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Finite Element Approximation of Ricci Curvature and Simulation of Ricci-Deturck Flow

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Finite Element Approximation of Ricci Curvature and Simulation of Ricci-Deturck Flow Finite Element Approximation of Ricci Curvature and Simulation of Ricci-DeTurck Flow by Hans Fritz Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨atFreiburg im Breisgau Freiburg im Breisgau 2013 2 Dekan: Prof. Dr. Michael Ru˚ziˇ ckaˇ Referent: Prof. Dr. Gerhard Dziuk Korreferent: Prof. Dr. Klaus Deckelnick Datum der Promotion: 26.03.2013 3 Acknowledgment I would like to express my gratitude to my advisor, Professor Gerhard Dziuk, for mentoring me, for teaching me so much and for encouraging me in my work. His advice has always been very important for me and helped me to complete this work. My thanks also belong to Professor Gerhard Huisken, since the idea to use Yano’s formula for the Ricci curvature goes back to him. I would like to thank Professor Miles Simon, who willingly shared his profound expertise in the Ricci Flow with me during the last years. I am also indebted to Claus-Justus Heine for his support concerning the Finite Element Toolbox ALBERTA. Furthermore, I would like to thank the reviewers for reading my thesis. Finally, I would like to thank all those who supported and encouraged me during the last years. 4 Abstract In this thesis, we present the definition of a discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension n that are isometrically embedded with codimension one into the Euclidean space Rn+1. Our definition is based on the discretization of a weak formulation of the Ricci curvature with isoparametric Finite Elements. In the case that a two- or three-dimensional, sufficiently smooth hypersurface Γ ⊂ Rn+1 is approximated by piecewise quadratic isoparametric hypersurfaces, this discrete Ricci curvature converges to the Ricci curvature of Γ with a linear order of convergence in the L2 (Γ)-norm. Using the discrete version of a smoothing operator, the discrete Ricci curvature of a piecewise linear Lagrange interpolant of Γ also 2 2 approximates the Ricci curvature of Γ in the L (Γ)-norm. The corresponding order of convergence is 3 . In 1 1;2 this case, the approximation error additionally converges with an order of 3 with respect to the W (Γ)- norm. Numerical tests confirm these error estimates for two- and three-dimensional hypersurfaces. Furthermore, we present a numerical method for the simulation of the Ricci-DeTurck Flow. This flow is a reparametrization of the Ricci Flow, which is an intrinsic geometric flow deforming a Riemannian metric in proportion to its Ricci curvature. In order to develop an algorithm for the Ricci-DeTurck Flow, the Finite Element discretization of the Ricci curvature is generalized to n-dimensional Riemannian manifolds that are merely embedded as differentiable manifolds into Rn+1. Since the embedding (Γ; g) is not assumed to be isometric anymore, we introduce a special matrix representation of the non-Euclidean Riemannian metric g on Γ only depending on the Cartesian coordinates of the ambient space. By this means, it is possible to derive a weak formulation of the Ricci-DeTurck Flow such that all appearing terms can be computed in Cartesian coordinates. A spatial discretization of this formulation with Finite Elements on polyhedral hypersurfaces and a semi-implicit time discretization lead to an algorithm for the computation of the Ricci-DeTurck Flow. Experimental tests for two- and three-dimensional manifolds show the convergence of this algorithm, albeit a convergence proof is missing so far. In the present work, the algorithm has been used to simulate the formation of singularities, such as neck pinch singularities, in the three-dimensional Ricci-DeTurck Flow. The following publication has evolved from this dissertation: Fritz (2013). 5 Zusammenfassung In dieser Arbeit fuhren¨ wir die Definition einer diskreten Ricci-Krummung¨ auf polyedrischen Hyperflachen¨ von beliebiger Dimension n ein, welche isometrisch mit Kodimension eins in den Euklidischen Raum Rn+1 eingebettet sind. Unsere Definition beruht auf der Diskretisierung einer schwachen Formulierung der Ricci-Krummung¨ mithilfe von isoparametrischen Finiten Elementen. Fur¨ den Fall, dass eine zwei- oder dreidimensionale, hinreichend glatte Hyperflache¨ Γ durch stuckweise¨ quadratische isoparametrische Hyperflachen¨ approximiert wird, konvergiert diese diskrete Ricci-Krummung¨ mit linearer Konvergenzord- nung in der L2 (Γ)-Norm gegen die Ricci-Krummung¨ von Γ. Verwendet man die diskrete Version eines Glattungsoperators,¨ so approximiert auch die diskrete Ricci-Krummung¨ einer stuckweise¨ linearen Lagrange Interpolierenden von Γ die Ricci-Krummung¨ von Γ in der L2 (Γ)-Norm. Die zugehorige¨ Konvergenzord- 2 1;2 nung ist 3 . Zusatzlich¨ konvergiert in diesem Fall der Approximationsfehler bezuglich¨ der W (Γ)-Norm 1 mit Ordnung 3 . Numerische Tests bestatigen¨ diese Fehlerabschatzungen¨ fur¨ zwei- und dreidimensionale Hyperflachen.¨ Des Weiteren prasentieren¨ wir eine numerische Methode zur Simulation des Ricci-DeTurck-Flusses. Dieser Fluss ist eine Reparametrisierung des Ricci-Flusses, welcher ein intrinsischer geometrischer Fluss ist, der eine Riemannsche Metrik proportional zu ihrer Ricci-Krummung¨ verformt. Um einen Algorithmus fur¨ den Ricci-DeTurck-Fluss zu entwickeln, wird die Finite Elemente Diskretisierung der Ricci-Krummung¨ auf n-dimensionale Riemannsche Mannigfaltigkeiten verallgemeinert, welche lediglich als differenzierbare Mannigfaltigkeiten in den Rn+1 einbettbar sind. Da die Einbettung (Γ; g) nicht mehr als isometrisch voraus- gesetzt wird, fuhren¨ wir eine spezielle Matrixdarstellung der nicht-Euklidischen Riemannschen Metrik g auf Γ ein, welche nur von den Kartesischen Koordinaten des umgebenden Raumes abhangt.¨ Auf diese Weise ist es moglich,¨ eine schwache Formulierung des Ricci-DeTurck-Flusses herzuleiten, sodass alle auftretenden Terme in Kartesischen Koordinaten berechnet werden konnen.¨ Eine Ortsdiskretisierung dieser Formulierung mit Finiten Elementen auf polyedrischen Hyperflachen¨ und eine semi-implizite Zeitdiskretisierung fuhren¨ auf einen Algorithmus zur Berechnung des Ricci-DeTurck-Flusses. Experimentelle Tests fur¨ zwei- und drei- dimensionale Mannigfaltigkeiten zeigen die Konvergenz dieses Algorithmus, auch wenn ein Konvergenzbe- weis bislang fehlt. In der vorliegenden Arbeit wurde der Algorithmus dazu verwendet, die Entstehung von Singularitaten,¨ wie beispielsweise Neck-Pinch-Singularitaten,¨ im dreidimensionalen Ricci-DeTurck-Fluss zu simulieren. Die folgende Vero¨ffentlichung ist aus dieser Dissertation hervorgegangen: Fritz (2013). 6 Contents 1 Introduction 9 1.1 Notations and Definitions . 14 1.2 Fundamentals of the Ricci Flow and of the Ricci-DeTurck Flow . 26 1.3 The Surface Finite Element Method . 34 2 Finite Element Approximation of the Ricci curvature 43 2.1 Yano’s formula for the Ricci curvature . 43 2.2 Weak formulation of Ricci curvature on embedded hypersurfaces . 44 2.3 Discrete Ricci curvature: Definition and error estimates . 47 2.4 Numerical results for the Approximation of the Ricci curvature . 55 3 Simulation of the Ricci-DeTurck Flow 63 3.1 A method for the simulation of intrinsic geometric flows . 63 3.2 Weak formulation of the Ricci-DeTurck Flow . 70 3.3 Discretization of the Ricci-DeTurck Flow . 79 3.4 Numerical results for the Simulation of the Ricci-DeTurck Flow . 82 Symbols 113 Bibliography 117 8 CONTENTS Chapter 1 Introduction The Ricci Flow is a geometric flow deforming a time-dependent Riemannian metric in proportion to its Ricci curvature. It was introduced by Richard Hamilton in 1982 (see Hamilton, 1982), who first proved short-time existence for solutions to the Ricci Flow on compact manifolds using the Nash-Moser Theorem. In 1983, Dennis DeTurck presented an alternative proof based on the idea of reparametrizing the original flow in order to get a strictly parabolic equation of second order (see DeTurck, 1983; Hamilton, 1995). A variant of this proof leads to the dual Ricci-Harmonic Map Flow (see Shi, 1989; Hamilton, 1995), which is often simply called the Ricci-DeTurck Flow1. It is a reparametrization of the Ricci Flow by a smooth family of diffeomorphisms solving the Harmonic Map Flow2. The short-time existence and uniqueness of solutions to the Ricci-DeTurck Flow can be used to prove the corresponding result for the Ricci Flow. In fact, the solution to the Ricci Flow is the pull-back of the solution to the Ricci-DeTurck Flow by a map that solves a certain system of ordinary differential equations. Since its introduction the Ricci Flow has served as an approach towards the geometrization conjecture (see Thurston, 1982). In 2002, Grigori Perelman finally proved this conjecture in a series of breathtaking papers by using the Ricci Flow (see Perelman, 2002, 2003a,b). This work also implied an affirmative answer to the Poincare´ conjecture (see Poincare,´ 1904) after this problem had been open for nearly one hundred years. Since then, the Ricci Flow has turned out to be a powerful tool solving also many other problems in Geometry and Topology (see for example Brendle & Schoen, 2009, 2011). In contrast to its outstanding impact on pure mathematics, the Ricci Flow has not attracted appropriate attention in Numerics. The hitherto existing numerical studies of the Ricci Flow are either restricted to two-dimensional manifolds (see e.g. Jin et al., 2008) or to manifolds of revolution (see e.g. Garfinkle & Isenberg, 2005, 2008; Rubinstein & Sinclair, 2005; Taft, 2010). This unsatisfactory situation is certainly due to the fact that there is up to now no general formulation of the Ricci Flow, which is open to a classical discretization. Unlike the computation of the Ricci Flow, the simulation of other geometric flows, such as the Mean Curvature Flow (see e.g. Deckelnick et al., 2005) and the Willmore Flow (see Dziuk, 2008), has 1Please note that some authors distinguish between the dual Ricci-Harmonic Map Flow and the Ricci-DeTurck Flow, see also Section 1.2. Since we are interested in both flows, we do not make this distinction here for the sake of simplicity. Hence, in the text, the Ricci-DeTurck Flow also means the dual Ricci-Harmonic Map Flow. 2The Harmonic Map Flow is sometimes also called the Harmonic Map Heat Flow. 10 CHAPTER 1. INTRODUCTION made significant progress during the last two decades.
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