Motion by Volume Preserving Mean Curvature Flow Near Cylinders

TL;DR

This paper uses center manifold analysis to study the stability of cylindrical solutions under volume-preserving mean curvature flow, showing that surfaces close to a cylinder evolve smoothly and exponentially converge to a cylinder, even without initial symmetry.

Contribution

It applies center manifold techniques to volume-preserving mean curvature flow near cylinders, demonstrating exponential convergence for a broad class of initial surfaces.

Findings

Flow exists for all time for surfaces close to cylinders

Surfaces converge exponentially to a cylinder

Convergence occurs even without initial axial symmetry

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).