Convergence of axially symmetric volume-preserving mean curvature flow

Abstract

We study the convergence of axially symmetric hypersurfaces evolving by volume-preserving mean curvature flow.Assuming the surfaces do not develop singularities along the axis of rotation at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that the flow converges to a hemisphere, when the initial hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.