Extrinsic Geometric Flows

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