Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

James Isenberg; Haotian Wu

doi:10.1515/crelle-2017-0019

Article

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

James Isenberg and Haotian Wu

Published/Copyright: May 5, 2017

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 754

Abstract

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ>12, there is a solution with the highest curvature blowing up at the rate (T-t)-(γ+12). (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

Funding source: National Science Foundation

Award Identifier / Grant number: PHY-1306441

Funding statement: James Isenberg is partially supported by NSF grant PHY-1306441.

Acknowledgements

We are grateful to Sigurd Angenent and Dan Knopf for their interest in and helpful discussions on this project. We also thank the referee for valuable comments on our manuscript.

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Received: 2016-03-25

Revised: 2017-01-07

Published Online: 2017-05-05

Published in Print: 2019-09-01

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https://doi.org/10.1515/crelle-2017-0019