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Motion by volume preserving mean curvature flow near cylinders. (English) Zbl 1312.53086
Summary: 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 {\it J. Escher} and {\it G. Simonett} [Proc. Am. Math. Soc. 126, No. 9, 2789--2796 (1998; Zbl 0909.53043)].

53C44Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
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