CMA Research Report

MRR02-015

The Surface Area Preserving Mean Curvature Flow

Abstract:  Let M₀ be a compact, strictly convex, n-dimensional hypersurface, n ≥ 2, without boundary, smoothly embedded in Rⁿ⁺¹. We consider a modified mean curvature flow, including a global term which keeps the surface area of the evolving hypersurface fixed under the flow. Given a short time existence result of Pihan, we show that the evolution of M₀ by the flow has a smooth solution for all time, which converges exponentially to a sphere with the same surface area as M₀. We follow the method of Huisken, incorporating the Sobolev inequality and Stampacchia iteration and we also use consequences of the Aleksandrov-Fenchel inequality for mixed volumes.

AMS Classification:  53C44
Date:  31 October 2002

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