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

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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 approx- imations of the quasihyperbolic, Ferrand, Kulkarni-Pinkall-Thurston, and hyperbolic distances via spaces with only finitely many boundary points. ii c 2019 by Abigail Richard. All rights reserved. Acknowledgments 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 Copyright iii 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 quasi- conformal 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 hy- perbolic metric, quasihyperbolic distance can indeed be challenging to compute in gen- eral 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 dis- tance, Kulkarni-Pinkall-Thurston distance, and hyperbolic distance. In determining approximations of non-Euclidean distances, we are led to explore rela- tionships 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 R^ := R [ f1g; 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 f1g [ (R n f0g); as this is the image of R after reflec- tion 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 demon- strate 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 R^ : 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 R^ : 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 2 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 R^ : For each x 2 R n A; n n there exists N 2 N so that whenever i ≥ N; x 2 R n Ai, and (R n Ai; x)i≥N pointed n Gromov-Hausdorff converges to (R n 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 2 R n A; there n n exists a sequence (xi) in R so that (R n Ai; xi) pointed Gromov-Hausdorff converges to n (R n 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 ^ n n R : Further (Ai) can be chosen so that for each i; Ai ⊂ A and for each x 2 R n A; the n n sequence (R n Ai; x) pointed Gromov-Hausdorff converges to (R n 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 con- text of complete length spaces. Theorem 1.0.4 provides an approximation of the quasi- hyperbolic 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 fur- ther 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.
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