Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces by Tracey Balehowsky A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2017 by Tracey Balehowsky (cid:13) Abstract Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces Tracey Balehowsky Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2017 Inthisthesis, weprovethatif(M,g)isaC3-smooth, 3-dimensionalRiemannianmanifold with mean convex boundary ∂M, which is additionally either a) C2-close to Euclidean or b) sufficiently thin, then knowledge of the least areas circumscribed by any simple closed curve γ ∂M uniquely determines the metric. In fact, given the least area data for a ⊂ much more restricted class of curves γ ∂M, we uniquely determine the metric. We also ⊂ prove a corresponding local result: assuming only that (M,g) has strictly mean convex boundary at a point p ∂M, we prove that knowledge of the least areas circumscribed ∈ by any simple closed curve γ in a neighbourhood U ∂M of p uniquely determines the ⊂ metric near p. ii Declaration of Originality The research in this thesis was conducted at the Department of Mathematics, Univer- sity of Toronto, in the period between June 2013 and June 2017. The material contained in the thesis is original and a product of both collaborative and independent research. Chapters 3, 4, and 5 contains joint work with Spyros Alexakis and Adrian Nachman. This joint research is in preparation for publication. iii To Minnie, Marilyn, and Megan iv Acknowledgements I would like to extend an enormous amount of gratitude to my supervisors, Spyros Alexakis and Adrian Nachman. Their incredible mathematical insight, guidance, and encouragement has been invaluable to me during my PhD research. They have been wonderful supervisors and I hope I will have the opportunity to work with them in the future. I wish to thank the University of Toronto and the Department of Mathematics at the UniversityofTorontoforitsfinancialsupportduringmyPhD.Additionally, thankstothe generous donation of John R. Levitt, I was partially supported by an Ontario Graduate Scholarship. Many thanks go to the Department of Mathematics administrative staff for all of their help. Finally, words cannot describe how much the friendship, support, and encouragement of my friends and family has helped me while I pursued my PhD degree. v Contents 1 Introduction 1 2 Preliminaries 10 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Area Minimizing Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Background on Inverse Problems . . . . . . . . . . . . . . . . . . . . . . 15 3 Asymptotically Flat Extension and Conformal Maps 17 3.1 Extension to an Asymptotically Flat Manifold . . . . . . . . . . . . . . . 17 3.2 Least Area Implies Dirichlet-to-Neumann Data . . . . . . . . . . . . . . . 26 4 Equations for the Components of the Inverse Metric 36 5 Proof of the Main Theorems 46 6 Appendix 82 6.1 First and Second Variations of Area . . . . . . . . . . . . . . . . . . . . . 82 6.2 Algebraic Relationships Between the Components of g and g−1 . . . . . . 90 Bibliography 92 vi List of Figures 1.1 Simple closed curve γ on ∂M. . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 DepictionofaregionAinan(n+1)-dimensionalCFTandaarea-minimizing surface Y in (n+2)-dimensional AdS. . . . . . . . . . . . . . . . . . . . 3 γ 1.3 Thin manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1 Depiction of the curves γ(t) and γ˜(t,(cid:15)). . . . . . . . . . . . . . . . . . . . 28 4.1 Depiction of the leaves Y(s,t). . . . . . . . . . . . . . . . . . . . . . . . . 38 5.1 Neighbourhood near a point on the manifold (M,g). . . . . . . . . . . . . 80 vii Chapter 1 Introduction The classical boundary rigidity problem in differential geometry asks whether knowledge of the distance between any two points on the boundary of a Riemannian manifold is sufficient to identify the metric up to isometries that fix the boundary. Manifolds for which this is the case are called boundary rigid. One motivation for the boundary rigidity problem comes from seismology, if one seeks to determine the interior structure of the Earth from measurements of travel times of seismic waves. Not all manifolds are boundary rigid. The manifolds which serve as counterexamples to boundary rigidity can be intuitively described as follows: if the manifold has a region of large positive curvature in the interior, length-minimizing geodesics between boundary points will never pass through this region. A way to rule out such counterexamples is to assume the manifold is simple, meaning that any two points can be joined by a unique minimizing geodesic and that the boundary is strictly convex. Michel [17] conjectured that simple manifolds are boundary rigid. Special cases have been proved by Michel [17], Mukhometov [18], Gromov [11], Croke [8], Burago and Ivanov [4, 5]. In two dimensions, the conjecture was settledbyPestovandUhlmann[20]. Movingawayfromthesimplicityassumption, recent work of Stefanov, Uhlmann and Vasy solved a local version of the rigidity problem in a neighbourhoodofanystrictlyconvexpointoftheboundary,andobtainedacorresponding global rigidity result for manifolds that admit a foliation satisfying a certain convexity condition. (see [23] and earlier references given there). In this thesis, we consider the following higher dimensional version of the boundary rigidity problem, where in lieu of lengths of geodesics the data consists of areas of area- minimizing surfaces. Question. Given any simple closed curve γ on the boundary of a Riemannian 3-manifold (M,g), supposetheareaofanyleast-areasurfaceY circumscribedbythecurveisknown. γ Does this information determine the metric g? 1 Chapter 1. Introduction 2 Under certain geometric conditions, we show that the answer is yes. In some cases, we only require the area data for a much smaller subclass of curves on the boundary. (M,g) Y� � Figure 1.1: Simple closed curve γ on ∂M. Theories posited by the AdS/CFT correspondence also provide strong physics moti- vation to consider the problem of using knowledge of the areas of certain submanifolds to determine the metric. Loosely speaking, the AdS/CFT correspondence states that the dynamics of (n + 2)-dimensional supergravity theories modelled on an Anti-de Sit- ter (AdS) space are equivalent to the quantum physics modelled by (n+1)-dimensional conformal field theories (CFT) on the boundary of the AdS space (see [14]). This equiv- alence is often referred to as holographic duality. Analogous to the problem of boundary rigidity, one of the main goals of the AdS/CFT correspondence is to use conformal field theory information on the boundary to determine the metric on the AdS spacetime which encodes information about the corresponding gravity dynamics. Towards this goal, Maldecena [15] has proposed that given a curve on the boundary of the AdS spacetime, the renormalized area of the minimal surface bounding the curve contains information about the expectation value of the Wilson loop associated to the curve. More recently, Ryu and Takayanagi [22] have conjectured that given a region A on the boundary of an (n+2)-dimensional AdS spacetime, the entanglement entropy of A is equivalent to the renormalized area of the area-minimizing surface Y with boundary γ γ = ∂A (see Figure 1.2). The AdS/CFT correspondence is often studied in Riemannian signature, where one must consider an asymptotically hyperbolic manifold (M,g), for which one knows information on the boundary and the renormalized area for minimal Chapter 1. Introduction 3 surfaces bounded by closed loops on the boundary. Hence, if given a collection of simple closed curve on the boundary-at-infinity of (M,g), does knowledge of the renormalized area of the area-minimizing surfaces bounded by these curves allow us to recover the metric? Answers in the affirmative may provide new methods to describe the relationship between gravity theories and conformal field theories. (n+1)-dim. CFT A Y � � (n+2)-dim. AdS Figure 1.2: Depiction of a region A in an (n+1)-dimensional CFT and a area-minimizing surface Y in (n+2)-dimensional AdS. γ In this thesis, we do not consider a collection of curves on a boundary-at-infinity, but rather a particular family of simple closed curves on a finite boundary. We use knowledge of the areas of the surfaces of least area bounded by such curves to find the metric. Althoughthequestionofrecoveringametricfromareadataisispurelygeometric in nature, our main approach to the proof is to reformulate the problem in terms of an inverse problem for a partial differential equation on the manifold. We note that the solution [20] of the Michel conjecture in two dimensions by Pestov and Uhlmann relied on a similar reformulation, as it involved the surprising discovery that for simple two dimensional manifolds one can obtain the Dirichlet-to-Neumann map for the Laplace Beltrami operator from knowledge of the boundary distance function. In our case, we start by showing the much simpler fact that the linearization of the area data encodes