Asymptotic convergence of evolving hypersurfaces
Carlo Mantegazza
Università di Napoli Federico II, ItalyMarco Pozzetta
Università di Napoli Federico II, Italy
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Abstract
If is a smooth immersed closed hypersurface, we consider the functional
where is a local unit normal vector along , is the Levi-Civita connection of the Riemannian manifold , with the pull-back metric induced by the immersion and the associated volume measure. We prove that if then the unique globally defined smooth solution to the -gradient flow of , for every initial hypersurface, smoothly converges asymptotically to a critical point of , up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional .
Cite this article
Carlo Mantegazza, Marco Pozzetta, Asymptotic convergence of evolving hypersurfaces. Rev. Mat. Iberoam. 38 (2022), no. 6, pp. 1927–1944
DOI 10.4171/RMI/1317