Smooth geometric evolutions of hypersurfaces

By Carlo Mantegazza. 2001

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Abstract. We consider the gradient flow associated to the following functionals:¶¶ $ {\cal F}_m(\varphi) = \int_M 1 + |\bigtriangledown^m\nu|^2\,d\mu $.¶The functionals are defined on hypersurfaces immersed in $ {\mathbb R}^{n+1} $ via a map $ \varphi : M \to {\mathbb R}^{n+1} $, where M is smooth closed and connected n-dimensional manifold without boundary. Here $ \mu $ and $ \bigtriangledown $ are respectively the canonical measure and the Levi—Civita connection of the Riemannian manifold (M,g), where the metric g is obtained by pulling back on M the usual metric of $ {\mathbb R}^{n+1} $ with the map $ \mu $. The symbol $ \bigtriangledown^m $ denotes the mth iterated covariant derivative and $ \nu $ is a unit normal local vector field to the hypersurface.¶Our main result is that if the order of derivation $ m \in {\mathbb N} $ is strictly larger than the integer part of n/2 then singularities in finite time cannot occur during the evolution.¶These geometric functionals are related to similar ones proposed by Ennio De Giorgi, who conjectured for them an analogous regularity result. In the final section we discuss the original conjecture of De Giorgi and some related problems.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f1e5b0215563cf1cc2e3cdf7fb76a59d02caf688

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In Applied Mathematics

In Geometry and Topology

X. Cabré

Citations in the corpus

Regularity of minimizers of semilinear elliptic problems up to dimension 4 (2009, background): “2 of [19], where references for it and related results are mentioned.”