Stability of spheres under volume-preserving mean curvature flow

Abstract

We give a new, elementary proof of the theorem, due to J. Escher and G. Simonett, that for the initial conditions close to Eucleadian spheres the solutions of the volume-preserving mean curvature flow converge to Eucleadian spheres (which, in general, differ from the initial spheres). Our result is in the metric given by Sobolev norms. While the proof by J. Escher and G. Simonett uses extensively rather involved results from the infinite-dimensional invariant manifold theory and quasilinear parabolic differential equations, our main point is to use an orthogonal decomposition of the solutions near the manifold of Euclidean spheres and differential inequalities for the Lyapunov functionals. Apart from local well-posedness, which is proven along standard lines, our proof is completely self-contained.