Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow / Gang Zhou, Dan Knopf, Israel Michael Sigal.

Format:

Book

Author/Creator:

Gang, Zhou (Mathematics professor), author.

Knopf, Dan, 1959- author.

Sigal, Israel Michael, 1945- author.

Series:

Memoirs of the American Mathematical Society ; Volume 253, Number 1210.

Memoirs of the American Mathematical Society ; Volume 253, Number 1210

Language:

English

Subjects (All):

Evolution equations--Asymptotic theory.

Evolution equations.

Asymptotic expansions.

Curvature.

Singularities (Mathematics).

Physical Description:

1 online resource (90 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are C^3-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. What we study

1.2. Basic evolution equations

1.3. Implied evolution equations

Chapter 2. The first bootstrap machine

2.1. Input

2.2. Output

2.3. Structure

Chapter 3. Estimates of first-order derivatives

Chapter 4. Decay estimates in the inner region

4.1. Differential inequalities

4.2. Lyapunov functionals of second and third order

4.3. Lyapunov functionals of fourth and fifth order

4.4. Estimates of second- and third-order derivatives

Chapter 5. Estimates in the outer region

5.1. Second-order decay estimates

5.2. Third-order decay estimates

5.3. Third-order smallness estimates

Chapter 6. The second bootstrap machine

6.1. Input

6.2. Output

6.3. Structure

Chapter 7. Evolution equations for the decomposition

Chapter 8. Estimates to control the parameters and

Chapter 9. Estimates to control the fluctuation

9.1. Proof of estimate (7.12)

9.2. Proof of estimate (7.13)

9.3. Proof of estimate (7.15)

9.4. Proof of estimate (7.14)

Chapter 10. Proof of the Main Theorem

Appendix A. Mean curvature flow of normal graphs

Appendix B. Interpolation estimates

Appendix C. A parabolic maximum principle for noncompact domains

Appendix D. Estimates of higher-order derivatives

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4415-1