دانلود کتاب Extrinsic Geometric Flows

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Contents Preface A Guide for the Reader The heat equation (Chapter 1) Curve shortening flow (Chapters 2–4) Mean curvature flow (Chapters 5–14) Gauß curvature flows (Chapters 15–17) Fully nonlinear curvature flows (Chapters 18–20) Acknowledgments Suggested Course Outlines Notation and Symbols Chapter 1. The Heat Equation §1.1. Introduction §1.2. Gradient flow §1.3. Invariance properties 1.3.1. Generating solutions from symmetries 1.3.2. Invariant solutions §1.4. The maximum principle §1.5. Well-posedness §1.6. Asymptotic behavior §1.7. The Bernstein method §1.8. The Harnack inequality §1.9. Further monotonicity formulae 1.9.1. The Nash entropy 1.9.2. Weighted monotonicity formulae 1.9.3. Semilinear heat equations §1.10. Sharp gradient estimates §1.11. Notes and commentary 1.11.1. The Omori–Yau maximum principle 1.11.2. Sturm’s theorem 1.11.3. The differential Harnack inequality 1.11.4. Ancient solutions 1.11.5. The heat equation on Riemannian manifolds §1.12. Exercises Chapter 2. Introduction to Curve Shortening §2.1. Basic geometric theory of planar curves 2.1.1. Immersed, embedded, and closed curves 2.1.2. Arc length, tangency, and normalcy 2.1.3. Curvature 2.1.4. Round circles 2.1.5. The normal angle 2.1.6. Arc length element, rotation index §2.2. Curve shortening flow §2.3. Graphs of functions 2.3.1. Grim Reaper solution §2.4. The support function §2.5. Short-time existence §2.6. Smoothing §2.7. Global existence §2.8. Notes and commentary §2.9. Exercises Chapter 3. The Gage–Hamilton–Grayson Theorem §3.1. The avoidance principle §3.2. Preserving embeddedness §3.3. Huisken’s distance comparison estimate §3.4. A curvature bound by distance comparison §3.5. Grayson’s theorem §3.6. Singularities of immersed solutions §3.7. Notes and commentary §3.8. Exercises Chapter 4. Self-Similar and Ancient Solutions §4.1. Invariance properties §4.2. Self-similar solutions §4.3. Monotonicity formulae 4.3.1. Isoperimetric ratio. The isoperimetric inequality 4.3.2. Differential Harnack estimate 4.3.3. Entropy monotonicity 4.3.4. A sketch of the Gage–Hamilton proof that convex embeddedcurves converge to round points 4.3.5. Huisken’s monotonicity formula 4.3.6. Monotonicity via Sturm’s theorem §4.4. Ancient solutions 4.4.1. The hairclip solution 4.4.2. The paperclip solution §4.5. Classification of convex ancient solutions on S^1 §4.6. Notes and commentary §4.7. Exercises Chapter 5. Hypersurfaces in Euclidean Space §5.1. Parametrized hypersurfaces §5.2. Alternative representations of hypersurfaces 5.2.1. Graphs of functions 5.2.2. Level sets at regular values 5.2.3. Starshaped hypersurfaces 5.2.4. Convex hypersurfaces §5.3. Dynamical properties 5.3.1. Geometry on the pullback bundle 5.3.2. Evolving orthonormal frames 5.3.3. First variation of area and volume §5.4. Curvature flows 5.4.1. The linearized flow 5.4.2. Local coordinate calculations for flows §5.5. Notes and commentary §5.6. Exercises Chapter 6. Introduction to Mean Curvature Flow §6.1. The mean curvature flow 6.1.1. Explicit solutions to the mean curvature flow §6.2. Invariance properties and self-similar solutions 6.2.1. Translators 6.2.2. Shrinkers and expanders 6.2.3. Rotators §6.3. Evolution equations 6.3.1. The gradient flow of the area functional 6.3.2. Normalized mean curvature flow §6.4. Short-time existence 6.4.1. Invariance under diffeomorphisms 6.4.2. Short-time existence §6.5. The maximum principle 6.5.1. Maximum principle for scalars 6.5.2. A maximum principle for tensors §6.6. The avoidance principle §6.7. Preserving embeddedness §6.8. Long-time existence §6.9. Weak solutions 6.9.1. The Brakke flow 6.9.2. The level set flow 6.9.3. The viscosity approach 6.9.4. The shadow flow of Sáez Trumper and Schnürer 6.9.5. Mean curvature flow with surgery §6.10. Notes and commentary 6.10.1. Maximum principles 6.10.2. Constrained mean curvature flows 6.10.3. Well-posedness 6.10.4. Blow-up of the mean curvature 6.10.5. The compact-open C^k topology 6.10.6. Regularity of the level set flow 6.10.7. Mean curvature flow of soap film clusters §6.11. Exercises Chapter 7. Mean Curvature Flow of Entire Graphs §7.1. Introduction §7.2. Preliminary calculations §7.3. The Dirichlet problem §7.4. A priori height and gradient estimates §7.5. Local a priori estimates for the curvature §7.6. Proof of Theorem 7.1 §7.7. Convergence to self-similarly expanding solutions §7.8. Self-similarly shrinking entire graphs §7.9. Notes and commentary §7.10. Exercises Chapter 8. Huisken’s Theorem §8.1. Pinching is preserved §8.2. Pinching improves: The roundness estimate §8.3. A gradient estimate for the curvature §8.4. Huisken’s theorem 8.4.1. Convergence to a point 8.4.2. Estimates in C^0 after rescalling 8.4.3. Estimates in C^1 after rescaling 8.4.4. Estimates in C^∞ after rescaling §8.5. Regularity of the arrival time §8.6. Huisken’s theorem via width pinching §8.7. Notes and commentary 8.7.1. Mean curvature flow in the sphere 8.7.2. Mean curvature flow in Riemannian ambient spaces 8.7.3. High-codimension mean curvature flow 8.7.4. Free boundary mean curvature flow §8.8. Exercises Chapter 9. Mean Convex Mean Curvature Flow §9.1. Singularity formation 9.1.1. The standard neckpinch 9.1.2. The degenerate neckpinch §9.2. Preserving pinching conditions §9.3. Pinching improves: Convexity and cylindrical estimates §9.4. A natural class of initial data §9.5. A gradient estimate for the curvature §9.6. Notes and commentary §9.7. Exercises Chapter 10. Monotonicity Formulae §10.1. Huisken’s monotonicity formula 10.1.1. Pointwise monotonicity formula 10.1.2. A maximum principle for noncompact solutions 10.1.3. Local area bounds §10.2. Hamilton’s Harnack estimate 10.2.1. The Harnack estimate as the nonnegativity of a quadratic 10.2.2. Saturation by expanding self-similar solutions 10.2.3. The Harnack calculation 10.2.4. The Harnack maximum principle argument 10.2.5. Convex eternal solutions 10.2.6. Space-time formulation of Hamilton’s Harnack estimate 10.2.7. Concavity of the arrival time §10.3. Notes and commentary 10.3.1. Huisken’s monotonicity formula for Brakke flows 10.3.2. General monotonicity formulae 10.3.3. Huisken’s monotonicity formula for mean curvature flow ingeneral ambient spaces 10.3.4. Ecker’s local monotonicity formula 10.3.5. Monotonicity formula for free boundary mean curvatureflow 10.3.6. Smockzyk’s Harnack estimate for flows with speed f(H) 10.3.7. Harnack estimates for fully nonlinear flows §10.4. Exercises Chapter 11. Singularity Analysis §11.1. Local uniform convergence of mean curvature flows 11.1.1. Hypersurfaces with bounded geometry 11.1.2. Mean curvature flows with bounded curvature §11.2. Neck detection §11.3. The Brakke–White regularity theorem §11.4. Huisken’s theorem revisited 11.4.1. Proper hypersurfaces with pinched principal curvatures §11.5. The structure of singularities 11.5.1. Singularity models for type-I singularities 11.5.2. The normalized flow about type-I singularities 11.5.3. Singularity models for type-II singularities 11.5.4. Tangent flows are shrinkers 11.5.5. Conjectures on singularity formation in mean curvature flow 11.5.6. Mean curvature flow with surgery 11.5.7. Piecewise smooth mean curvature flow §11.6. Notes and commentary 11.6.1. A local compactness theorem 11.6.2. A local Brakke–White regularity theorem 11.6.3. Curvature pinching and compactness 11.6.4. Generic singularities and uniqueness of tangent flows 11.6.5. Singularities in free boundary mean curvature flow §11.7. Exercises Chapter 12. Noncollapsing §12.1. The inscribed and exscribed curvatures §12.2. Differential inequalities for the inscribed and exscribed curvatures 12.2.1. Simons-type differential inequalities §12.3. The Gage–Hamilton and Huisken theorems via noncollapsing §12.4. The Haslhofer–Kleiner curvature estimate §12.5. Notes and commentary §12.6. Exercises Chapter 13. Self-Similar Solutions §13.1. Shrinkers — an introduction §13.2. The Gaußian area functional 13.2.1. Differential identities §13.3. Mean convex shrinkers §13.4. Compact embedded self-shrinking surfaces 13.4.1. Compact, embedded shrinkers of genus 0 13.4.2. Compact embedded shrinkers of higher genus §13.5. Translators — an introduction §13.6. The Dirichlet problem for graphical translators §13.7. Cylindrical translators §13.8. Rotational translators §13.9. The convexity estimates of Spruck, Sun, and Xiao §13.10. Asymptotics §13.11. X.-J. Wang’s dichotomy §13.12. Rigidity of the bowl soliton §13.13. Flying wings §13.14. Bowloids §13.15. Notes and commentary 13.15.1. Classification of shrinkers 13.15.2. Open problems related to shrinkers 13.15.3. Classification of translators 13.15.4. Open problems related to translators 13.15.5. Expanders 13.15.6. Self-similar solutions to other flows §13.16. Exercises Chapter 14. Ancient Solutions §14.1. Rigidity of the shrinking sphere §14.2. A convexity estimate §14.3. A gradient estimate for the curvature §14.4. Asymptotics §14.5. X.-J. Wang’s dichotomy §14.6. Ancient solutions to curve shortening flow revisited §14.7. Ancient ovaloids §14.8. Ancient pancakes §14.9. Notes and commentary 14.9.1. Classification of ancient solutions 14.9.2. Ancient solutions to further extrinsic curvature flows 14.9.3. Open problems related to ancient solutions §14.10. Exercises Chapter 15. Gauß Curvature Flows §15.1. Invariance properties and self-similar solutions 15.1.1. Homothetic solutions §15.2. Basic evolution equations §15.3. Chou’s long-time existence theorem 15.3.1. A parabolic Monge–Ampère equation 15.3.2. Chou’s Gauß curvature estimate 15.3.3. Estimate on the radii of curvature 15.3.4. Higher regularity and convergence to a point §15.4. Differential Harnack estimates §15.5. Firey’s conjecture §15.6. Variational structure and entropy formulae 15.6.1. Gradient flow 15.6.2. The Firey entropy 15.6.3. The Gaußian entropy 15.6.4. The general Gaußian entropies §15.7. Notes and commentary §15.8. Exercises Chapter 16. The Affine Normal Flow §16.1. Affine invariance §16.2. Affine-renormalized solutions 16.2.1. Gauß curvature bound 16.2.2. Gauß curvature lower bound 16.2.3. Radius of curvature bound 16.2.4. Higher regularity §16.3. Convergence and the limit flow §16.4. Self-similarly shrinking solutions are ellipsoids §16.5. Convergence without affine corrections §16.6. Notes and commentary §16.7. Exercises Chapter 17. Flows by Superaffine Powers of the Gauß Curvature §17.1. Bounds on diameter, speed, and inradius 17.1.1. The Gauß curvature flow 17.1.2. The α-Gauß curvature flow §17.2. Convergence to a shrinking self-similar solution §17.3. Shrinking self-similar solutions are round 17.3.1. Preliminary calculations 17.3.2. Powers \frac{1}{n+2} ≤ α ≤ 1/2 17.3.3. Powers α > 1/2 §17.4. Notes and commentary §17.5. Exercises Chapter 18. Fully Nonlinear Curvature Flows §18.1. Introduction §18.2. Symmetric functions and their differentiability properties 18.2.1. Functions of curvature 18.2.2. Homogeneity 18.2.3. Concavity 18.2.4. Inverse-concavity §18.3. Examples §18.4. Short-time existence §18.5. The avoidance principle §18.6. Differential Harnack estimates §18.7. Entropy estimates §18.8. Alexandrov reflection 18.8.1. Alexandrov reflection of convex hypersurfaces 18.8.2. Alexandrov reflection of embedded hypersurfaces §18.9. Notes and commentary §18.10. Exercises Chapter 19. Flows of Mean Curvature Type §19.1. Convex hypersurfaces contract to round points 19.1.1. Preliminaries 19.1.2. Preserving pinching §19.2. Evolving nonconvex hypersurfaces 19.2.1. Convexity and cylindrical estimates 19.2.2. Noncollapsing §19.3. Notes and commentary §19.4. Exercises Chapter 20. Flows of Inverse-Mean Curvature Type §20.1. Convex hypersurfaces expand to round infinity 20.1.1. Preliminaries 20.1.2. Geometric estimates 20.1.3. Convergence §20.2. Notes and commentary 20.2.1. Geometric inequalities 20.2.2. Flows with free boundary §20.3. Exercises Bibliography 1-14 15-35 36-55 56-75 76-93 94-116 117-136 137-158 159-177 178-199 200-220 221-242 243-263 264-285 286-306 307-327 328-349 350-371 372-390 391-411 412-431 432-453 454-474 475-496 497-518 519-539 Index