Asymptotic convergence of evolving hypersurfaces (Q2104854)

Summary: If \(\psi: M^n \to \mathbb{R}^{n + 1}\) is a smooth immersed closed hypersurface, we consider the functional \[ \mathcal{F}_m (\psi) = \int_M 1 + |{\nabla}^m \nu|^2 d \mu, \] where \(\nu\) 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 \(\mathcal{F}_m\).