Asymptotic convergence of evolving hypersurfaces

Abstract

If $\psi :M^n\to {\mathbb{R}}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional

where $\nu$ is a local unit normal vector along $\psi$, $\nabla$ is the Levi-Civita connection of the Riemannian manifold $(M,g)$, with $g$ the pull-back metric induced by the immersion and $\mu$ the associated volume measure. We prove that if $m>\lfloor n/2\rfloor$ then the unique globally defined smooth solution to the $L^2$-gradient flow of ${\mathcal{F}}_m$, for every initial hypersurface, smoothly converges asymptotically to a critical point of ${\mathcal{F}}_m$, up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional ${\mathcal{F}}_m$.