Asymptotic convergence of evolving hypersurfaces

Publication date: 11 January 2021

Abstract: If

p s i : M^{n} o m a t h b b R^{n + 1}

is a smooth immersed closed hypersurface, we consider the functional

m a t h c a l F_{m} (p s i) = i n t_{M} 1 + | a b l a^{m} u |^{2}, d m u

, where

u

is a local unit normal vector along

p s i

,

a b l a

is the Levi-Civita connection of the Riemannian manifold

(M, g)

, with

g

the pull-back metric induced by the immersion and

m u

the associated volume measure. We prove that if

m > l f l o o r n / 2 f l o o r

then the unique globally defined smooth solution to the

L^{2}

-gradient flow of

m a t h c a l F_{m}

, for every initial hypersurface, smoothly converges asymptotically to a critical point of

m a t h c a l F_{m}

, up to diffeomorphisms. The proof is based on the application of a Lojasiewicz-Simon gradient inequality for the functional

m a t h c a l F_{m}

.

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