Stability of Geodesic Spheres in $\mathbb{S}^{n+1}$ under Constrained Curvature Flows

TL;DR

This paper investigates the stability of geodesic spheres in spherical spaces under constrained curvature flows, extending previous Euclidean results to Riemannian manifolds and establishing conditions for stability under volume-preserving perturbations.

Contribution

It generalizes stability results of curvature flows from Euclidean spaces to spherical manifolds, considering volume-preserving constraints.

Findings

Geodesic spheres are stable under certain curvature flows in $\

$ ext{S}^{n+1}$ with volume-preserving perturbations.

Stability results extend previous Euclidean findings to Riemannian manifolds.

Abstract

In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.