The quermassintegral-preserving mean curvature flow in the sphere

Cabezas-Rivas, Esther; Scheuer, Julian

doi:10.2140/apde.2024.17.3589

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The quermassintegral-preserving mean curvature flow in the sphere

Esther Cabezas-Rivas and Julian Scheuer

Vol. 17 (2024), No. 10, 3589–3621

DOI: 10.2140/apde.2024.17.3589

Abstract

We introduce a mean curvature flow with global term of convex hypersurfaces in the sphere, for which the global term can be chosen to keep any quermassintegral fixed. Then, starting from a strictly convex initial hypersurface, we prove that the flow exists for all times and converges smoothly to a geodesic sphere. This provides a workaround to an issue present in the volume-preserving mean curvature flow in the sphere introduced by Huisken (1987). We also classify solutions for some constant curvature-type equations in space forms, as well as solitons in the sphere and in the upper branch of the De Sitter space.

Keywords

volume-preserving mean curvature flow, spherical geometry

Mathematical Subject Classification

Primary: 53C21, 53E10

Milestones

Received: 5 December 2022

Accepted: 31 August 2023

Published: 21 November 2024

Authors

Esther Cabezas-Rivas
Departament de Matemàtiques Universitat de València València Spain

Julian Scheuer
Institut für Mathematik Goethe-Universität Frankfurt Germany

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Vol. 17, No. 10, 2024