Smooth Geometric Evolutions of Hypersurfaces

We consider the gradient flow associated to the following functionals Fm(φ) = ∫ M 1 + |∇ν| dμ . The functionals are defined on hypersurfaces immersed in R via a map φ : M → R, where M is a smooth closed and connected n–dimensional manifold without boundary. Here μ and ∇ are respectively the canonical measure and the Levi–Civita connection on the Riemannian manifold (M, g), where the metric g is obtained by pulling back on M the usual metric of R with the map φ. The symbol ∇ denotes the m–th iterated covariant derivative and ν is a unit normal local vector field to the hypersurface. Our main result is that if the order of derivation m ∈ 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.