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Existence and Regularity of Branched Minimal Submanifolds

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Existence and Regularity of Branched Minimal Submanifolds EXISTENCE AND REGULARITY OF BRANCHED MINIMAL SUBMANIFOLDS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Brian Krummel August 2011 © 2011 by Brian James Krummel. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/rc085mz1473 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leon Simon, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Richard Schoen I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Brian White Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract We consider two-valued solutions to elliptic problems, which arise from the study branched minimal submanifolds. Simon and Wickramasekera constructed in [10] examples of two-valued solutions to the Dirichlet problem for the minimal surface ˘2 n−2 equation on the cylinder C = B1 (0) × R with H¨oldercontinuity estimates on the gradient assuming the boundary data satisfies a symmetry condition. However, their method was specific to the minimal surface equation. We generalize Simon and Wickramasekera's result to an existence theorems for a more general class elliptic equations and for a class of elliptic systems with small data. In particular, we extend Simon and Wickramasekera's result to the minimal surface system. Our approach uses techniques for elliptic differential equations such as the Leray-Schauder theory and contraction mapping principle, which have the advantage of applying in more general contexts than codimension 1 minimal surfaces. We also show that for two- valued solutions to elliptic equations with real analytic data, the branch set of their graphs are real analytic (n − 2)-dimensional submanifolds. This is a consequence of using the Schauder estimate for two-valued functions and a technique involving majorants due to Friedman [2] to inductively get estimates on the derivatives of the two-valued solutions. iv Acknowledgments I am grateful for my advisor Leon Simon, whose ideas and encouragement were in- valuable to my dissertation work. Leon's insightful way of looking at mathematics has had a positive influence on me as a mathematician. I consider myself fortunate to have been his student. I also thank Rick Schoen, Brian White, Rafe Mazzeo, and Simon Brendle for tak- ing time to meet with me to discuss mathematics throughout my time at Stanford. Their ideas have often been interesting and helpful in my work. In particular, I thank Rick Schoen, Brian White, and Rafe Mazzeo for being on my dissertation committee and Simon Brendle for providing financial support for my quarter spent at Cambridge. I appreciate Neshan Wickramasekera showing his interest in my research and tak- ing time to talk with me about it while I was visiting Cambridge. Finally I am thankful to my friends and family for their support and interest in my academic endeavors. In particular, I am grateful to my parents for teaching me the value of education from a young age and for their encouragement during my graduate work. v Contents Abstract iv Acknowledgments v 1 Introduction and Statement of Main Results 1 1.1 Notation . 1 1.2 Two-valued functions . 2 1.3 The function spaces Vk and Vk,µ ..................... 10 1.4 Minimal submanifolds . 18 1.5 Existence and regularity theorems . 21 2 Elliptic theory for two-valued functions 28 2.1 Overview . 28 2.2 Maximum principles . 29 2.3 Schauder estimates . 35 2.4 Global estimates . 45 2.5 H¨oldercontinuity estimates . 50 3 Existence theorems 60 3.1 Overview . 60 3.2 Poisson equation . 60 3.3 Elliptic systems . 68 3.4 Elliptic equations . 73 vi 4 Regularity theorems 88 4.1 C1,µ regularity of C1 solutions . 88 4.2 Smoothness of two-valued solutions . 112 4.3 Bounding gγ ................................ 118 i 4.4 Bounding the H¨oldercoefficient of fγ . 123 4.5 Analyticity of two-valued solutions . 129 Bibliography 133 vii Chapter 1 Introduction and Statement of Main Results 1.1 Notation We shall adopt the following notation and conventions throughout this thesis. n ≥ 3 is a fixed integer. m m BR (X0) denotes the closed ball of radius R centered at X0 in R . ˘m m BR (X0) denotes the open ball of radius R centered at X0 in R . n ˘ ˘n BR(X0) = BR(0), BR(X0) = BR(X0). ˘2 n−2 n C = B1 (0) × R denotes an open cylinder in R . 2 n−2 X = (x; y) denotes a point in C, where x 2 B1 (0) and y 2 R . We identify x with the point reiθ in C, where r 2 [0; 1] and θ 2 R. 1 CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 2 iθ iθ+iφ Rφ denotes the n × n matrix such that Rφ(re ; y) = (re ; y). We write 0 1 cos φ − sin φ 0 0 ··· 0 B C Bsin φ cos φ 0 0 ··· 0C B C B 0 0 1 0 ··· 0C i B C Rφ = (R )i;j=1;:::;n = B C j B 0 0 0 1 ··· 0C B C B . .C B . .. .C @ A 0 0 0 0 ··· 1 j where Ri denotes the entry in the i-th row and j-th column of Rφ. Rm×n denotes the space of m × n real matrices. Hk denotes the k-th dimensional Hausdorff measure. Ln denotes the Lebesgue measure in Rn. 1.2 Two-valued functions Let Ω be a domain in Rn. Given an integer q ≥ 2, we say u is a q-valued function m on Ω taking values in R if at each point X 2 Ω, u(X) = fu1(X); : : : ; uq(X)g is an m unordered q-tuple, where uj(X) 2 R [1]. We can define a metric G on the space of unordered q-tuples by q !1=2 X 2 G(u; v) = sup inf juj − vσ(j)j X2Ω σ j=1 for unordered q-tuples u = fu1; : : : uqg and v = fv1; : : : vqg, where σ is a permutation of f1; 2; : : : ; qg. We say a q-valued function u is continuous at X 2 Ω if lim G(u(X); u(Y )) = 0: Y !X CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 3 We say a q-valued function u from Ω ⊆ Rn to unordered q-tuples in Rm is differentiable at X 2 Ω if for some m × n matrices A1;:::Aq, G(u(X + h); fu (X) + A hg) lim j j = 0; h!0 jhj in which case we say Du(X) = fA1;:::;Aqg is the derivative of u at X. We define k-th order derivatives for integers k ≥ 2 inductively by Dku(X) = D(Dk−1u)(X). For integers k ≥ 0, Ck(Ω; Rm) denotes the space of q-valued functions u on Ω taking values in Rm such that Dju exists and is continuous for 0 ≤ j ≤ k. We say u is H¨oldercontinuous with exponent µ 2 (0; 1] on Ω if G(u(X); u(Y )) [u]µ,Ω = sup µ < 1: (1.1) X;Y 2Ω;X6=Y jX − Y j For k ≥ 0 and µ 2 (0; 1], Ck,µ(Ω; Rm) denotes the space of two-valued functions u 2 Ck(Ω; Rm) such that Dku is H¨oldercontinuous with exponent µ. For Ω ⊆ Rn open, k m k m given k ≥ 0 we let Cc (Ω; R ) denote the space of two-valued functions u 2 C (Ω; R ) such that u = f0; 0g on Ω n Ω0 for some Ω0 ⊂⊂ Ω and given k ≥ 0 and µ 2 (0; 1] we k,µ m k,µ m let Cc (Ω; R ) denote the space of two-valued functions u 2 C (Ω; R ) such that u = f0; 0g on Ω n Ω0 for some Ω0 ⊂⊂ Ω. For Ω ⊆ Rn open, k ≥ 0, and µ 2 (0; 1], we k,µ m k m k,µ 0 m let Cloc (Ω; R ) denote the space of u 2 C (Ω; R ) such that u 2 C (Ω ; R ) for all Ω0 ⊂⊂ Ω. For Ω ⊆ Rn, we let kukC0(Ω) = sup G(u(X); 0) X2Ω for each two-valued function u 2 C0(Ω; Rm), X α kukCk(Ω) = kD ukC0(Ω) jα|≤k for each two-valued function u 2 Ck(Ω; Rm) for k ≥ 1, and X α X α kukCk,µ(Ω) = kD ukC0(Ω) + [D u]µ,Ω jα|≤k jαj=k CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 4 for each two-valued function u 2 Ck,µ(Ω; Rm) for k ≥ 0 and µ 2 (0; 1). We define a metric on C0(Ω; Rm) by ku − vkC0(Ω) = sup G(u(X); v(X)) X2Ω for two-valued functions u; v 2 C0(Ω; Rm) and we define a metric on Ck(Ω; Rm) for k ≥ 1 by X α α ku − vkCk(Ω) = kD u − D vkC0(Ω) jα|≤k for two-valued functions u; v 2 Ck(Ω; Rm).
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