Asymptotic convergence of evolving hypersurfaces

If $\psi\colon M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $$ \mathcal{F}\_m(\psi) = \int\_M 1 + |\nabla^m \nu |^2 , d\mu, where $\nu$ local unit normal vector along $\psi$, $\nabla$ Levi-Civita connection of Riemannian manifold $(M,g)$, with $g$ pull-back metric induced by immersion and $\mu$ associated volume measure. We prove that if $m>\lfloor n/2 \rfloor$ then unique globally defined solution to $L^2$-gradient flow $\mathcal{F}\_m$, for every initial smoothly converges asymptotically critical point up diffeomorphisms. The proof based on application Łojasiewicz–Simon gradient inequality $\mathcal{F}\_m$.