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Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow


About this Title

Zhou Gang, Dan Knopf and Israel Michael Sigal

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1210
ISBNs: 978-1-4704-2840-2 (print); 978-1-4704-4415-0 (online)
DOI: https://doi.org/10.1090/memo/1210
Published electronically: March 29, 2018
Keywords:Mean Curvature Flow, Neckpinch, Asymptotics, Rotational Symmetry

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. The first bootstrap machine
  • Chapter 3. Estimates of first-order derivatives
  • Chapter 4. Decay estimates in the inner region
  • Chapter 5. Estimates in the outer region
  • Chapter 6. The second bootstrap machine
  • Chapter 7. Evolution equations for the decomposition
  • Chapter 8. Estimates to control the parameters $a$ and $b$
  • Chapter 9. Estimates to control the fluctuation $\phi $
  • 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

Abstract


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

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