Motion by Mixed Volume Preserving Curvature Functions Near Spheres

TL;DR

This paper studies the evolution of surfaces under curvature-driven flows with volume constraints, proving stability and convergence to spheres for surfaces close to spherical shape.

Contribution

It extends stability analysis of volume-preserving curvature flows to a broader class of symmetric functions beyond mean curvature.

Findings

Surfaces close to spheres exist for all time under the flow.

Such surfaces exponentially converge to spheres.

The stability of spheres as stationary solutions is established.

Abstract

In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time existence theorem for a large class of surfaces that are sufficiently close to a sphere and, using center manifold analysis, the stability of the sphere as a stationary solution to the flow is determined. We will find that for initial surfaces sufficiently close to a sphere, the flow will exist for all time and converge exponentially to a sphere. This result was shown for the case where the symmetric function is the mean curvature and the constraint is on the (n+1)-dimensional enclosed volume by Escher and Simonett (1998).