Asymptotic convergence of evolving hypersurfaces

If (Formula Presented) is a smooth immersed closed hypersurface, we consider the functional (Formula Presented); where ν is a local unit normal vector along, r is the Levi-Civita connection of the Riemannian manifold .M; g/, with g the pull-back metric induced by the immersion and μ the associated volume measure. We prove that if m > bn=2c then the unique globally defined smooth solution to the L2-gradient flow of Fm, for every initial hypersurface, smoothly converges asymptotically to a critical point of Fm, up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional Fm