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The Inverse Mean Curvature Flow: Singularities, Dynamical Stability, and Applications to Minimal Surfaces

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The Inverse Mean Curvature Flow: Singularities, Dynamical Stability, and Applications to Minimal Surfaces The Inverse Mean Curvature Flow: Singularities, Dynamical Stability, and Applications to Minimal Surfaces By BRIAN DONALD HARVIE DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Adam Jacob, Chair Joel Hass Andrew Waldron Committee in Charge 2021 i Source: The Far Side by Gary Larson ii To my friends, family, and teachers. iii Contents Abstract vi Acknowledgments vii List of Figures ix Chapter 1. Introduction 1 1.1. A Brief Overview of Geometric Flows 1 1.2. The Contraction and Expansion of Hypersurfaces by their Mean Curvature 3 1.3. Singularities of MCF and IMCF 6 1.4. The Dynamical Stability of Round Spheres under MCF and IMCF 13 1.5. Weak Solutions, Outward-Minimizing Sets, and Applications in General Relativity 16 Chapter 2. The Eventual Star Shape of Classical and Weak Solutions 22 2.1. The Main Theorems 22 2.2. An Aleksandrov Reflection Approach to Variational Solutions 25 2.3. Consequences for Classical Solutions 31 2.4. Intersections and Singularities in the Spherical Topology 38 Chapter 3. The Singular Limit of IMCF on a Torus 45 3.1. The Main Theorems 45 3.2. The Generating Curve 47 3.3. An Energy Estimate on Gauss Curvature 51 3.4. Rescaling the Singularity 53 3.5. Convergence at Tmax 57 3.6. The General Case 58 Chapter 4. The Flow of Rotationally Symmetric Hypersurfaces into Spheres 61 iv 4.1. The Main Theorem 61 4.2. Preserving Embeddedness 64 4.3. Non-Cylindrical Spacetime Domains 67 4.4. Evolution Equations 74 4.5. A Priori Height Estimates 76 4.6. The Bridge Region 78 4.7. The Cap Region 83 4.8. Global Existence and Convergence for Admissible Data 86 Chapter 5. An Application of IMCF to Plateau's Problem, and Vice Versa 87 5.1. Statement of the Problem 87 5.2. A Barrier Method via IMCF 89 5.3. Singularity Detection using Minimal Disks 93 Appendix A. Non-Cylindrical Maximum Principle 97 Appendix B. Non-Star-Shaped Admissible Initial Data 102 Bibliography 107 v Abstract This dissertation concerns the Inverse Mean Curvature Flow of closed hypersurfaces in Eu- clidean Space, and its relationship with minimal surfaces. Inverse Mean Curvature Flow is an extrinsic geometric flow which has become prominent in differential geometry because of its appli- cations to geometric inequalities and general relativity, but deep questions persist about its analytic and geometric structure. The first four chapters of this dissertation focus on singularity formation in the flow, the flow behavior near singularities, and the dynamical stability of round spheres under mean-convex perturbations. On the topic of singularities, I establish the formation of a singularity for all embedded flow solutions which do not have spherical topology within a prescribed time interval. I later show that mean-convex, rotationally symmetric tori undergo a flow singularity wherein the flow surfaces converge to a limit surface without rescaling, contrasting sharply with the singularities of other extrinsic geometric flows. On the topic of long-time behavior, I show that all flow solutions which exist and remain embedded for some minimal time depending only on initial data must exist for all time and asymptotically converge to round spheres at large times. In the fourth chapter, I utilize this characterization to establish dynamical stability of the round sphere under certain mean-convex, axially symmetric perturbations that are not necessarily star-shaped. In the last chapter, I relate questions of singularities and dynamical stability for the Inverse Mean Curvature Flow to the mathematics of soap films. Specifically, I show that certain families of solutions to Plateau's problem do not self-intersect and remain contained within a given region of Euclidean space. I accomplish this using a barrier method arising from global embedded solutions of Inverse Mean Curvature Flow. Conversely, I also use minimal disks to establish that a singularity likely forms in the flow of a specific mean-convex embedded sphere. vi Acknowledgments I would like to thank my advisor, Professor Adam Jacob, for his continued support and guidance throughout my time as a graduate student. Early in the program I realized that my mathematical interests diverged from most of the research in the math department at Davis, and I feared that this would make realizing my ambitions difficult or impossible. In a very fortunate occurrence for me, the department hired Adam the same year I began the PhD. program. Through my years as his student, he has consistently provided encouragement and helpful input in our weekly meetings. He has supported my career and my development as a mathematician through using his grant money to fund my conference trips, writing several recommendation letters on my behalf, and providing extensive feedback on all of my research papers. A disproportionate number of the conversations about differential geometry and the Simpsons in my life have been with him, and I am going to miss these. There are a number of other professors at Davis and elsewhere that I would like to thank. Thank you to Professors Joel Hass and Andrew Waldron for serving on both my qualifying exam committee and my dissertation committee. Professor Hass also wrote a letter of recommendation for me and inspired the last chapter of this dissertation, so I would like to give him a special thanks for this. Thank you to my other letter writers, Professor John Hunter of UC Davis and Professor Eric Woolgar of the University of Alberta, the latter of whom also invited me to the Workshop on Asymptotically Hyperbolic Manifolds at the Banff International Research Station. Several current and former graduate students in this department made the potentially isolating experience of research in mathematics far more manageable for me. I have been blessed over the past six years to meet so many kind and colorful individuals who relieved my stress through difficult times, reinvigorated my enthusiasm for mathematics, and taught me much about myself. I will certainly think back to some of these people the next time I find myself at a Blaze Pizza or a farmer's market. In no particular order, to Michael Ponce, Wencin Poh, Norman Sheu, Jianping Pan, Lily Silverstein, Shanon Rubin, Kyle Chickering, Tonie Scroggin, and Karry Wong: thank you all. The administration and staff in the mathematics department at UCD have done an outstanding job making the bureaucratic aspects of completing a PhD. much easier for me to navigate. With vii classes, teaching, and research already being as exhausting as they are, I am grateful to be saved stress on the administrative side of my work. I would like to thank all past and present staff members of the math department for a job well done, especially Tina Denena, Sarah Driver, and Victoria Whistler. I would like to give a special thanks to the excellent teachers from prior to graduate school who encouraged me to pursue a career in science. In particular, thank you to Thomas Noviello of Leominster High School, whose dedication to physics education has positively affected my life tra- jectory and the life trajectories of many others, and to Professors Jennie Traschen and David Kastor of the University of Massachusetts Amherst, whose willingness to mentor a naive undergraduate in general relativity paid off eventually. Last but certainly not least, I give thanks to my family. To my sisters Katie and Julie, we had our face-offs as siblings growing up, but to now understand one another and to have each of you rooting for me means so much. To my parents Patricia and Donald, I am forever grateful for your love and support. I realized in my teenage years that I was interested in the less practical aspects of science, and dad, you were understandably skeptical about the career prospects in this direction. But you believed in me enough to put those concerns aside and fully support my life decision, and I could not have accomplished this without both of you. Throughout many busy weeks crammed with difficult assignments and never-ending edits to papers, I always appreciated being able to call and talk about life during a walk through the campus arboretum. I hope I find as good of a spot in Taipei to chat with each of you. But until that next chapter begins, thank you for everything. viii List of Figures 1.1 Finite-Time Extinction under MCF 9 1.2 Evolution of the Thin Torus under IMCF 12 1.3 Weak IMCF of Two Spheres [HI01] 20 2.1 The Moving Plane Method for Variational Solutions 27 2.2 A Non-Outward-Minimizing, H > 0 Sphere 41 3.1 Profile of the Generating Curve 48 3.2 Slope Control on @St 51 4.1 Neckpinching in the Mean Curvature Flow [AMC+13] 62 4.2 The Generating Curve of Rotationally Symmetric Spheres 65 4.3 Different Regions of N0 68 4.4 The Reduced Parabolic Boundary of U 70 4.5 The Left and Right Support Functions 84 5.1 Self-Intersecting Soap Films [Cos12] 88 5.2 Comparison with Et0 91 5.3 Stable Minimal Disks with the Same Boundary [Mea] 94 A.1 The Cut-off in Time 100 B.1 A Non-Star-Shaped Admissible N0 105 ix CHAPTER 1 Introduction Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. -Jean-Baptiste Joseph Fourier 1.1. A Brief Overview of Geometric Flows The heat equation is one of the most thoroughly-studied partial differential equations in math- ematics. Beginning as a model for diffusive behavior in physical systems, it has since inspired a n large body of mathematical research in analysis and geometry.
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