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The Bianchi Identity and the Ricci Curvature Equation Godfrey Dinard Smith Bachelor of Finance, Master of Science

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The Bianchi Identity and the Ricci Curvature Equation Godfrey Dinard Smith Bachelor of Finance, Master of Science The Bianchi Identity and the Ricci Curvature Equation Godfrey Dinard Smith Bachelor of Finance, Master of Science A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2016 School of Mathematics and Physics Abstract The main result of this Ph.D. thesis is to present conditions on the curvature and boundary of an orientable, compact manifold under which there is a unique global solution to the Dirichlet boundary value problem (BVP) for the prescribed Ricci curvature equation. This Dirichlet BVP is a determined, non-elliptic system of 2nd-order, quasilinear partial differential equations, which is supplemented with a constraint equation in the form of the so-called Bianchi identity. Indeed, in order to prove the main result of this Ph.D. thesis, we are also required to prove that the kernel of the Bianchi operator is a smooth tame Fréchet submanifold of the space of Riemannian metrics on a compact Riemannian manifold with boundary. After presenting an overview of the literature in Chapter 1 and the required notation and background in Riemannian geometry and geometric analysis in Chapter 2, the main results of this thesis are organised into the following two chapters. In Chapter 3 we present conditions under which the kernel of the Bianchi operator is globally a smooth tame Fréchet submanifold of the space of Riemannian metrics on a Riemannian compact manifold both with or without boundary. This global submanifold result for the kernel of the Bianchi operator is central to the proof of global existence and uniqueness for the Dirichlet BVP for the Ricci curvature equation, and extends the analogous local submanifold result by Dennis DeTurck in [18] which was fundamental to the proof in [18] of local existence and uniqueness of the prescribed Ricci curvature equation on a compact manifold without boundary. Furthermore, the material in Chapter 3 of this thesis sits more generally within the literature on linearisation stability of nonlinear PDE. Indeed, the method by which we prove that the kernel of the Bianchi operator is a global submanifold of the space of metrics on a compact manifold is an application of the more general technique of proving the linearisation stability of a system of nonlinear PDE, and yields that the Bianchi operator is itself linearisation stable in a sense made precise in Chapter 3. In Chapter 4 we then use the fact that the kernel of the Bianchi operator is globally a smooth tame Fréchet manifold, and the Nash-Moser implicit function theorem (IFT) in the smooth tame category, to find and conditions under which globally there is a unique Riemannian metric satisfying the Dirichlet BVP for the Ricci curvature equation. This global existence and uniqueness result is motivated by, and can be viewed as a modification of, an analogous result for Einstein manifolds of negative sectional curvature and convex and umbilical boundary presented by Jean-Marc Schlenker in [50]. Loosely speaking, in the context of Einstein manifolds the kernel of the Bianchi operator is the entire space of Riemannian metrics and thus automatically a smooth tame Fréchet manifold; however, in the context of the prescribed Ricci curvature equation, in which this Ph.D. thesis is interested, we are required to prove that the kernel is a smooth tame Fréchet submanifold in order to apply the Nash-Moser IFT. In general, global existence for the prescribed Ricci curvature equation on a manifold with or without boundary is difficult and results in this direction are few; the main goal of this thesis makes an important contribution in this area and the techniques used to prove it have potential applications in other areas of geometric analysis such as Yang-Mills gauge theory and curvature flows. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis. Publications during candidature No publications. Publications included in this thesis No publications included. Contributions by others to the thesis No contributions by others. Statement of parts of the thesis submitted to qualify for the award of another degree None. Acknowledgements I first and foremost sincerely thank my Ph.D. advisors Dr. Artem Pulemotov and Prof. Joseph Grotowski for their advice, encouragement, and support. My journey into geometric analysis started when, after having completed my Masters, Prof. Joseph Grotowski recommended that I attend the 2012 winter school in geometric PDE hosted by the University of Queensland. The speakers at this winter school, including Richard Schoen, Gang Tian, Tristian Riviere, and Michael Struwe, introduced the participants to many foundational areas of geometric analysis and their importance to mathematical physics. It was via this winter school, and my own personal study in the following months, that I was able to orient myself in the parts of differential geometry and geometric analysis that are central to important areas of theoretical physics, such as Yang-Mills gauge theory, semi- Riemannian geometry, and Kahler and Calabi-Yau manifolds. It was also at this winter school that I originally approached my then-to-become principal advisor Dr. Artem Pulemotov about the possibility of him supervising a Ph.D. in geometric analysis. I particularly thank Dr. Artem Pulemotov for suggesting the prescribed Ricci curvature problem as an area via which to learn about and make an initial contribution to the vast literature on geometric analysis. I believe this has built a strong foundation upon which to continue down this path. A number of people also provided support and encouragement, either direct or indirect, throughout the course of my Ph.D. My twin brother and his family have, of course, always been a major part of my life, and things would be very different without them. My other brothers and my parents also continue to give unconditional love and support. I thank my lifelong childhood friends from my hometown of Maroochydore for their typical frank advice and opinion, which more often than not was initially hard to swallow but beneficial in the long run. I would like to thank fellow Ph.D. student Nirav Shah for his friendship, advice, and for acting as a nonjudgmental sounding board during my studies. I thank the varied and colourful characters of the Moffat Beach community for openly accepting me after I moved town (to escape the distractions of my childhood friends!) and suddenly appeared at the top of their lineup with the clear intention of getting often more than my fair share. Finally, I thank the student Nok-Man (Gloria) Cheng for her efforts as a teaching assistant (tutor) in reducing the work required in delivering various courses I taught at the University of Queensland during the course of my Ph.D. Keywords Ricci curvature, partial differential equation, boundary value problem, linearisation stability, Nash-Moser Australian and New Zealand Standard Research Classifications (ANZSRC) 010110 Partial Differential Equations 50% 010102 Algebraic and Differential Geometry 50% Fields of Research (FoR) Classification 0101 Pure Mathematics 100% Contents 1 Introduction and Statement of Results1 2 Background and Notation7 2.1 Riemannian Manifolds.........................7 2.2 Differential Operators.......................... 13 2.3 Smooth Tame Fr´echet Manifolds.................... 15 2.4 Linearisation Stability......................... 21 3 Kernel of the Bianchi Operator 23 3.1 Under-Determined Elliptic Operators................. 23 3.2 Formal Adjoint L∗ ........................... 25 3.3 Proof of Theorem 1.3.......................... 27 4 Ricci Curvature Equation 29 4.1 Outline of the Proof of Theorem 1.6.................. 29 4.2 Additional Notation and Terminology................. 32 4.3 Step 1: Covariant Derivative...................... 34 4.4 Step 2: Approximate Inverse...................... 35 4.4.1 Unique Solvability of (4.5)................... 37 4.4.2 Approximate Right Inverse................... 41 4.4.3 Approximate Left Inverse................... 43 Bibliography 46 vii Chapter 1 Introduction and Statement of
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