Extrinsic Geometric Flows (Graduate Studies in Mathematics)

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauss curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter. Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter. Contents Preface A Guide for the Reader Suggested Course Outlines Notation and Symbols 1. The Heat Equation 2. Introduction to Curve Shortening 3. The Gage–Hamilton–Grayson Theorem 4. Self-Similar and Ancient Solutions 5. Hypersurfaces in Euclidean Space 6. Introduction to Mean Curvature Flow 7. Mean Curvature Flow of Entire Graphs 8. Huisken’s Theorem 9. Mean Convex Mean Curvature Flow 10. Monotonicity Formulae 11. Singularity Analysis 12. Noncollapsing 13. Self-Similar Solutions 14. Ancient Solutions 15. Gauß Curvature Flows 16. The Affine Normal Flow 17. Flows by Superaffine Powers of the Gauß Curvature 18. Fully Nonlinear Curvature Flows 19. Flows of Mean Curvature Type 20. Flows of Inverse-Mean Curvature Type Bibliography Index The heat equation -- Introduction to curve shortening -- The Gage-Hamilton and Grayson theorems -- Self-similar and ancient solutions -- Hypersurfaces in Euclidean space -- Introduction to mean curvature flow -- Mean curvature flow of entire graphs -- Huisken's theorem -- Mean convex mean curvature flow -- Monotonicity formulae -- Singularity analysis -- Noncollapsing -- Self-similar solutions -- Ancient solutions -- Gauss curvature flows -- The affine normal flow -- Flows by super-affine powers of the Gauss curvature -- Fully nonlinear curvature flows -- Flows of mean curvature type -- Flows of inverse-mean curvature type