Communications in Analysis and Geometry

Volume 21 (2013)

Number 5

Motion by volume preserving mean curvature flow near cylinders

Pages: 873 – 889



David Hartley (School of Mathematical Sciences, Monash University, Canberra, Victoria, Australia)


We investigate the volume preserving mean curvature flow with Neumann boundary condition for hypersurfaces that are graphs over a cylinder. Through a center manifold analysis we find that initial hypersurfaces sufficiently close to a cylinder of large enough radius, have a flow that exists for all time and converges exponentially fast to a cylinder. In particular, we show that there exist global solutions to the flow that converge to a cylinder, which are initially non-axially symmetric. A similar case where the initial hypersurfaces are spherical graphs has previously been investigated by Escher and Simonett [8].

Published 27 December 2013