Extrinsic Geometric Flows

791 Pages·2020·18.893 MB·English

by  Ben Andrews, Bennett Chow, Christine Guenther, Mat Langford

206
GRADUATE STUDIES
IN MATHEMATICS
Extrinsic
Geometric Flows
Ben Andrews
Bennett Chow
Christine Guenther
Mat Langford
Extrinsic
Geometric Flows

206
GRADUATE STUDIES
IN MATHEMATICS
Extrinsic
Geometric Flows
Ben Andrews
Bennett Chow
Christine Guenther
Mat Langford
EDITORIAL COMMITTEE
Daniel S. Freed (Chair)
Bjorn Poonen
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 53C44, 58J35, 53A07, 52A20, 35K20.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-206
Library of Congress Cataloging-in-Publication Data
Names: Andrews,Ben,author. |Chow,Bennett,author. |Guenther,Christine(ChristineMarie),
1966-author. |Langford,Mat(Mathew),1987-author.
Title: Extrinsic geometricflows / Ben Andrews, Bennett Chow, Christine Guenther, Mat Lang-
ford.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2020]|Series: Gradu-
atestudiesinmathematics,1065-7339;206|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2019059835|ISBN9781470455965(v. 206;hardcover)|ISBN9781470456863
(v. 206;ebook)
Subjects: LCSH:Globaldifferentialgeometry. |Differentialequations,Parabolic. |Flows(Differ-
entiabledynamicalsystems). |Curvature. |Geometricanalysis. |AMS:Differentialgeometry
– Global differential geometry. | Global analysis, analysis on manifolds – Partial differential
equations on manifolds; differential operators. | Differential geometry – Classical differen-
tial geometry – Higher-dimensional and -codimensional. | Convex and discrete geometry –
General convexity – Convex sets in n dimensions (including convex hypersurfaces). | Partial
differentialequations–Parabolicequationsandsystems–Initial-boundaryvalueproblemsfor
second-orderparabolicequations.
Classification: LCCQA670.A532020|DDC516.3/62–dc23
LCrecordavailableathttps://lccn.loc.gov/2019059835
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Contents
Preface xiii
A Guide for the Reader xv
The heat equation (Chapter 1) xv
Curve shortening flow (Chapters 2–4) xv
Mean curvature flow (Chapters 5–14) xvi
Gauß curvature flows (Chapters 15–17) xix
Fully nonlinear curvature flows (Chapters 18–20) xx
Acknowledgments xx
Suggested Course Outlines xxiii
Notation and Symbols xxv
Chapter 1. The Heat Equation 1
§1.1. Introduction 1
§1.2. Gradient flow 3
§1.3. Invariance properties 3
§1.4. The maximum principle 8
§1.5. Well-posedness 12
§1.6. Asymptotic behavior 14
§1.7. The Bernstein method 17
§1.8. The Harnack inequality 17
§1.9. Further monotonicity formulae 19
§1.10. Sharp gradient estimates 21
v
vi Contents
§1.11. Notes and commentary 29
§1.12. Exercises 33
Chapter 2. Introduction to Curve Shortening 37
§2.1. Basic geometric theory of planar curves 38
§2.2. Curve shortening flow 42
§2.3. Graphs of functions 45
§2.4. The support function 48
§2.5. Short-time existence 50
§2.6. Smoothing 51
§2.7. Global existence 54
§2.8. Notes and commentary 59
§2.9. Exercises 59
Chapter 3. The Gage–Hamilton–Grayson Theorem 63
§3.1. The avoidance principle 64
§3.2. Preserving embeddedness 66
§3.3. Huisken’s distance comparison estimate 68
§3.4. A curvature bound by distance comparison 74
§3.5. Grayson’s theorem 82
§3.6. Singularities of immersed solutions 88
§3.7. Notes and commentary 90
§3.8. Exercises 92
Chapter 4. Self-Similar and Ancient Solutions 95
§4.1. Invariance properties 95
§4.2. Self-similar solutions 96
§4.3. Monotonicity formulae 102
§4.4. Ancient solutions 110
§4.5. Classification of convex ancient solutions on S1 116
§4.6. Notes and commentary 122
§4.7. Exercises 123
Chapter 5. Hypersurfaces in Euclidean Space 125
§5.1. Parametrized hypersurfaces 125
§5.2. Alternative representations of hypersurfaces 143
§5.3. Dynamical properties 152
§5.4. Curvature flows 164
Contents vii
§5.5. Notes and commentary 169
§5.6. Exercises 169
Chapter 6. Introduction to Mean Curvature Flow 173
§6.1. The mean curvature flow 173
§6.2. Invariance properties and self-similar solutions 176
§6.3. Evolution equations 179
§6.4. Short-time existence 184
§6.5. The maximum principle 189
§6.6. The avoidance principle 192
§6.7. Preserving embeddedness 196
§6.8. Long-time existence 197
§6.9. Weak solutions 206
§6.10. Notes and commentary 215
§6.11. Exercises 219
Chapter 7. Mean Curvature Flow of Entire Graphs 223
§7.1. Introduction 223
§7.2. Preliminary calculations 224
§7.3. The Dirichlet problem 227
§7.4. A priori height and gradient estimates 228
§7.5. Local a priori estimates for the curvature 232
§7.6. Proof of Theorem 7.1 238
§7.7. Convergence to self-similarly expanding solutions 239
§7.8. Self-similarly shrinking entire graphs 240
§7.9. Notes and commentary 240
§7.10. Exercises 241
Chapter 8. Huisken’s Theorem 243
§8.1. Pinching is preserved 244
§8.2. Pinching improves: The roundness estimate 246
§8.3. A gradient estimate for the curvature 256
§8.4. Huisken’s theorem 259
§8.5. Regularity of the arrival time 266
§8.6. Huisken’s theorem via width pinching 267
§8.7. Notes and commentary 274
§8.8. Exercises 278
viii Contents
Chapter 9. Mean Convex Mean Curvature Flow 281
§9.1. Singularity formation 281
§9.2. Preserving pinching conditions 284
§9.3. Pinching improves: Convexity and cylindrical estimates 294
§9.4. A natural class of initial data 301
§9.5. A gradient estimate for the curvature 303
§9.6. Notes and commentary 308
§9.7. Exercises 309
Chapter 10. Monotonicity Formulae 311
§10.1. Huisken’s monotonicity formula 311
§10.2. Hamilton’s Harnack estimate 319
§10.3. Notes and commentary 338
§10.4. Exercises 342
Chapter 11. Singularity Analysis 345
§11.1. Local uniform convergence of mean curvature flows 345
§11.2. Neck detection 354
§11.3. The Brakke–White regularity theorem 363
§11.4. Huisken’s theorem revisited 366
§11.5. The structure of singularities 371
§11.6. Notes and commentary 389
§11.7. Exercises 394
Chapter 12. Noncollapsing 395
§12.1. The inscribed and exscribed curvatures 395
§12.2. Differential inequalities for the inscribed and exscribed
curvatures 402
§12.3. The Gage–Hamilton and Huisken theorems via
noncollapsing 412
§12.4. The Haslhofer–Kleiner curvature estimate 415
§12.5. Notes and commentary 421
§12.6. Exercises 422
Chapter 13. Self-Similar Solutions 425
§13.1. Shrinkers — an introduction 425
§13.2. The Gaußian area functional 426
§13.3. Mean convex shrinkers 431
Contents ix
§13.4. Compact embedded self-shrinking surfaces 443
§13.5. Translators — an introduction 452
§13.6. The Dirichlet problem for graphical translators 454
§13.7. Cylindrical translators 455
§13.8. Rotational translators 456
§13.9. The convexity estimates of Spruck, Sun, and Xiao 462
§13.10. Asymptotics 468
§13.11. X.-J. Wang’s dichotomy 469
§13.12. Rigidity of the bowl soliton 470
§13.13. Flying wings 477
§13.14. Bowloids 490
§13.15. Notes and commentary 492
§13.16. Exercises 499
Chapter 14. Ancient Solutions 503
§14.1. Rigidity of the shrinking sphere 504
§14.2. A convexity estimate 509
§14.3. A gradient estimate for the curvature 511
§14.4. Asymptotics 513
§14.5. X.-J. Wang’s dichotomy 516
§14.6. Ancient solutions to curve shortening flow revisited 525
§14.7. Ancient ovaloids 531
§14.8. Ancient pancakes 533
§14.9. Notes and commentary 536
§14.10. Exercises 540
Chapter 15. Gauß Curvature Flows 543
§15.1. Invariance properties and self-similar solutions 545
§15.2. Basic evolution equations 546
§15.3. Chou’s long-time existence theorem 548
§15.4. Differential Harnack estimates 558
§15.5. Firey’s conjecture 560
§15.6. Variational structure and entropy formulae 570
§15.7. Notes and commentary 578
§15.8. Exercises 578