Motion by Volume Preserving Mean Curvature Flow Near Cylinders

Abstract: Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such a technique. We will consider surfaces with boundary contained within two parallel planes such that the surface meets these planes orthogonally. With this set up the stationary solution is a cylinder. We will find that for initial surfaces that are sufficiently close to a cylinder the flow will exist for all time and converge to a cylinder exponentially. In particular, we show that there exists global solutions to the flow that converge to a cylinder, which are initially non-axially symmetric. A similar case where the initial surfaces are compact without boundary has previously been investigated by Escher and Simonett (1998).