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.
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Höhe: 254 mm
Breite: 178 mm
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ISBN-13
978-1-4704-2840-2 (9781470428402)
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Schweitzer Klassifikation
Gang Zhou, California Institute of Technology, Pasadena, California.
Dan Knopf, University of Texas at Austin, Texas.
Israel Michael Sigal, University of Toronto, Ontario, Canada.
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.
