). Thus (Ωi,Si (a), ki)i≥N pointed Gromov-Hausdorﬀ converges to (Ω, a, k).

If X is a locally compact, complete length space, we can replace chordal Hausdorﬀ convergence with Carath´eodory convergence in the preceding theorem.

Corollary 5.1.3. Suppose X is a locally compact, complete length space. Let Ω, Ωi ( X be non-empty and open. Suppose (Ωi) Carath´eodory converges to core(Ωi) = Ω. Let a ∈ Ω and ai ∈ Ωi so that ai → a. Then dGH∗ ((Ωi, ai, ki), (Ω, a, k)) → 0.

ˆ Proof. By Fact 2.2.1, X is proper. Then dH (Xˆ \Ωi, Xˆ \Ω) → 0 by Theorem 4.3.3. Then dGH∗ ((Ωi, ai, ki), (Ω, a, k)) → 0 by Theorem 5.1.1.

In order to approximate quasihyperbolic distance by utilizing the previous theorem, we focus on determining a sequence of ﬁnite subsets of A which chordal Hausdorﬀ converge.

Proposition 5.1.4. Suppose (X, d) is a locally compact, complete length space, and ˆ ∅ 6= A ⊂ X is closed. Then there exist ﬁnite sets Ai ⊂ A so that dH (Ai,A) → 0 and

(X \ Ai) Carath´eodory converges to core(X \ Ai) = X \ A.

Proof. By Fact 2.2.1, X is proper. Let o ∈ A. Let (Ri) be a sequence of positive numbers so that Ri → ∞. Since X is proper, A ∩ Bd[o, Ri] is compact. By compactness, we can

i let Ai be a ﬁnite (1/2) -net for A ∩ Bd[o, Ri] with Ai ⊂ A.

55 ˆ We show that dH (Ai,A) → 0. First suppose that A is bounded. Let M ∈ N so that i whenever i ≥ M,A ⊂ Bd[o, Ri]. Note that whenever i ≥ M, dH (Ai,A) ≤ (1/2 ). Then ˆ dH (Ai,A) → 0, and so dH (Ai,A) → 0 by Remark 2.4.2.

Now suppose that A is unbounded. Let ε > 0. Since Ri → ∞, the chordal diameter ˆ ˆ diamdˆ(X \ Bd[o, Ri]) → 0. Pick N ∈ N so that whenever i ≥ N, diamdˆ(X \ Bd[o, Ri]) is less than ε/3. Let a ∈ A \ Bd[o, RN ]. Pick P ∈ N so that whenever i ≥ P, a ∈ Bd[o, Ri]. i Pick S ∈ N so that whenever i ≥ S, (1/2) < ε/3. Let i ≥ max{N,P,S}. Suppose i x ∈ A ∩ Bd[o, Ri]. Then there exists y ∈ Ai so that d(x, y) ≤ (1/2) < ε/3. Then by ˆ Remark 2.4.1, the chordal distance d(x, y) ≤ d(x, y) < ε/3. Now suppose z ∈ A\Bd[o, Ri].

i Let ai ∈ Ai so that d(a, ai) ≤ (1/2) . Then 2ε 2ε dˆ(a , z) ≤ dˆ(a , a) + dˆ(a, ∞) + dˆ(∞, z) < d(a , a) + ≤ (1/2)i + < ε. i i i 3 3 ˆ Then Ai ⊂ A ⊂ Ndˆ(Ai, ε), and so dH (Ai,A) ≤ ε whenever i ≥ max{N,P,S}. Then ˆ dH (Ai,A) → 0. Then (X \ Ai) Carath´eodory converges to core(X \ Ai) = (X \ A) by Theorem 4.3.3.

The previous proposition is a very nice result. Coupled with Theorem 4.1.1, Exam- ple 3.2.2, and Theorem 3.1.2, we see that pointed Gromov-Hausdorff convergence can be attained via spaces with only finitely many boundary points. The quasihyperbolic metric on these “nice” spaces can then serve as a means to approximate quasihyperbolic distance in general domains of a complete, locally compact, length space. This is especially helpful when the metric boundary of the domain is complex, as for instance in the well-known Koch snowflake (see for instance [S1] for a description of the Koch snowflake).

n We now provide a couple examples of domains Ωi, Ω (R so that there exists ai ∈ Ωi and a ∈ Ω for which dGH∗ ((Ωi, ai, ki), (Ω, a, k)) → 0. Here ki, k is quasihyperbolic

56 distance in Ωi, Ω respectively. However, in each of the following examples in this section, ˆ ˆ n ˆ n dH (R \ Ωi, R \ Ω) 6→ 0, and (Ωi) does not Carath´eodory converge to Ω.

Example 5.1.5. Let

2 2 2 U1 := {(x, y) ∈ R | (x + 1) + y < 1},

2 U2 := {(x, y) ∈ R | −1/n < x < 1/n, −1/n < y < 1/n}, and

2 2 2 U3 := {(x, y) ∈ R | (x − 1) + y < 1}.

Let p denote the point (1, 0) ∈ U3. For each i ∈ N, let Ωi := U1 ∪ U2 ∪ U3, and let

Ω := U3. Then (Ωi, p, ki) pointed Gromov-Hausdorﬀ converges to (Ω, p, k). However, ˆ ˆ n ˆ n dH (R \ Ωi, R \ Ω) does not converge to zero. Further (Ωi) Carath´eodory converges to core(Ωi) = U1 ∪ U3 rather than Ω.

n Example 5.1.6. For each i, let Ωi := B(0, 1/i) ⊂ R be the open ball of radius 1/i centered at the origin 0. Let Ω := B(0, 1). Then dGH∗ ((Ωi, 0, ki), (Ω, 0, k)) → 0. However ˆ ˆ n ˆ n dH (R \ Ωi, R \ Ω) 6→ 0, and also (Ωi) Carath´eodory converges to core(Ωi) = ∅ rather than Ω.

5.2 SNCF Distance

Unfortunately, if X is not locally compact, it may be impossible to attain pointed

Gromov-Hausdorﬀ convergence by “nice” spaces having only ﬁnitely many boundary points. We provide an example to demonstrate this. For this example, we let (X, d) be

2 the open unit disk in R with the SNCF distance. The SNCF distance is deﬁned so that whenever x and y are points in the open unit disk, the distance d between x and y is

  |x − y| if x, y, and the origin are collinear d(x, y) :=  |x| + |y| otherwise

57 Note that in this setting, (X, d) is not locally compact. For each natural number j, let

Qj be a ﬁnite subset of ∂X = X \ X. Let Yj := X \ Qj. Let k be the quasi-hyperbolic distance on X, and let kj be the quasi-hyperbolic distance on Yj. We will show that the pointed metric spaces (Yj, 0) do not pointed Gromov-Hausdorﬀ converge to (X, 0) with respect to quasi-hyperbolic distance. Here, recall that 0 denotes the origin. Let δ(z) denote the distance in the SNCF metric from z ∈ X to ∂X, and let δj(z) denote the distance in the SNCF metric from z ∈ Yj to the nearest point in Qj.

Lemma 5.2.1. Whenever x ∈ X, k(0, x) = − log(1 − |x|).

Proof. Let x ∈ X. Let γ : [0, 1] → C be given by γ(t) = tx. Note that by the way the SNCF metric is deﬁned, γ is most certainly a quasi-hyperbolic geodesic from 0 to x.

Then Z 1 Z 1 |dz| Z 1 |dtγ0(t)| k(0, x) = `k(γ) = |dz| = = . γ δ(z) 0 1 − |z| 0 1 − t|x|

Using the substitution w = 1 − t|x|, we get that Z 1 |dtγ0(t)| Z 1−|x| −1 1 = |x|dw = − log(1 − |x|). 0 1 − t|x| 1 |x| w

y y Lemma 5.2.2. Let y ∈ Yj. If |y| ∈ Qj, then kj(0, y) = − log(1−|y|). If |y| is not a point in Qj, then kj(0, y) < 1.

y Proof. If |y| ∈ Qj, then the calculation from Lemma 5.2.1 shows that kj(0, y) is equiva- y lent to − log(1 − |y|). So suppose |y| is not a point in Qj. Let γ : [0, 1] → C be given by γ(t) = ty. Then Z |dz| Z 1 |dtγ0(t)| kj(0, y) ≤ = . γ δ(z) 0 1 + t|y|

Using the substitution w = 1 + t|y|, we get that Z 1 |y||dt| Z 1+|y| 1 1 = |y| dw = log(1 + |y|) < 1. 0 1 + t|y| 1 |y| w

58 1 Proposition 5.2.3. For each j, distGH∗ ((Yj, 0, kj), (X, 0, k)) ≥ 3 .

1 Proof. Suppose there is j so that distGH∗ ((Yj, 0, kj), (X, 0, k)) < 3 . By Fact 2.3.11, there 2 2 is a map f : Bkj [0, 3] → X that is a 3 -rough isometry from (Bkj [0, 3], 0) to (Bk[0, 3− 3 ], 0) so that f(0) = 0. The set of points on the unit circle is uncountable, and only ﬁnitely many of them are in Qj. So we can choose a sequence of distinct points (ζn) on the unit

2 −3 circle so that for all n ∈ N, ζn ∈/ Qj. Now let xn = (1 − e 3 )ζn, and note that then 2 −3 xn ∈ X, xn/|xn| = ζn, and |xn| = 1 − e 3 . For each n, we have from Lemma 5.2.1 that 2 k(0, x ) = − log(1 − |x |) = 3 − . n n 3 2 2 2 Then for each n, xn ∈ Bk[0, 3 − 3 ]. Then as f(Bkj [0, 3]) is a 3 -net for Bk[0, 3 − 3 ], for 2 each n ∈ N, there exists yn ∈ Bkj [0, 3] so that k(xn, f(yn)) < 3 . (*)

Then for each n, k(xn, f(yn)) < k(0, xn). Note that whenever a and b are points in

X so that ca is the point in ∂X closest to a and cb is the point in ∂X closest to b, then if ca 6= cb, the quasi-hyperbolic geodesic from a to b includes the line segment with endpoints a and 0 as well as the line segment with endpoints 0 and b. Then as k(xn, f(yn)) < k(0, xn), we know that f(yn) lies on the interval (0, xn/|xn|) for each n.

2 Now since f is a 3 -rough isometry, 2 |k (0, y ) − k(f(0), f(y ))| < j n n 3 for each n, and so 2 2 k (0, y ) > k(f(0), f(y )) − = k(0, f(y )) − . (∗∗) j n n 3 n 3 Now for each n, 2 k(0, x ) ≤ k(0, f(y )) + k(f(y ), x ) < k(0, f(y )) + by (*). n n n n n 3 That is, 2 k(0, f(y )) > k(0, x ) − . n n 3

59 Then for each n, 2 k (0, y ) > k(0, f(y )) − by (**) j n n 3 2 2 > k(0, x ) − − by the preceding sentence n 3 3 2 4 = 3 − − = 1. 3 3

Then we know by the preceding lemma that for each n, the point in ∂X closest to yn is an element of Qj. Then as there are only ﬁnitely many points in Qj, there exists b0 ∈ Qj so that the sequence {yn} falls inﬁnitely often on the line segment L from 0 to b0. Then there is a subsequence {ynα } which lies on L. Note that {|ynα |} is a bounded sequence of real numbers. Then there is a subsequence {|yn` |} of {|ynα |} which is convergent. Note that

lim kj(0, z) = lim − log(1 − |z|) = ∞. L3z→b0 L3z→b0

Then as each yn ∈ Bkj [0, 3], the sequence {|yn` |} does not converge to |b0| (i.e., the sequence does not converge to one). Then the sequence {|yn` |} converges to an element of [0, 1), and so {yn` } converges with respect to the quasi-hyperbolic distance kj. Then there exists n 6= n so that k (y , y ) < 1 . Then remembering that f is a 2 -rough `m `p j n`m n`p 3 3 isometry, 2 1 2 k(f(yn ), f(yn )) < kj(yn , yn ) + < + = 1. `m `p `m `p 3 3 3

Then

k(x , x ) ≤ k(x , f(y )) + k(f(y ), f(y )) + k(f(y ), x ) n`m n`p n`m n`m n`m n`p n`p n`p 2 2 2 < + 1 + = 3 − = k(0, xn ) = k(0, xn ). 3 3 3 `m `p Then there is a natural number i so that x and x both lie on the line segment n`m n`p 2 −3 from 0 to ζ . Then x = (1 − e 3 )ζ = x , and so n = n . i n`m i n`p `m `p

However, this contradicts the idea that n`m 6= n`p . Then by contradiction, there does 1 not exist j so that distGH∗ ((Yj, 0, kj), (X, 0, k)) < 3 . Then note that the pointed metric spaces (Yj, 0, kj) do not pointed Gromov-Hausdorﬀ converge to (X, 0, k).

60 5.3 Ferrand, Kulkarni-Pinkall-Thurston, and Hyperbolic

Distances

n Throughout this section X is Euclidean space R unless otherwise noted, d is Euclidean n n distance in R , and dˆ is the metric chordal distance in Rˆ as described in Section 2.4.1. ˆ Also, A, Ai ( X are closed sets each having at least two points, Ω is the open set X \ A, and Ωi := X \ Ai. We show a relationship between chordal Hausdorﬀ convergence and bounded uniform convergence of the Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic metrics. In our discussion of the Ferrand metric, we let ϕ and ϕi denote the Ferrand metrics on Ω and Ωi respectively, and we let µ and µi denote the Kulkarni-Pinkall- ˆ Thurston metrics on Ω and Ωi respectively. When X = C, and Ω, Ωi are hyperbolic regions, we let λ, λi represent the hyperbolic metrics on Ω, Ωi respectively. Also δ, δi represent the distances to A, Ai respectively. We use the following characterizations of

n the Ferrand and Kulkarni-Pinkall-Thurston metrics in Ω ∩ R .

n Lemma 5.3.1. For each x ∈ Ω ∩ R , ϕ(x) = diam[Jx(A)] and µ(x) = cdiam[Jx(A)].

n n Proof. Let x ∈ Ω ∩ R , and let U be the component of Ω in Rˆ containing x. Recall that c c ϕ(x) = diam[Jx(U )] = diam[Jx(∂U)], and µ(x) = cdiam[Jx(U )] = cdiam[Jx(∂U)] (see

n Sections 2.2.2 and 2.2.3). Note that ∂U ⊂ A ⊂ (Rˆ \ U). So

ˆ n diam[Jx(∂U)] ≤ diam[Jx(A)] ≤ diam[Jx(R \ U)] = diam[Jx(∂U)], and

ˆ n cdiam[Jx(∂U)] ≤ cdiam[Jx(A)] ≤ cdiam[Jx(R \ U)] = cdiam[Jx(∂U)].

Thus ϕ(x) = diam[Jx(∂U)] = diam[Jx(A)] and µ(x) = cdiam[Jx(∂U)] = cdiam[Jx(A)].

Note that we can similarly characterize the Ferrand and Kulkarni-Pinkall-Thurston metrics in Ωi by simply replacing Ω with Ωi and A with Ai in the preceding lemma.

61 In addition to using the preceding characterizations of the Ferrand and Kulkarni-

Pinkall-Thurston metrics, we also use the Poincar´eextension of M¨obiustransformations.

n Poincaréobserved that any Möbiustransformation on Rˆ can be extended to a Möbius n+1 transformation acting on Rˆ . This extension is used and described in various literature n (See for example section 3.3 of [B]). Suppose f is a Möbiustransformation on Rˆ . n+1 The Poincaréextension f˜ of f is a Möbiustransformation on Rˆ so that whenever x1, x2, ..., xn, y1, y2, ..., yn ∈ R so that f(x1, x2, ..., xn) = (y1, y2, ..., yn),

f˜(x1, x2, ..., xn, 0) = (y1, y2, ..., yn, 0).

Suppose z1, z2, ..., zn ∈ R. In the case that f is reﬂection across the sphere centered at

(z1, z2, ..., zn) of radius r > 0, f˜ is reﬂection across the sphere centered at (z1, z2, ..., zn, 0) of radius r. If f is translation by (z1, z2, ..., zn), then f˜ is translation by (z1, z2, ..., zn, 0).

For ease of notation, if z = (z1, z2, ..., zn) and w ∈ R, we let (z, w) denote the point n+1 (z1, z2, ..., zn, w) ∈ R . We use Poincar´eextensions with regards to the following fact from Theorem 3.6.1 of [B].

n n n Fact 5.3.2. Suppose f : Rˆ → Rˆ is a M¨obiustransformation. Whenever x, y ∈ Rˆ ,

χ(f(x), f(y)) ≤ Hχ(x, y)

˜ n+1 where H = H(f) := exp(h(en+1, f(en+1))). Here en+1 := (0, 0, ..., 0, 1) ∈ R and h is n+1 hyperbolic distance in the upper half-space {z ∈ R | the dot product z · en+1 > 0}.

Because the Möbius transformation Jx is used in the characterizations of the Ferrand and Kulkarni-Pinkall-Thurston metrics, it is useful to determine H(Jx) for each x ∈ Ω. (−x, 1) Lemma 5.3.3. Whenever x ∈ Ω,H(J ) = exp(h(e , x0)) where x0 := . x n+1 1 + |x|2 ˆ n ˆ n Proof. Suppose x ∈ Ω. Let g : R → R be given by g(z) := z − x. Note that Jx = J0 ◦ g where 0 denotes the origin. The Poincaréextensiong ˜ of g is given byg ˜(z) := z − (x, 0). ˆ n+1 Also the Poincaréextension of J0 is reflection across the unit sphere (i.e., J0 for R ).

Then the Poincar´eextension of Jx is J0 ◦ g˜ = J(x,0). Now

62 (−x, 1) J (e ) = J (˜g(e )) = J ((−x, 1)) = = x0. (x,0) n+1 0 n+1 0 1 + |x|2 (−x, 1) Then H(J ) = exp(h(e , x0)) where x0 := and h is hyperbolic distance in x n+1 1 + |x|2 n+1 {z ∈ R | the dot product z · en+1 > 0}.

n We remind the reader that whenever x, y ∈ Ω ∩ R , ! d(x, y) k(x, y) ≥ log 1 + , min{δ(x), δ(y)} n where k denotes quasihyperbolic distance in Ω ∩ R . The logarithmic quantity on the right-hand side of the preceding inequality is often denoted as j(x, y) in various scholarly works, and it is known as the j-metric. Multiple connections exist between the j-metric and the quasihyperbolic and hyperbolic metrics.

The connection that we use is the well-known fact that the hyperbolic metric is no more

n+1 than 2j in the upper half-space {z ∈ R | the dot product z · en+1 > 0} (See Lemma 2.6 of [KVZ] for further details).

We now demonstrate a relationship between the Ferrand/Kulkarni-Pinkall-Thurston metrics and Hausdorﬀ convergence with respect to the distance χ. In the following the- ˆ n orem, A, Ai ( R are non-empty closed sets each having at least two distinct points. ˆ n ˆ n Also Ω := R \ A and Ωi := R \ Ai. Additionally, δ, δi are the Euclidean distances to n n A∩R ,Ai ∩R respectively. The Ferrand metrics on Ω and Ωi are ϕ and ϕi respectively, and the Kulkarni-Pinkall-Thurston metrics on Ω and Ωi are µ and µi respectively.

χ Theorem 5.3.4. Suppose dH (Ai,A) → 0 and that A contains at least two distinct points. Then (ϕi ds) and (µi ds) converge locally uniformly respectively to ϕ ds and µ ds in Ω.

n n Proof. Note that rotations of Rˆ = S are isometries with respect to the chordal, Fer- rand, and Kulkarni-Pinkall-Thurston distances. Thus we may assume that ∞ ∈ A. Also we may assume that each Ai contains at least two distinct points.

63 Let z ∈ Ω, and let 0 < R < δ(z)/2. Then the Euclidean ball B[z, 2R] ⊂ Ω, and since

χ dH (Ai,A) → 0, we can let N ∈ N be so that whenever i ≥ N,B[z, 2R] ⊂ Ωi. We show that ϕi → ϕ and µi → µ uniformly in B[z, R]. Let h denote hyperbolic distance in the

n+1 upper half-space U := {z ∈ R | the dot product z · en+1 > 0}, and let j := jU denote the j-metric distance in U. Let x ∈ B[z, R]. Following the notation of Lemma 5.3.3, let (−x, 1) (−z, 1) x0 := and z0 := . 1 + |x|2 1 + |z|2 Then

0 0 0 0 |h(en+1, x ) − h(en+1, z )| ≤ h(x , z ) = h(J0[(−x, 1)],J0[(−z, 1)]).

Note that J0 is an isometry with respect to the hyperbolic metric in U. So

h(J0[(−x, 1)],J0[(−z, 1)]) = h((−x, 1), (−z, 1)).

Remember that h ≤ 2j within U, and so

h((−x, 1), (−z, 1)) ≤ 2j((−x, 1), (−z, 1)) = 2 log(1 + |x − z|) ≤ 2 log(1 + R).

Then

H(Jx) 0 0 0 0 2 = exp[h(en+1, x ) − h(en+1, z )] ≤ exp [h(en+1, x ) − h(en+1, z )] ≤ (1 + R) . H(Jz)

Note that whenever i ≥ N,B[x, R] ⊂ B[z, 2R] is contained in both Ω and Ωi. Then

−1 −1 whenever i ≥ N,Jx(A) ⊂ B(0,R ) and Jx(Ai) ⊂ B(0,R ). Then recall from Sec- tion 2.1, the following inequality holds whenever i ≥ N and a, b ∈ A ∪ Ai, 2|J (a) − J (b)| x x ≤ χ(J (a),J (b)). 1 + (R−1)2 x x Then whenever i ≥ N,

|ϕi(x) − ϕ(x)| = |diam[Jx(Ai)] − diam[Jx(A)]| ≤ 2dH (Jx(Ai),Jx(A))

−2 χ −2 χ ≤ (1 + R )dH (Jx(Ai),Jx(A)) ≤ (1 + R )H(Jx)dH (Ai,A)

−2 2 χ ≤ (1 + R )H(Jz)(1 + R) dH (Ai,A).

64 χ Then as dH (Ai,A) → 0, ϕi → ϕ uniformly in B[z, R]. By replacing diam with cdiam,

ϕ with µ, and ϕi with µi in this proof, it can also be shown that µi → µ uniformly in B[z, R].

n Let dˆ be the metric chordal distance in Rˆ as in section 2.4.1. Using the inequalities given in sections 2.1 and 2.4.1, it is not hard to see that the Euclidean chordal distance

χ is bilipschitz equivalent to the metric chordal distance d.ˆ As a result, the following ˆ n observation regarding chordal Hausdorﬀ convergence holds. Here A, Ai ⊂ R . ˆ χ Observation 5.3.5. dH (Ai,A) → 0 iﬀ dH (Ai,A) → 0.

We use Theorem 5.3.4 to attain the following approximation with pointed Gromov- n ˆ n ˆ Hausdorﬀ convergence. Here, X is either R , R , C, or C. Also A, Ai ( X are non-empty and closed, Ω := X \ A, and Ωi := X \ Ai.

ˆ Theorem 5.3.6. Suppose dH (Ai,A) → 0. Let a ∈ Ω (Note a is then also in Ωi for

n all suﬃciently large i). If X = Rˆ and A contains at least two distinct points, then

(Ωi, a) pointed Gromov-Hausdorﬀ converges to (Ω, a) with respect to both the Ferrand ˆ and Kulkarni-Pinkall-Thurston distances. If X = C and Ω, Ωi are hyperbolic spaces, then

(Ωi, a) pointed Gromov-Hausdorﬀ converges to (Ω, a) with respect to hyperbolic distance.

ˆ n Proof. First consider the case that X = R . By Theorem 5.3.4, both ϕi → ϕ and

µi → µ boundedly uniformly in Ω. Then we attain the desired pointed Gromov-Hausdorﬀ convergence by Theorem 3.1.2.

Now consider the case that X = Cˆ. By [BM] and Theorem 3.1.2, we attain the desired pointed Gromov-Hausdorﬀ convergence with respect to the hyperbolic metric.

As a result of Theorem 1.0.1, chordal Hausdorﬀ convergence in the preceding theorem

n can be replaced with Gromov-Hausdorﬀ convergence. Here, X is either R or C, and ˆ ˆ Ai,A ( X are non-empty and closed. Let Ai, A denote the closure of Ai,A respectively in X.ˆ Let Ω := Xˆ \ Aˆ and Ωi := Xˆ \ Aˆi.

65 n ˆ Corollary 5.3.7. Suppose dGH (Ai,A) → 0, and let a ∈ Ω. If X = R and A contains at least two distinct points, then there exist ai ∈ Ωi so that (Ωi, ai) pointed Gromov- Hausdorﬀ converges to (Ω, a) with respect to both the Ferrand and Kulkarni-Pinkall-

Thurston distances. If X = C and Ω, Ωi are hyperbolic spaces, then there exist ai ∈ Ωi so that (Ωi, ai) pointed Gromov-Hausdorﬀ converges to (Ω, a) with respect to hyperbolic distance.

66 Chapter 6

More General Results

In this chapter, we look at more general spaces than in the preceding chapters. In chapter

5, we determined an approximation of the quasihyperbolic metric in domains of a locally compact, complete length space. Here, we find an alternative approximation for spaces which are locally compact, but are not necessarily length spaces. Then we explore the relationship between Gromov-Hausdorff and pointed Gromov-Hausdorff convergence in general metric spaces.

6.1 An Alternative Approximation

Throughout this section, (X, d) is a metric space, and ∅= 6 A ( X is a closed set. In chapter 5, we showed that if (X, d) is a locally compact, complete, quasihyperbolic super space that is also a length space, then there is a sequence of finite sets (Ai) so that ˆ dH (Ai,A) → 0. This then gives us an approximation of the quasihyperbolic distance on X \ Ω by Theorem 4.1.1, Example 3.2.2, and Theorem 3.1.2. The following example illustrates that we must assume that X is a length space in order to ensure the existence of such a sequence of finite sets (Ai) which chordal Hausdorff converge to A.

67 2 Example 6.1.1. Let X = R , and let d be the distance function on X given by

d(x, y) := min{|x − y|, 1}

2 whenever x, y ∈ R . Consider the origin to be the ﬁxed base point o used in deﬁning the chordal distance dˆ (See Section 2.4.1). We remind the reader that whenever 0 < r ≤ R, r < d(x, o) < R, and r < d(y, o) < R, d(x, y) d(x, y) ≤ dˆ(x, y) ≤ . 4(1 + R)2 (1 + r)2

2 Let A := {(a, b) | a, b ∈ Z}. For each i, let Ai be a ﬁnite subset of R , and then 2 ˆ ˆ 1 = dH (Ai,A) ≤ 4(1 + 2) dH (Ai,A). Hence dH (Ai,A) does not converge to 0.

The preceding example indicates that when X is not a length space, we might not be able to use chordal Hausdorﬀ convergence as a tool for attaining pointed Gromov-

Hausdorﬀ convergence of the quasihyperbolic metric via spaces with only ﬁnitely many boundary points. As a consequence, we focus on determining an alternative means for estimating quasihyperbolic distance.

By an alternative method, we approximate the quasihyperbolic metric with only

ﬁnitely many boundary points in locally compact quasihyperbolic super spaces (no as- sumption as to whether or not X is a length space is needed). This is useful because quasihyperbolic distance is often hard to compute.

In order to approximate the quasihyperbolic metric with only ﬁnitely many boundary points, we will use a countable union of compact spaces. This union forms a space which is said to be a σ−compact space. That is, a σ-compact space is a space which is a countable union of compact spaces. We have the following fact regarding σ−compact spaces from page 2 of appendix A in [S2].

Fact 6.1.2. A connected locally compact metric space is σ−compact.

The next remark easily follows.

Remark 6.1.3. Any locally compact quasihyperbolic super space is σ−compact.

68 The next remark directly follows from the deﬁnition of a locally compact metric space

(see Section 2.1).

Observation 6.1.4. In a locally compact metric space, each open set is locally compact.

We also use the following fact, which is well-known as Dini’s Theorem (See for instance

[H5, pg. 175], [RF, pg. 64], [BS, Theorem 8.2.6], or [R, Theorem 7.13]).

Fact 6.1.5. Let X be a compact topological space and for each i ∈ N, let fi : X → R be a continuous function. Suppose f : X → R is continuous and fi → f pointwise on

X. If (fi) is either a monotonically increasing sequence or a monotonically decreasing sequence, then fi → f uniformly in X.

For the remainder of this section, (X, d) is a locally compact quasihyperbolic super space, and ∅= 6 Ω ( X is a domain. Also δ denotes distance to the metric boundary ∂Ω. We let A := X \ Ω, and note that in this setting, δ is not necessarily equal to the distance to A. As in Section 2.2.1, we let bd(Ω) denote the topological boundary of Ω.

Note that ∂Ω is isometrically equivalent to bd(Ω). Thus δ is additionally equal to the distance to the topological boundary of Ω. Note that the bounded uniform convergence that we achieve in the proof of the following theorem does not necessarily hold for general bounded sets in (X, d). Instead, the bounded uniform convergence refers to sets that are bounded in the space (Ω, k).

Theorem 6.1.6. Suppose (X, d) is a locally compact quasihyperbolic super space, and

∅= 6 Ω ( X is a domain. There exists a sequence of ﬁnite subsets (Ai) of bd(Ω) so that the distance to Ai converges boundedly uniformly to δ in (Ω, k).

Proof. By Observation 6.1.4, (Ω, d) is locally compact. Recall that the identity map from (Ω, k) to (Ω, d) is a homeomorphism (See Section 2.2.1). Then (Ω, k) is also locally compact and complete, hence proper. By Remark 6.1.3, (X, d) is σ-compact. Thus let

69 C1,C2,C3, ... be compact subsets of X so that [ X = Ci. i∈N Note that by not specifying that each Ci is distinct, we ensure that such a countably inﬁnite sequence (Ci) exists even if X is compact. For each i, let Di := C1 ∪C2 ∪...∪Ci ; each Di is compact. Then also Ki := bd(Ω)∩Di is compact. Thus we can let A1 ⊂ bd(Ω)

i+1 be a finite (1/2)-net for K1. Inductively for each i, let Ai+1 ⊂ bd(Ω) be a finite (1/2) - net for Ki+1 so that Ai ⊂ Ai+1. For each i, let Ωi := X \ Ai, and let δi denote distance to ∂Ωi. We note that since Ai is a finite set, δi is equivalent to distance to Ai.

Since A1 ⊂ A2 ⊂ A3 ⊂ ... ⊂ bd(Ω), δ1(x) ≥ δ2(x) ≥ δ3(x) ≥ ... ≥ δ(x) for each x ∈ Ω.

We show that δi → δ pointwise in Ω. Let x ∈ Ω and ε > 0. Let a ∈ bd(Ω) so that

i δ(x) ≥ d(x, a) − (ε/2). Let N ∈ N so that whenever i ≥ N, a ∈ Di and (1/2) < (ε/2).

Then for each i ≥ N, we can let ai ∈ Ai so that d(a, ai) < (ε/2). Then for each i ≥ N, ε ε δ(x) ≥ d(x, a) − ≥ d(x, a ) − d(a , a) − ≥ δ (x) − ε. 2 i i 2 i

Then δi → δ pointwise in X. Let R > 0. Since (Ω, k) is proper, Bk[x, R] is compact.

Then δi → δ uniformly in Bk[x, R] by Fact 6.1.5. Thus δi → δ boundedly uniformly in (Ω, k).

As a direct consequence of the preceding theorem, we obtain an approximation of the quasihyperbolic distance on Ω via pointed Gromov-Hausdorﬀ convergence.

Corollary 6.1.7. Suppose (X, d) is a locally compact quasihyperbolic super space, and

∅= 6 Ω ( X is a domain. There exists a sequence of ﬁnite subsets (Ai) of bd(Ω) so that for each x ∈ Ω, (X \ Ai, x, ki) pointed Gromov-Hausdorﬀ converges to (Ω, x, k), where ki and k are quasihyperbolic distance in X \ Ai and Ω respectively.

Proof. This directly follows from Theorem 6.1.6 and Theorem 3.1.2.

70 We note that Proposition 5.2.3 indicates that the preceding corollary does not necessarily hold in general quasihyperbolic super spaces which are not locally compact.

6.2 Gromov-Hausdorﬀ Convergence of General Complete

Metric Spaces

In Theorem 1.0.1, we studied the connection between Gromov-Hausdorff convergence and chordal Hausdorff convergence in Euclidean space. Combining this with the results from chapter five, we have relationships between Gromov-Hausdorff convergence and pointed Gromov-Hausdorff convergence in Euclidean space. In this section, we investigate Gromov-Hausdorff convergence in general complete metric spaces. To this end, we let Ω, Ωi be quasihyperbolic metric spaces, and we let k, ki denote quasihyperbolic distance in Ω, Ωi respectively. We note that the following question is non-interesting, as it is easy to see that its answer is “sometimes but not always.”

If Ω, Ωi are quasihyperbolic metric spaces with dGH (∂Ωi, ∂Ω) → 0, does there exist ai ∈ Ωi and a ∈ Ω so that (Ωi, ai, ki) pointed Gromov-Hausdorﬀ converges to (Ω, a, k)?

In this section, we give an example of quasihyperbolic metric spaces Ω, Ωi so that dGH (∂Ωi, ∂Ω) → 0, but (Ωi, ai, ki) very clearly does not pointed Gromov-Hausdorﬀ converge to (Ω, a, k) for any a ∈ Ω and any ai ∈ Ωi. After describing our example, we answer the following related question.

Suppose that (Ai, |.|i) and (A, |.|) are non-empty, complete metric spaces for which dGH (Ai,A) → 0. Does there exist pointed quasihyperbolic metric spaces (Ωi, ai, ki), (Ω, a, k) so that Ai ≡ ∂Ωi, A ≡ ∂Ω, and (Ωi, ai, ki) pointed Gromov-Hausdorﬀ converges to (Ω, a, k)?

We now introduce some notation that we will use in our example and throughout the rest of this section. Whenever ∼ is an equivalence relation on a set S, we say that x, y ∈ S are related if x ∼ y. Whenever r > 0 and A is a complete metric space, we

71 consider the standard `1 product distance on A × [0, r). Note here that A ≡ A × {0}.

Hence in this context for ease of notation, we think of A as being “the same as” A × {0} and say that A = A × {0}.

Whenever (A, |.|) is a complete metric space, o ∈ A, and r > 0, let !, G Z(A, o, r) := A × [0, r) t Ia ∼, a∈A∗ ∗ with quotient distance. Here A = A \{o}, each Ia = [0, |a − o|] ⊂ R, and ∼ is the smallest equivalence relation satisfying the following two conditions:

∗ (1) Whenever a ∈ A , a is related to the endpoint |a − o| of the interval Ia.

∗ (2) Whenever 0 ∈ Ia for some a ∈ A , 0 ∼ o. Here, recall that o ∈ A = A × {0} and a ∈ A∗ = (A \{o}) × {0}. In the following ﬁgure, we present a visualization of the “gluing” given by the two above conditions for one of the intervals Ia.

A = A × {0} a ∈ A*

Ia = [0, |a - o|] ⊂ ℝ with 0 ~ o and |a - o| ~ a

Figure 6.1: Gluing Illustration

Whenever u ∈ A × [0, r), we simply let u denote the equivalence class [u] ∈ Z(A, o, r) for ease of notation, and in using this notation, we note that then A = A × {0} can be considered as a subset of Z(A, o, r). Similarly whenever 0 < t ≤ r, Z(A, o, t) can be considered to be a subset of Z(A, o, r).

72 By using the closed interval [0, r] rather than [0, r) in the above construction, we also define a similar space Z[A, o, r] with quotient distance. Using this notation, we provide the following example of quasihyperbolic metric spaces (Ωi, ki), (Ω, k) so that dGH (∂Ωi, ∂Ω) → 0 but for each ai ∈ Ωi and a ∈ Ω, (Ωi, ai, ki) does not pointed Gromov- Hausdorff converge to (Ω, a, k). In such cases where basepoints cannot be found to yield pointed Gromov-Hausdorff convergence, we say that (Ωi, ki) does not pointed Gromov- Hausdorff converge to (Ω, k).

Example 6.2.1. For each i, let the space Ai := {−1/i, +1/i} ⊂ R with standard Eu- clidean distance, and let A := {0}. Let r > 0, and let oi ∈ Ai. Note that the metric boundaries of Z(A, 0, r) and Z(Ai, oi, r) are isometric to A and Ai respectively. Also note that Z(A, 0, r) and Z(Ai, oi, r) are each quasihyperbolic metric spaces. Let k˜ and k˜i denote quasihyperbolic distance in Z(A, 0, r) and Z(Ai, oi, r) respectively. Then ˜ dGH (Ai,A) → 0. However Z(Ai, oi, r), ki does not pointed Gromov-Hausdorﬀ converge to Z(A, 0, r), k˜ .

Note that whenever (A, |.|) is a complete metric space with o ∈ A, the metric boundary of Z(A, o, 1) is isometric to A, and Z(A, o, 1) is a quasihyperbolic metric space. To answer the second question that we posed in this section, we will construct quasihyperbolic metric spaces (Ωi, ki) which pointed Gromov-Hausdorﬀ converge to the quasihyperbolic space Z(A, o, 1). We do this in the proof of the following proposition.

Proposition 6.2.2. Suppose that (Ai, |.|i) and (A, |.|) are non-empty, complete metric spaces so that dGH (Ai,A) → 0. Then there exist pointed quasihyperbolic metric spaces

(Ωi, ai, ki) and (Ω, a, k) with Ai ≡ ∂Ωi and A ≡ ∂Ω so that (Ωi, ai, ki) pointed Gromov- Hausdorﬀ converges to (Ω, a, k).

Proof. Let (εi) be a sequence of real numbers so that εi → 0 and for each i, the Gromov-

Hausdorﬀ distance dGH (Ai,A) < εi/2. When determining convergence of a sequence, we need only be concerned with what happens to a “tail” of the sequence. Hence without

73 loss of generality, assume that for each i, εi/2 ≤ 1/4. For each i, let mi be the largest

m integer satisfying εi/2 ≤ 1/2 i . Note that necessarily mi ≥ 2. Let oi, o be ﬁxed base points in Ai,A respectively. We will choose (Ω, k) to be the quasihyperbolic metric space Z(A, o, 1), noting that the metric boundary of Z(A, o, 1) is isometric to A.

We now give a brief sketch of the proof of this proposition, and describe the idea

−m behind the construction of the spaces (Ωi, ki). We will “glue” Z(Ai, oi, 2 i ) to the

−m +1 space Z A, o, 1 − 2 i to form quasihyperbolic metric spaces (Ωi, ki). Note that

−m −m +1 as i → ∞, mi → ∞ and Z(Ai, oi, 2 i ) becomes “smaller” while Z A, o, 1 − 2 i becomes “larger.” Thus as i → ∞,Z A, o, 1 − 2−mi+1 takes up a “larger” portion of the space (Ωi, ki) and so the quasihyperbolic balls in Ωi become more like those in Ω.

−m Also because we keep Z(Ai, oi, 2 i ) “near” the metric boundary of Ωi in our “gluing,” we ensure that ∂Ωi ≡ Ai. Now that we have discussed an outline for the proof, we provide more speciﬁc details.

Fix i ∈ N. Let fi : A → Ai be an εi-rough isometry. Note that such a rough isometry exists since dGH (Ai,A) < εi/2 (See Fact 2.3.4). For each a ∈ A, we will connect the point a, 1 − 2−mi+1 ∈ Z A, o, 1 − 2−mi+1 to the point

−mi fi(a) ∈ Ai = Ai × {0} ⊂ Z(Ai, oi, 2 )

m by “gluing” in the interval [0, 1/2 i ] ⊂ R. For a point a ∈ A, we visualize this “gluing” in the ﬁgure below:

−m +1 −m +1 For each a ∈ A, let za := (a, 1 − 2 i ) ∈ Z[A, o, 1 − 2 i ]. Also let (Ωi, di) be the space resulting from the “gluing” of the intervals described above, where di is the quotient distance. In order to ensure that di satisﬁes the triangle inequality, we chose the length 1/2mi for each of the intervals which is “glued” in the manner displayed in the preceding ﬁgure. In particular for a, b ∈ A, note that due to the length 1/2mi chosen

74 −m +1 m −m Z[A, o, 1 − 2 i ] 0 1/2 i Z(Ai, oi, 2 i ) [0, 1/2mi ]

−m +1 −m +1 m −m 0 ∼ (a, 1 − 2 i ) ∈ Z A, o, 1 − 2 i and 1/2 i ∼ fi(a) ∈ Z(Ai, oi, 2 i ).

Figure 6.2: The Quasihyperbolic Space (Ωi, ki) for the “glued” intervals,

mi di(za, zb) = |a − b| ≤ |fi(a) − fi(b)|i + εi ≤ |fi(a) − fi(b)|i + 2(1/2 )

= di(za, fi(a)) + |fi(a) − fi(b)|i + di(fi(b), zb)

= di(za, fi(a)) + di(fi(a), fi(b)) + di(fi(b), zb), and similarly

mi di(fi(a), fi(b)) = |fi(a) − fi(b)|i ≤ |a − b| + εi ≤ |a − b| + 2(1/2 )

= di(fi(a), za) + |a − b| + di(zb, fi(b))

= di(fi(a), za) + di(a, b) + di(zb, fi(b)).

Note that Ωi is a quasihyperbolic metric space. We let ki be the quasihyperbolic

−m distance in Ωi. Also note that ∂Ωi is isometric to the metric boundary of Z(Ai, oi, 2 i ).

That is, ∂Ωi ≡ Ai.

We now show that (Ωi, ki) pointed Gromov-Hausdorﬀ converges to (Ω, k). Let r > 0 and α = (a, t) ∈ A × [0, 1) with a ∈ A and t ∈ [0, 1). By the inequalities in (2) from

−r −r section 2.2.1, Bk[α, r] ⊂ Z[A, o, 1 − e + te ] ⊂ Ω.

−r −r −M+1 Let M ∈ N \{1} so that 1 − e + te < 1 − 2 . Let P ∈ N so that whenever M −r −r i ≥ P, εi/2 ≤ (1/2) . Then whenever i ≥ P,Z[A, o, 1 − e + te ] ⊂ Ωi. Recall

75 −r −r that Z[A, o, 1 − e + te ] is also a subset of Ω. Further note that k = ki within

−r −r Z[A, o, 1 − e + te ] ⊂ Ω ∩ Ωi. In particular Bki [α, r] = Bk[α, r]. Then whenever ε is such that 0 < ε < r and i ≥ P, the identity map is an ε-rough isometry from (Bk[α, r], α) to (Bki [α, r − ε], α). Then by Fact 2.3.10, dGH∗ ((Ωi, α, ki), (Ω, α, k)) → 0.

76 Chapter 7

Quasihyperbolic Distance in Euclidean Domains

In Theorem 3.1.2, we established a relationship between pointed Gromov-Hausdorff convergence and bounded uniform convergence. To explore this relationship further, now we investigate properties of quasihyperbolic isometries between domains in Euclidean space. In particular, we discuss Hästö’sresults from [H1], and we use his results to further develop a connection between pointed Gromov-Hausdorff convergence and bounded uniform convergence of quasihyperbolic metrics.

7.1 Quasihyperbolic Isometries

0 n n Unless stated otherwise, Ω, Ω (R are domains and n ≥ 2. Whenever x ∈ R , δ(x) 0 n n 0 0 and δ (x) denote the distance from x to R \ Ω and R \ Ω respectively. Also k and k denote quasihyperbolic distance with respect to Ω and Ω0 respectively.

We begin by reviewing some of Hästö’sresults from [H1]. In [H1], Hästöspeaks of

m n m domains whose boundaries are C -smooth, and he states that a domain Ω ⊂ R has C - smooth boundary if and only if ∂Ω “is locally the graph of a Cm function.” We take this

77 m n−1 to mean that whenever x ∈ ∂Ω, there exists R > 0, a C function f : B (0,R) → R, n n and an Euclidean isometry S : R → R so that f(0) = 0,S(0) = x, and

n−1 n S({(t, f(t)) t ∈ B (0,R)}) = B (x, R) ∩ ∂Ω.

2 H¨ast¨oproves the following result for domains in R .

Fact 7.1.1. Suppose n = 2 and Ω is a domain whose boundary is C3-smooth. If Ω is not a half-plane and f :Ω → Ω0 is a quasihyperbolic isometry, then f is a similarity.

H¨ast¨oalso claims to prove the following result for higher dimensions of Euclidean space:

Suppose n ≥ 3 and Ω is a domain with C1-smooth boundary. If Ω is not a half-space and f :Ω → Ω0 is a quasihyperbolic isometry, then f is a similarity.

However, in proving the above claim, H¨ast¨ostates that every domain with C1-smooth boundary is ball accessible, which means that whenever ζ ∈ ∂Ω, there exists an open ball B ⊂ Ω so that ζ ∈ ∂B. But, the following example illustrates that this is not always true.

3/2 2 Example 7.1.2. Let f : R → R be the function f(x) = |x| . Let Ω ⊂ R be the domain lying above the image of f. Note that there is no open disk in Ω having the origin as a boundary point. Thus, although Ω is a domain whose boundary is C1-smooth, it is not ball accessible.

Because domains with C1-smooth boundary are not always ball accessible, H¨ast¨o’s proof only holds for domains with C1,1-smooth boundary. However, C1,1 is a strong smoothness condition, and so we give a new proof which does not utilize the C1,1 smoothness condition. In order to remove the C1,1 condition, we make use of the following trivial observation.

Observation 7.1.3. Suppose f :Ω → Ω0 is a quasihyperbolic isometry that is an

n inversion centered at a point p ∈ R . Then p∈ / Ω, and every inversion g centered at p is a quasihyperbolic isometry from Ω to its image under g.

78 Although Hästöerroneously relied on ball accessibility, his technique can be applied in a different manner to obtain Lemma 7.1.4 below (without assuming ball accessibility).

In this result, J is as in Section 2.2.2.

Lemma 7.1.4. Suppose J :Ω → Ω0 is a quasihyperbolic isometry. For each x ∈ Ω and each y ∈ ∂Ω ∩ B[x, δ(x)],

y 6= 0 =⇒ [x, y] ∩ Bn(0, |y|) = ∅, −→ and therefore the angle determined by y 0 and y−→ x is at least π/2.

Proof. Let x ∈ Ω and y ∈ ∂Ω so that δ(x) = |x−y|. Let z ∈ (x, y). Say that z := y +2ru where u := (x − y)/|y − x| = (x − y)/δ(x) and 0 < r < δ(x)/2. Put w := y + ru and let x0, y0, z0, w0 be the images under J of x, y, z, w respectively.

Then |z − w| r |z0 − w0| = = . |z||w| |z||w|

Also, y0 ∈ ∂Ω0, and so |z − y| 2r |w − y| r δ0(z0) ≤ |z0 − y0| = = and δ0(w0) ≤ |w0 − y0| = = . |z||y| |z||y| |w||y| |w||y| Then ! |z0 − w0| log(2) = k(z, w) = k(z0, w0) ≥ log 1 + δ0(z0) ∧ δ0(w0)    r  ≥ log 1 +   2r r  |z||w| ∧ |y||z| |y||w| |y| |y| = log 1 + ≥ log(1 + ), (2|w|) ∧ |z| |z| whence |z| ≥ |y|. −→ Note that if the angle determined by y 0 and y−→ x were acute, then we could pick r to be suﬃciently small enough that |z| < |y| (See Figure 7.1 to visualize this for an acute angle).

79 y

y + 2ru

x 0

Figure 7.1: Acute Angle

However, it is instead the case that for all suﬃciently small r, |z| ≥ |y|, and therefore −→ the angle determined by y 0 and y−→ x is not acute.

Using the previous lemma, we prove the following theorem.

Theorem 7.1.5. Suppose J :Ω → Ω0 is a quasihyperbolic isometry. Then Ω = Ω0,

n n 0 ∈ ∂Ω, and either Ω = R \{0} or R \Ω is a union of closed Euclidean rays emanating from 0.

n Proof. Suppose Γ is a closed Euclidean ray emanating from 0 so that Γ 6⊂ R \ Ω. We 0 n n will show that Γ \{0} ⊂ Ω. Since J(Ω) = Ω ⊂ R , 0 ∈/ Ω. Since 0 ∈/ Ω and Γ 6⊂ R \ Ω, we can let a ∈ Γ ∩ ∂Ω and b ∈ Γ ∩ Ω so that (a, b] ⊂ Ω and |a| < |b|.

Suppose for contradiction that a 6= 0. Let a0 := J(a) and b0 := J(b). Note that

J(Ω) = Ω0 as surjectivity was included in our deﬁnition of a quasihyperbolic isometry (see

Section 2.2.1). Also note that Γ0 := J(Γ) = {∞}∪(Γ \{0}) since Γ is a closed Euclidean ray emanating from 0. Also |a0| > |b0| as shown in Figure 7.2. Let c := (a0 + b0)/2, and let w ∈ (c, a0) so that δ0(w) < δ0(c). As we trace Γ0 from c to w, let p be the ﬁrst point that we come to for which δ0(p) = δ0(w).

Let L be the aﬃne hyperplane which intersects Γ orthogonally at p. Note that 0 ∈/ L.

n Thus R \ L contains two components (one which contains 0 and one which does not

80 0 0 0 0 n 0 a0+b0 0 0 0 0 (a , b ] ⊂ Ω , a ∈ R \ Ω , c := 2 , δ (c) > δ (w), δ (p) = δ (w) 0 b0 c p w a0

Figure 7.2: The Ray Γ

contain 0). Let U be the component containing 0, and let V be the other component

(See Figure 7.3).

U V

0 b0 c p w a0

Figure 7.3: L, U, and V

Let ζ ∈ ∂Ω0 with |ζ − p| = δ0(p). Note that since p ∈ (c, a0), |a0 − p| < |p − 0|, and −→ −→ so ζ 6= 0. By Lemma 7.1.4, the angle formed by ζ 0 and ζ p is at least π/2. So ζ ∈ U, and we can project the point ζ orthogonally onto the ray Γ. Say y := projΓ(p). Since δ0(c) > δ0(p), |c − ζ| > |p − ζ|. Thus |y| > |c|. Then y lies on the segment (c, p), and

δ0(y) ≤ |ζ − y| < |ζ − p| = δ0(p) < δ0(c). Because δ0 is continuous on Γ, there is then a point z ∈ (c, y) for which δ0(z) = δ0(p), contradicting the manner in which we picked p.

Then a = 0.

Now suppose for contradiction that Γ \{0} 6⊂ Ω. Then there exists s ∈ Γ ∩ ∂Ω so that

(0, s) ⊂ Ω. Let t := s/4. By replacing a0 with s, b0 with t,Ω0 with Ω, and δ0 with δ, we can arrive at a contradiction in the same manner as before. Thus Γ \{0} ⊂ Ω.

n n n Then either Γ ⊂ R \ Ω or Γ \{0} ⊂ Ω. Then Ω = R \{0} or R \ Ω is a union of closed Euclidean rays emanating from 0. Then it is also clear that Ω = Ω0 and 0 ∈ ∂Ω.

81 We are now ready to characterize many situations where the only possible isometries with respect to the quasihyperbolic metric are Euclidean similarity maps. It is already known that isometries with respect to the quasihyperbolic metric on Euclidean domains are conformal (see [MO, Theorem 2.6]). When n ≥ 3, a conformal map is a Möbius transformation; see for instance [BP, Theorem A.3.7]. Thus when n ≥ 3, every isometry with respect to the quasihyperbolic metric is a Möbius transformation. We use Theorem 7.1.5 to further characterize these Möbius transformations in the following manner.

Theorem 7.1.6. Suppose n ≥ 3. Then one of the following holds:

n n (a) Ω = R \{a} for some a ∈ R , n (b) R \ Ω is a union of closed Euclidean rays emanating from one single point, or (c) every quasihyperbolic isometry of (Ω, k) is an Euclidean similarity map.

Proof. This directly follows from Theorem 7.1.5, from the fact that any isometry with

n respect to the quasihyperbolic metric on a domain in R is conformal, and from the fact n that the only conformal maps in R are M¨obiustransformations whenever n ≥ 3.

Suppose n ≥ 3. Note that in the case that ∂Ω is as in (a) or (b) of the preceding theorem, the ‘shape’ of ∂Ω is left unchanged by any isometry of the quasihyperbolic metric. In (c), the only quasihyperbolic isometries which change the ‘shape’ of ∂Ω are compositions of similarity maps which include a dilation. That is, whenever a map f : (Ω, k) → (Ω0, k0) is an isometry so that ∂Ω is not Euclidean isometric to ∂Ω0, then f is a composition of an Euclidean isometry with a dilation. Note also that dilations are the only Euclidean similarity maps which change Euclidean distance between points.

n n In particular if z ∈ Ω and g : R → R is given by g(x) = rx for some r > 0, then dist(g(z), g(∂Ω)) = rδ(z). Thus if h : (Ω, k) → (Ω0, k0) is an isometry for which there exists z ∈ Ω so that δ(z) = δ0(h(z)), then ∂Ω ≡ ∂Ω0 via an Euclidean isometry.

This idea will prove useful in the next section, where we further extend the connection

82 between pointed Gromov-Hausdorﬀ convergence and bounded uniform convergence of the quasihyperbolic metric.

7.2 Bounded Uniform Convergence of the Quasihyperbolic

Metric in Euclidean Domains

In this section, we use Theorem 7.1.6 to prove an additional relationship between pointed

Gromov-Hausdorﬀ convergence and bounded uniform convergence, beyond that which

n was observed in Theorem 3.1.2. Throughout this section, Ω, Ωi (R are domains, δ, δi n n denote Euclidean distance to R \Ω, R \Ωi respectively, and k, ki denote quasihyperbolic ˆ n ˆ n ˆ n distance in Ω, Ωi respectively. Also, A := R \ Ω, and Ai := R \ Ωi. Whenever F ⊂ R n is closed, we let δF denote Euclidean distance to F ∩ R , and note that in Ω, δ = δ∂Ω.

Also in Ωi, δi = δ∂Ωi . Using quasihyperbolic isometries, we assume a normalization in the next theorem.

Because Euclidean translations and dilations are quasihyperbolic isometries, we assume that 0 ∈ Ω and that δi(0) → δ(0). These conditions give the proper normalization for the following theorem to hold.

Theorem 7.2.1. Let n ≥ 3. Suppose 0 ∈ Ω with δi(0) → δ(0). If dGH∗ ((Ωi, 0, ki), (Ω, 0, k)) n n converges to 0, then there exist Euclidean isometries Φi : R → R so that

−1 δΦi(Ai) := δi ◦ Φi → δ locally uniformly in Ω. Here, each Φi can be chosen to be a composition of rotations and reﬂections which ﬁx the origin.

Proof. Suppose dGH∗ ((Ωi, 0, ki), (Ω, 0, k)) → 0, and let (Ap) be a subsequence of (Ai). n n Let H denote the collection of all closed sets in Rˆ . Note that since Rˆ is compact, ˆ the metric space (H, dH ) is also compact (see for instance Fact 2.5 in [H2]). Then there

83 ˆ n ˆ exists a subsequence (Aj) of (Ap) and a closed set C ⊂ R so that dH (Aj,C) → 0. Then n δj → δC boundedly uniformly in R by Theorem 4.1.1.

Note that since 0 ∈ Ω and δi(0) → δ(0), 0 ∈ Ωi for all suﬃciently large i, and

n n n 0 ∈ R \ C. Then let D be the component of R \ C containing 0, and let B := Rˆ \ D.

Note that dist(0, ∂D) = δ(0), and in D, δB = δC . Since δj → δB boundedly uniformly in

(D, kD), dGH∗ ((Ωj, 0, kj), (D, 0, kD)) → 0 by Theorem 3.1.2. Then (D, 0, kD) ≡ (Ω, 0, k) by Fact 2.3.12. Then by the discussion in the paragraph directly following Theorem 7.1.6,

n n there is an Euclidean isometry Φ : R → R so that Φ(D) = Ω and Φ(0) = 0. In particular Φ is a composition of rotations and reﬂections which ﬁx the origin. Extend Φ

n to Rˆ by setting Φ(∞) = ∞. n Note that Φ(B) = Rˆ \Ω = A. So ∂Ω = Φ(∂D) ⊂ Φ(C) ⊂ Φ(B) = A, and so whenever x ∈ Ω, dist(x, ∂Ω) ≤ dist(x, Φ(C)) ≤ dist(x, A) = dist(x, ∂Ω). In particular, δ = δΦ(C) in Ω.

n Remember that δj → δC boundedly uniformly in R . Then

−1 δΦ(Aj ) = δj ◦ Φ → δΦ(C)

n boundedly uniformly in R . But since δΦ(C) = δ in Ω, we then have that δΦ(Aj ) → δ locally uniformly in Ω.

n n Let G denote the set of Euclidean isometries from R onto R which ﬁx the origin. Note that each map in G is a composition of reﬂections and rotations. Whenever g ∈ G,

n letg ˆ be the extension of g to Rˆ given by settingg ˆ(∞) = ∞. Let

n n F := {F ⊂ Rˆ | F ∈ H,F ⊂ Rˆ \ Ω, ∂Ω ⊂ F }.

Also for each i, let

ˆ ∆i := inf dH (ˆg(Ai),F ). g∈G,F ∈F

84 Note that C ∈ F. Hence it can be shown that each subsequence (∆j) of (∆i) has a further subsequence which converges to 0, and so ∆i → 0. Then there exist sequences ˆ (Φi) in G and (Fi) in F so that dH (Φˆ i(Ai),Fi) → 0.

Let (Fp) be a subsequence of (Fi). There is a subsequence (Fj) of (Fp) and a closed ˆ n ˆ set F ⊂ R so that dH (Fj,F ) → 0. Now for each j,

ˆ ˆ ˆ dH (Φˆ j(Aj),F ) ≤ dH (Φˆ j(Aj),Fj) + dH (Fj,F ).

Hence dˆ (Φˆ (A ),F ) → 0. Then δ → δ boundedly uniformly in n. Since H j j Φˆ j (Aj ) F R ˆ ∂Ω ⊂ Fj ⊂ A and since dH (Fj,F ) → 0, ∂Ω ⊂ F ⊂ A. Then δ = δF in Ω. Then δ → δ locally uniformly in Ω. We have shown that each subsequence δ of Φˆ j (Aj ) Φˆ p(Ap) δ has a further subsequence which converges to δ locally uniformly in Ω. Thus Φˆ i(Ai) δ → δ locally uniformly in Ω. Φˆ i(Ai)

Using Remark 2.2.4 and Fact 7.1.1, we prove the following result regarding domains

2 3 2 in R with C -smooth boundary. We keep the same notation, letting Ω, Ωi (R be domains. The quasihyperbolic distances on Ω, Ωi are k, ki respectively. Also δ, δi are

2 2 distance to R \ Ω, R \ Ωi respectively.

2 3 Theorem 7.2.2. Let n = 2, and suppose Ω (R is a domain with C -smooth boundary.

Suppose 0 ∈ Ω and δi(0) → δ(0). If dGH∗ ((Ωi, 0, ki), (Ω, 0, k)) → 0, then there exist n n Euclidean isometries Φi : R → R so that

−1 δΦi(Ai) := δi ◦ Φi → δ locally uniformly in Ω. Here, each Φi can be chosen to be a composition of rotations and reﬂections which ﬁx the origin.

2 Proof. Suppose D (R is a domain, and suppose Φ : D → Ω is a quasihyperbolic isometry. If Ω is a half-plane, Φ is M¨obiusby Remark 2.2.4, and by Theorem 7.1.5,

D must also be a half-plane. In particular, ∂D is Euclidean isometric to ∂Ω if Ω is a

85 half-plane. If Ω is not a half-plane, then Φ is a similarity map by Fact 7.1.1. Also by

Theorem 7.1.6, ∂D is not Euclidean isometric to ∂Ω only if Φ is a composition of an

Euclidean isometry and a dilation. Then as compositions of similarity maps involving a dilation are the only isometries which potentially alter the ‘shape’ of ∂D, this theorem can be proven in the same manner that Theorem 7.2.1 was proven.

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