Volume-preserving mean curvature flow of hypersurfaces in space forms

Abstract

The purpose of this paper is to investigate the convergence of the volume-preserving mean curvature flow of closed hypersurfaces in space forms. Assume that the initial hypersurface satisfies suitable integral curvature pinching conditions. We prove that along the volume-preserving mean curvature flow the initial hypersurface will be deformed to a totally umbilical sphere.The basic property of flow (1.1) is its isoperimetric nature. Namely, in the case that M t is embedded along the flow, the (n + 1)-dimensional volume |Ω t | of the region Ω t enclosed by M t is preserved:while the n-dimensional volume of M t strictly decreases unless the mean curvature is constant:1450021-2 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by KANSAS STATE UNIVERSITY on 06/25/14. For personal use only.Volume-preserving mean curvature flow of hypersurfaces in space forms then the volume-preserving mean curvature flow with M as initial hypersurface will converge exponentially fast to a round sphere.When the ambient space is the hyperbolic space, we obtain the following convergence theorem. Theorem 1.2. Let M be an n-dimensional smooth closed and orientable hypersurface immersed in H n+1 . Suppose the volume Vol(M ), the second fundamental form A and the averaged mean curvature h satisfy Vol(M ) ≤ V, |A| ≤ Λ, h ≥ n + γ for positive constants V, Λ, γ. There is a positive constant ε = ε(n, V, Λ, γ) depending only on n, V, Λ and γ such that if M |Å| 2 dµ < ε, (1.3)then the volume-preserving mean curvature flow with M as initial hypersurface will converge exponentially fast to a round sphere.

Evolution of non-simple closed curves in the area-preserving curvature flow

“…For example, there are Huisken's volume-preserving mean-curvature flow [12] and McCoy's surface-area-preserving mean-curvature flow [17]. Recently, the study of non-local flow was extended to the case of the Riemannian manifold; see [25].…”