**Graduate Studies in Mathematics**

Volume: 206;
2020;
790 pp;
Hardcover

MSC: Primary 53; 58; 52; 35;

**Print ISBN: 978-1-4704-5596-5
Product Code: GSM/206**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

**Electronic ISBN: 978-1-4704-5686-3
Product Code: GSM/206.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

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#### Supplemental Materials

# Extrinsic Geometric Flows

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*Ben Andrews; Bennett Chow; Christine Guenther; Mat Langford*

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.

#### Readership

Graduate students and researchers interested in mean curvature flow.

#### Reviews & Endorsements

This textbook, written by four experts in the field, offers an authoritative introduction and overview to the topic of extrinsic geometric flows. It will serve well as a primary text for a graduate student who already has background knowledge of differential geometry and (some) partial differential equations. It will also serve as a useful reference for experts in the field.

-- John Ross, Southwestern University

#### Table of Contents

# Table of Contents

## Extrinsic Geometric Flows

- Cover Cover11
- Title page iii4
- Preface xiii14
- A Guide for the Reader xv16
- Suggested Course Outlines xxiii24
- Notation and Symbols xxv26
- Chapter 1. The Heat Equation 130
- 1.1. Introduction 130
- 1.2. Gradient flow 332
- 1.3. Invariance properties 332
- 1.4. The maximum principle 837
- 1.5. Well-posedness 1241
- 1.6. Asymptotic behavior 1443
- 1.7. The Bernstein method 1746
- 1.8. The Harnack inequality 1746
- 1.9. Further monotonicity formulae 1948
- 1.10. Sharp gradient estimates 2150
- 1.11. Notes and commentary 2958
- 1.12. Exercises 3362

- Chapter 2. Introduction to Curve Shortening 3766
- Chapter 3. The Gage–Hamilton–Grayson Theorem 6392
- Chapter 4. Self-Similar and Ancient Solutions 95124
- Chapter 5. Hypersurfaces in Euclidean Space 125154
- Chapter 6. Introduction to Mean Curvature Flow 173202
- 6.1. The mean curvature flow 173202
- 6.2. Invariance properties and self-similar solutions 176205
- 6.3. Evolution equations 179208
- 6.4. Short-time existence 184213
- 6.5. The maximum principle 189218
- 6.6. The avoidance principle 192221
- 6.7. Preserving embeddedness 196225
- 6.8. Long-time existence 197226
- 6.9. Weak solutions 206235
- 6.10. Notes and commentary 215244
- 6.11. Exercises 219248

- Chapter 7. Mean Curvature Flow of Entire Graphs 223252
- 7.1. Introduction 223252
- 7.2. Preliminary calculations 224253
- 7.3. The Dirichlet problem 227256
- 7.4. A priori height and gradient estimates 228257
- 7.5. Local a priori estimates for the curvature 232261
- 7.6. Proof of Theorem 7.1 238267
- 7.7. Convergence to self-similarly expanding solutions 239268
- 7.8. Self-similarly shrinking entire graphs 240269
- 7.9. Notes and commentary 240269
- 7.10. Exercises 241270

- Chapter 8. Huisken’s Theorem 243272
- 8.1. Pinching is preserved 244273
- 8.2. Pinching improves: The roundness estimate 246275
- 8.3. A gradient estimate for the curvature 256285
- 8.4. Huisken’s theorem 259288
- 8.5. Regularity of the arrival time 266295
- 8.6. Huisken’s theorem via width pinching 267296
- 8.7. Notes and commentary 274303
- 8.8. Exercises 278307

- Chapter 9. Mean Convex Mean Curvature Flow 281310
- Chapter 10. Monotonicity Formulae 311340
- Chapter 11. Singularity Analysis 345374
- Chapter 12. Noncollapsing 395424
- 12.1. The inscribed and exscribed curvatures 395424
- 12.2. Differential inequalities for the inscribed and exscribed curvatures 402431
- 12.3. The Gage–Hamilton and Huisken theorems via noncollapsing 412441
- 12.4. The Haslhofer–Kleiner curvature estimate 415444
- 12.5. Notes and commentary 421450
- 12.6. Exercises 422451

- Chapter 13. Self-Similar Solutions 425454
- 13.1. Shrinkers —an introduction 425454
- 13.2. The Gaußian area functional 426455
- 13.3. Mean convex shrinkers 431460
- 13.4. Compact embedded self-shrinking surfaces 443472
- 13.5. Translators —an introduction 452481
- 13.6. The Dirichlet problem for graphical translators 454483
- 13.7. Cylindrical translators 455484
- 13.8. Rotational translators 456485
- 13.9. The convexity estimates of Spruck, Sun, and Xiao 462491
- 13.10. Asymptotics 468497
- 13.11. X.-J. Wang’s dichotomy 469498
- 13.12. Rigidity of the bowl soliton 470499
- 13.13. Flying wings 477506
- 13.14. Bowloids 490519
- 13.15. Notes and commentary 492521
- 13.16. Exercises 499528

- Chapter 14. Ancient Solutions 503532
- 14.1. Rigidity of the shrinking sphere 504533
- 14.2. A convexity estimate 509538
- 14.3. A gradient estimate for the curvature 511540
- 14.4. Asymptotics 513542
- 14.5. X.-J. Wang’s dichotomy 516545
- 14.6. Ancient solutions to curve shortening flow revisited 525554
- 14.7. Ancient ovaloids 531560
- 14.8. Ancient pancakes 533562
- 14.9. Notes and commentary 536565
- 14.10. Exercises 540569

- Chapter 15. Gauß Curvature Flows 543572
- 15.1. Invariance properties and self-similar solutions 545574
- 15.2. Basic evolution equations 546575
- 15.3. Chou’s long-time existence theorem 548577
- 15.4. Differential Harnack estimates 558587
- 15.5. Firey’s conjecture 560589
- 15.6. Variational structure and entropy formulae 570599
- 15.7. Notes and commentary 578607
- 15.8. Exercises 578607

- Chapter 16. The Affine Normal Flow 581610
- Chapter 17. Flows by Superaffine Powers of the Gauß Curvature 607636
- Chapter 18. Fully Nonlinear Curvature Flows 639668
- 18.1. Introduction 639668
- 18.2. Symmetric functions and their differentiability properties 641670
- 18.3. Examples 650679
- 18.4. Short-time existence 655684
- 18.5. The avoidance principle 658687
- 18.6. Differential Harnack estimates 660689
- 18.7. Entropy estimates 664693
- 18.8. Alexandrov reflection 670699
- 18.9. Notes and commentary 682711
- 18.10. Exercises 683712

- Chapter 19. Flows of Mean Curvature Type 687716
- Chapter 20. Flows of Inverse-Mean Curvature Type 711740
- Bibliography 727756
- Index 753782
- Back Cover Back Cover1791