CMA Research Report

MRR02-014

The Mixed Volume Preserving Mean Curvature Flow

Abstract:  Let M₀ be a compact, strictly convex, n-dimensional hypersurface without boundary, smoothly embedded in Rⁿ⁺¹. We consider a modified mean curvature flow, which includes a global term to keep any particular mixed volume V_n-k fixed under the flow. The flow includes the volume preserving and surface area preserving mean curvature flows as special cases. We show that the evolution of M₀ by such a flow has a smooth solution for all time, which converges exponentially to a sphere with the same value of V_n-k as M₀. We avoid the Sobolev inequality and Stampacchia iteration by obtaining a pointwise bound on the mean curvature via the support function of the evolving hypersurfaces.

AMS Classification:  53C44
Date:  31 October 2002

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