Asymptotic convergence of evolving hypersurfaces

created by pozzetta1 on 12 Jan 2021

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Submitted Paper

Inserted: 12 jan 2021
Last Updated: 12 jan 2021

Year: 2021

Abstract:

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

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