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THE UNIVERSITY of CALIFORNIA, IRVINE on Singularities and Weak

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THE UNIVERSITY of CALIFORNIA, IRVINE on Singularities and Weak THE UNIVERSITY OF CALIFORNIA, IRVINE On Singularities and Weak Solutions of Mean Curvature Flow DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in mathematics by Alexander Everest Mramor Dissertation Committee: Professor Richard Schoen, Chair Associate Professor Li-Sheng Tseng Assistant Professor Xiangwen Zhang 2019 • Portions of chapter 4 of this thesis were originally published in Entropy and the generic mean curvature flow in curved ambient spaces, Proc. Amer. Math. Soc. 146, 2663-2677, American Mathematical Society, Providence, RI, 2018. • Portions of chapter 6 of this thesis were originally published in A finiteness the- orem via the mean curvature flow with surgery, Journal of geometric analysis, December 2018, Volume 28, Issue 4, pp 3348{3372, Springer Nature Switzer- land AG. • Portions of chapter 7 of this thesis are to appear in Regularity and stability results for the level set flow via the mean curvature flow with surgery in Com- munications in Analysis and Geometry, International Press of Boston, Boston, Mass • Portions of chapter 8 of this thesis are to appear in On the topological rigidity of self shrinkers in R3 in Int. Math. Res. Not., Oxford univerisity press, Oxford, UK, 2019. All other materials c 2019 Alexander Mramor ii Table of contents List of Figures . vi Acknowledgements . vii Curriculum Vitae . ix Abstract of the dissertation . .x Preface 1 1 Introduction to the mean curvature flow 4 1.1 Classical formulation of the mean curvature flow . .5 1.2 Mean curvature flow with surgery for compact 2-convex hypersurfaces 11 1.2.1 Mean curvature flow with surgery according to Huisken and Sinestrari . 11 1.2.2 Haslhofer-Kleiner theory . 13 1.3 Background on Colding and Minicozzi's entropy. 17 1.4 Using Brakke regularity theorem to control curvature blowup . 19 2 The simplest phenomena: surfaces which shrink to round points 23 2.1 First attempts at providing nonconvex surfaces which flow to round points ................................... 24 2.1.1 A compactness contradiction argument . 25 2.1.2 Neighborhoods of curves which shrink to round points . 28 2.2 Constructing new flows from old via barriers . 32 2.2.1 Quickly disappearing spikes . 34 2.2.2 Quickly disappearing pancakes . 41 2.3 Pathological examples of surfaces which flow to round points . 45 3 A first application of Colding and Minicozzi theory 53 3.1 The first technical hurdle: failure of Huisken monotonicity. 54 3.2 A weighted monotonicity formula for forced flows. 54 3.3 Almost monotonicity of entropy. 57 3.4 Lipschitz continuity of entropy in certain cases. 60 3.5 Generic flow to round points . 62 iii 4 The mean curvature flow with surgery under a low entropy assump- tion 68 4.1 Structure of α-noncollapsed ancient flows of low entropy . 70 4.2 Existence of small entropy mean convex cap . 70 4.3 Estimation of entropy across surgeries. 76 5 Topological applications of the mean curvature flow with surgery 84 5.1 A finiteness theorem via the mean curvature flow . 85 5.1.1 Path to round sphere . 87 5.1.2 Torus to knot theorem . 95 5.1.3 Hypersurface to skeleton theorem . 97 5.1.4 Finiteness theorem . 98 5.2 An application to self shrinkers of low entropy . 101 6 Using the mean curvature flow with surgery to study the level set flow 109 6.1 Localizing the mean curvature flow with surgery. 114 6.2 Convergence to level set flow. 117 6.3 A Variant of the Local Brakke Regularity Theorem for the LSF. 121 6.4 Corrollaries to the regularity theorem . 127 6.4.1 Rapid smoothing . 127 6.4.2 LSF long time convergence to a plane . 127 6.5 Explicit examples of theorem 6.0.3 . 128 7 The existence and structure of \exotic" singularity models 130 7.1 Ancient and eternal solutions to the mean curvature flow from minimal surfaces . 134 7.1.1 Ancient flows out of a wide class of minimal surfaces . 134 7.1.2 An eternal solution out of the catenoid . 138 7.1.3 Partial uniqueness of the reapernoid . 145 7.2 An unknotedness theorem for self shrinking surfaces . 148 Conclusion 153 iv List of figures 1.1 The 4 possible types of high curvature regions for the MCF with surgery 12 2.1 The spike construction iterated several times . 33 2.2 A barrier to force the model spike down . 36 2.3 A diagram of planting the model spike onto the surface . 38 2.4 Positioning of slightly bent barriers as sub and supersolutions to the flow .................................... 39 2.5 Cross-section of \pancake" construction . 41 2.6 Edge of pancake being smoothed out by flow . 45 2.7 A sequence of flows shrinking to a round point yet the sequence of initial surfaces has no Gromov-Hausdorff limit. 46 2.8 A surface that is close in Hausdorff distance to a sphere and shrinks to a round point yet has high entropy and large area. 47 2.9 A (slightly inaccurate) sketch of one of the first elements in the space filling sequence of surfaces which shrink to points. 49 4.1 A slowly forming neckpinch which gives a low entropy cap . 74 4.2 A potential high curvature region under the surgery flow . 81 5.1 A diagram of a possible skelton found via theorem 5.1.3. 86 5.2 Decomposition of the hypersurface along standard surgery spots . 89 5.3 Isotopy to round sphere in inductive step . 90 5.4 Preconditioning for neck extension . 91 5.5 Extending the neck along the isotopies of A and B ........... 92 5.6 Bumping A to be able to smooth the isotopy . 94 5.7 A junction of a skeleton in a small cube . 99 5.8 A junction point of a skeleton after conditioning . 99 5.9 A deformation of a skeleton to a distinguished skeleton . 100 6.1 An illustration of a neighborhood in definition 6.0.1. 111 6.2 α noncollapsedness for sets with boundary . 116 6.3 Using enclosed volume to control height in a noncollapsed flow . 122 v 7.1 A sketch of the regimes of the profile curves of the higher-dimensional reapernoid, the eternal solution of Theorem 7.0.3. For t 0, the eternal solution has a profile curve close to that of the catenoid, and for t 0, it has a profile curve close to that of a grim reaper. 132 7.2 An approximate profile of ut along with the tilted support plane Hθ. Note that in reality, Hθ is not rotationally symmetric about the x-axis.141 vi Acknowledgements As I mulled over the contents of this thesis, I realized that my modest accom- plishment would not have been achieved without the help and insipration of a great number of mathematicians and otherwise, both near and afar, who I thank here. I owe a huge debt of gratitude to my advisor, Richard Schoen, for setting me on my way in geometric analysis, his great courses filled with his one-of-a-kind insight, and constant support and encouragement. His dedication to geometric analysis and penchant for creativity are truly inspirational. Special thanks are also due to my wonderful collaborators Shengwen Wang and Alec Payne, with whom much of the work in this thesis has been done. I also wish to warmly thank thank in particular professors Jacob Bernstein, Or Hershkovits, Bruce Kleiner, and Longzhi Lin for their very helpful suggestions, to either me or my collaborators, which directly impacted the work presented in this thesis. Their own work often served as an important source of inspiration as well. I've also had numerous long mathematical discussions concerning geometry and learned a lot from Jean-Francois Arbour, professor Theodora Bourni, professor Reto Buzano, professor Steven Cutkosky, professor Dan Edidin, professor Adam Helfer, Sven Hirsch, Andrew Holbrook, Yucheng Ji, professor Mat Langford, Chao Li, Martin Lisourd, Chris Lopez, Christos Mantoulidis, Peter McGrath, Henri Roesch, Hongyi Sheng, professor Jan Segert, professor Jeff Streets, professor Hung Tran, professor Li-Sheng Tseng, Ryan Unger, David Wiygul, Lisandra Vasquez, Hang Xu, professor Xiangwen Zhang, Kai-Wei Zhao, and Jonathan Zhu (and, if this list included people who I've discussed not just geometry with but all math, would be much longer). There is a similarly long, if not longer list of all the people who I had less of a chance to talk to but whose work was very inspirational; those whose work is built on time and again on this thesis not already mentioned are Ben Andrews, Bing-Long Chen, Tobias Colding, Klaus Ecker, Bob Haslhofer, Gerhard Huisken, Tom Illmanen, Bill Minicozzi, Lu Wang, Brian White, and Le Yin. The coursework and mentorship apart from my advisor I recieved at UC Irvine and my undergraduate school, University of Missouri, also had an important impact on my development and I'm thankful to the many great lecturers I've had along the way. Of course, this thesis wouldn't have been possible without the love and support of my parents, my in-laws, and my wife, Alica Gonzalez, who helpped me navigate particularly confusing vagueries of life I encountered in my time in graduate school. vii Finally, I'd like to thank the generosity of the American Mathematical Society, Springer Nature Switzerland-AG, International press of Boston, and Oxford Univer- sity press for allowing me to include passages found in chapters 4, 6, 7, and 8 of this thesis which originally appeared in Proceedings of the AMS, Journal of Geometric Analysis, Communications in Analysis and Geometry, and International Mathemat- ics Research Notices, respectively.
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