Asymptotic convergence of evolving hypersurfaces

TL;DR

This paper proves that the gradient flow of a geometric functional involving higher derivatives of the normal vector on hypersurfaces converges to a critical point, using a Lojasiewicz-Simon inequality, for all initial shapes.

Contribution

It establishes asymptotic convergence of the hypersurface evolution under the gradient flow of a higher-order geometric functional, extending previous results to a broader class of functionals.

Findings

Gradient flow solutions converge to critical points asymptotically.

Convergence holds for all initial hypersurfaces when m > floor(n/2).

Application of Lojasiewicz-Simon inequality to geometric evolution.

Abstract

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 Lojasiewicz-Simon gradient inequality for the functional $\mathcal{F}_m$.