Communications in Analysis and Geometry

Volume 21 (2013)

Number 5

Motion by volume preserving mean curvature flow near cylinders

Pages: 873 – 889

DOI: http://dx.doi.org/10.4310/CAG.2013.v21.n5.a1

Author

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

Abstract

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].

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