COMPLETE NONNEGATIVELY CURVED SPHERES AND PLANES A Thesis Presented to The Academic Faculty by Jing Hu In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mathematics Georgia Institute of Technology August 2015 Copyright c 2015 by Jing Hu COMPLETE NONNEGATIVELY CURVED SPHERES AND PLANES Approved by: Igor Belegradek, Advisor Dan Margalit School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology John Etnyre Arash Yavari School of Mathematics School of Civil and Environmental Georgia Institute of Technology Engineering Georgia Institute of Technology Mohammad Ghomi Date Approved: May 22, 2015 School of Mathematics Georgia Institute of Technology To my parents, iii PREFACE This dissertation is submitted for the degree of Doctor of Philosophy at Georgia Institute of Technology. The research described herein was conducted under the supervision of Professor Igor Belegradek in the School of Mathematics, Georgia Institute of Technology, between August 2010 and April 2015. This work is to the best of my knowledge original, except where acknowledgements and references are made to previous work. Part of this work has been presented in the following publication: Igor Belegradek, Jing Hu.: Connectedness properties of the space of complete nonnega- tively curved planes, Math. Ann., (2014), DOI 10.1007/s00208-014-1159-7. iv ACKNOWLEDGEMENTS I would like to express my special appreciation and thanks to my advisor Professor Igor Belegradek, who have been a tremendous mentor for me. I would like to thank him for encouraging my research. All results in this thesis are joint work with him. I would also like to thank my committee members, Professor John Etnyre, Professor Dan Margalit, Professor Mohammad Ghomi and Professor Arash Yavari for serving as my com- mittee members. A special thanks to my family. Words cannot express how grateful I am to my mother, my father and my younger brother for all of the sacrifices that you’ve made on my behalf. I would also like to thank my friends Ling Ling, Qingqing Liu, Helen Lin and others gave me inspiration toward my research and life. v TABLE OF CONTENTS DEDICATION ..................................... iii PREFACE ....................................... iv ACKNOWLEDGEMENTS .............................. v SUMMARY ...................................... viii I INTRODUCTION ................................ 1 1.1 Introduction . .1 II NON NEGATIVELY CURVED SPHERES .................. 8 k+α 2 2.1 Interpretation of R≥0 (S )..........................8 1 2 2.2 R≥0(S )...................................9 k+α k+α 2 2.3 L1 with C (k ≥ 2) topology and R≥0 (S ) are not completely metrizable . 10 2.4 Beltrami equation on S 2 ........................... 16 k+α 2.5 L1 equipped with C topology is homogenous . 21 + 2 k+1+α 2.6 Diff0;1;1(S ) equipped with the C topology is contractible . 26 Rk+α 2 n ≥ 2.7 >0 (S ) K is weakly contractible for k 2................ 27 III NON NEGATIVELY CURVED PLANES ................... 31 3.1 Non negatively curved planes . 31 3.2 Complete non negatively curved planes . 33 1 2 3.3 Properties of S1 = fu 2 C (R )j ∆u ≥ 0; α(u) ≤ 1g ............. 37 k+α 2 3.4 Connectedness properties of R≥0 (R )................... 48 3.5 Moduli space of complete non negatively curved metrics . 52 k+α 2 3.6 R≥0 (R ) n K is weakly contractible for α 2 (0; 1) . 56 APPENDIX A — DIMENSION .......................... 59 APPENDIX B — CK- TOPOLOGY AND HOLDER¨ SPACE .......... 61 APPENDIX C — BELTRAMI EQUATION ................... 65 APPENDIX D — METRIZABILITY ...................... 67 vi REFERENCES ..................................... 72 VITA .......................................... 75 vii SUMMARY First we study the space of complete Riemannian metrics of nonnegative curva- ture on the sphere equipped with Ck+α topology. In this thesis, unless stated otherwise we assume α 2 [0; 1). When α = 0, it means the compact open Ck topology. We show the space is homogenous for k ≥ 2 and α 2 (0; 1). If k is infinite, we show that the space is homeomorphic to the separable Hilbert space. We also prove for finite k and α 2 (0; 1), + 2 the space minus any compact subset is weakly contractible. We also study Diff0;1;1(S ), the group of self-diffeomorphism of the sphere fixing the complex numbers 0, 1, 1 and + 2 k+α isotopic to the identity. For any k + α, we show Diff0;1;1(S ) equipped with C topology is not completely metrizable. Then we study the space of complete Riemannian metrics of nonnegative curvature on the plane equipped with the Ck+α topology. If k is infinite, we show that the space is home- omorphic to the separable Hilbert space. For any finite k and α 2 (0; 1), we prove that the space minus a compact subset is weakly contractible. For any finite k + α, we show the space cannot be made disconnected by removing a finite dimensional subset. A similar re- sult holds for the associated moduli space. The proof combines properties of subharmonic functions with results of infinite dimensional topology and dimension theory. A key step is a characterization of the conformal factors that make the standard Euclidean metric on the plane into a complete metric of nonnegative sectional curvature. viii CHAPTER I INTRODUCTION 1.1 Introduction The spaces of Riemannian metrics have been studied under various geometric assumptions such as positive scalar [34, 35, 37], negative sectional [18, 19, 20, 21], positive Ricci [42], and nonnegative sectional [10] curvatures. In this thesis we study the spaces of Riemannian metrics under the assumption that the metric is both complete and nonnegativey curved. k+α 1 Let M be any m-dimensional manifold. Let R≥0 (M) denote the set of C complete Rie- mannian metrics on M of nonnegative sectional curvature equipped with the Ck+α topology, where k is a finite integer or 1 and α 2 [0; 1). When α = 0, it is the compact open Ck topol- ogy. When α 2 (0; 1), it is the Holder¨ Ck+α topology. For the sphere, Earle-Schwartz [17] implies that φ depends in Ck+α (α 2 (0; 1)) on the Beltrami dilatation, i.e. φ depends in Ck+α (α 2 (0; 1)) on w in the Beltrami equation w w k+α φz¯ = wφz . This easily implies that φ depends in C (α 2 (0; 1)) on the metric g . For the sphere, we consider a bijection + 2 2 Π : L1 × Diff0;1;1(S ) !R≥0(S ); f j 2 1 2 ≤ g + 2 where L1 = u u C (S ); ∆ug1 1 and Diff0;1;1(S ) is the group of self-diffeomorphism of the sphere fixing the complex numbers 0, 1, 1 and isotopic to the identity. For the plane, we consider a bijection + 2 2 Π : S1 × Diff0;1(R ) !R≥0(R ): 1 k+α 2 k+α 2 In this thesis, we mainly study topology of R≥0 (S ), R≥0 (R ) and how their topologi- cal properties vary with k. The cases of spheres and planes are considered in chapter 2 and chapter 3, respectively. k+α 2 2 In chapter 2 we study topology of R≥0 (S ). Any metric on S is conformal to the round 2 ∗ 2u metric g1. So up to normalization a metric g 2 R≥0(S ) can be written uniquely as φ e g1 where φ is a C1 self-diffeomorphism of S 2 that fixes 0, 1, 1, and u is a C1 function on S 2. −2u The sectional curvature of g equals e (1−4g1 u), so nonnegatively curved means 4g1 u 6 1. Consider the bijection + 2 2 Π : L1 × Diff0;1;1(S ) !R≥0(S ); f j 2 1 2 ≤ g + 2 where L1 = u u C (S ); ∆ug1 1 and Diff0;1;1(S ) is the group of self-diffeomorphism of the sphere fixing the complex numbers 0, 1, 1 and isotopic to the identity. k+α 2 To better understand the topology of R≥0 (S ) we prove the following k+α + 2 (1) The map Π is a homeomorphism if L1 is given the C topology and Diff0;1;1(S ) is given the Ck+1+α topology, where k is nonnegative integer or 1 and α 2 (0; 1) [Theo- rem 2.4.1]. We prove the following + 2 2 1 2 (2) For k = 1 the spaces Diff0;1;1(S ) and L1 are both homeomorphic to l , so R≥0(S ) is homeomorphic to l2 [Theorem 2.2.1]. k+α k+α 2 We also study the metrizability of L1 with C topology and R≥0 (S ). We prove the fol- lowing theorems. k+α (3) If k , 1 and k ≥ 2, then L1 equipped with C topology is not completely metriz- k+α 2 + 2 able [Theorem 2.3.8]. The space R≥0 (S ) is not completely metrizable since Diff0;1;1(S ) equipped with Ck+1+α topology is not completely metrizable [Theorem 2.3.9, Theorem 2.3.10]. 2 k+α For finite k we discuss the homeomorphism type of L1 equipped with C topology. k+α One natural guess is that L1 equipped with C topology are of the same homeomorphism type for different k’s. k+α 2 A natural question is whether R≥0 (S ) is homogenous. In general, convex subset of Banach spaces need not be homogenous and their homeomorphism classification is wide k+α 2 open. We prove R≥0 (S ) is homogenous for k ≥ 2 as follows k+α k+α 2 (4) L1 equipped with C topology and R≥0 (S )(α 2 (0; 1); k ≥ 2) are homogenous [Theorem 2.5.5].