Motion by Mixed Volume Preserving Curvature Functions Near Spheres

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).