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Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

Mat Langford and Stephen Lynch

Abstract

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature (“cylindrical estimates”) for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of the shrinking sphere amongst ancient solutions of these flows.

Funding statement: Mat Langford was supported by an Alexander von Humboldt fellowship. Stephen Lynch was supported by the Berlin Mathematical School.

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Received: 2017-05-08
Revised: 2018-12-24
Published Online: 2019-06-14
Published in Print: 2020-08-01

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