The surface area preserving mean curvature flow

Abstract

Let MQ be a compact, strictly convex hypersurface of dimension n > 2, without boundary, smoothly embedded in E n+1 and represented locally by some diffeomor-phism FQ : R n D U-> FQ (U) C MQ C R n+1. Under the surface area preserving mean curvature flow, formulated by Pihan in [P], the family of maps Ft = F(' } t) evolves according to (1)-F(x,t) = {l-h{t)H(x,t)}v{x,t) iXGU, 0 -%**■ where dfi t is the surface area element on M t. Pihan studied basic properties of this flow for general n and showed that (1) has a unique solution for a short time. He also proved for n = 1 that an initially closed, convex curve in the plane converges exponentially to a circle with the same length as the initial curve. Analogous to this result and those of Huisken in [Hul] and [Hu2] for the mean curvature flow and the volume preserving mean curvature flow, we show here a similar result for the surface area preserving flow, when n > 2.