Asymptotic convergence of evolving hypersurfaces

Carlo Mantegazza ^[1] ; Marco Pozzetta ^[1]
1. [1] Università di Napoli Federico II
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 6, 2022, págs. 1927-1944
Idioma: inglés
DOI: 10.4171/RMI/1317
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Resumen
- If ψ:Mn→Rn+1 is a smooth immersed closed hypersurface, we consider the functional,Fm(ψ)=∫M1+∣∇mν∣2dμ, where ν is a local unit normal vector along ψ, ∇∇ is the Levi-Civita connection of the Riemannian manifold (M,g), with g the pull-back metric induced by the immersion and μ the associated volume measure. We prove that if m>⌊n/2⌋ then the unique globally defined smooth solution to the 2L2-gradient flow of Fm, for every initial hypersurface, smoothly converges asymptotically to a critical point of Fm, up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional Fm