On the classification of ancient solutions to curvature flows on the sphere

Bryan, Paul, Ivaki, Mohammad and Scheuer, Julian ORCID: https://orcid.org/0000-0003-2664-1896 2022. On the classification of ancient solutions to curvature flows on the sphere. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 25 (1) , pp. 53-76. 10.2422/2036-2145.202105_006

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Abstract

We consider the evolution of hypersurfaces on the unit sphere by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Aleksandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Scuola Normale Superiore
ISSN:	0391-173X
Funders:	EPSRC, EP/K00865X/1, Austrian Science Fund (FWF), M1716-N25, European Research Council (ERC), 306445
Date of First Compliant Deposit:	29 June 2022
Date of Acceptance:	29 June 2022
Last Modified:	10 Nov 2024 10:45
URI:	https://orca.cardiff.ac.uk/id/eprint/150866

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