The quermassintegral preserving mean curvature flow in the sphere

TL;DR

This paper introduces a mean curvature flow in the sphere that preserves quermassintegrals, ensuring long-term existence and smooth convergence to a geodesic sphere, addressing previous limitations in volume-preserving flows.

Contribution

It develops a new global term in mean curvature flow that preserves quermassintegrals and proves convergence, expanding understanding of geometric flows in spherical spaces.

Findings

Flow exists for all time from convex initial hypersurfaces.

Flow converges smoothly to a geodesic sphere.

Classifies solutions and solitons in space forms.

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 in 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.