Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow

ISBN-13: 9781470428402 / Angielski

Gang Zhou;Dan Knopf;Israel Michael Sigal

Opis

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.

Spis treści

• Introduction
• The first bootstrap machine
• Estimates of first-order derivatives
• Decay estimates in the inner region
• Estimates in the outer region
• The second bootstrap machine
• Evolution equations for the decomposition
• Estimates to control the parameters $a$ and $b$
• Estimates to control the fluctuation $\phi $
• 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.