Calculus of Variations and Geometric Measure Theory

M. Eminenti - C. Mantegazza

Some Properties of the Distance Function and a Conjecture of De Giorgi

created on 09 Jun 2003
modified by root on 05 Jun 2013


Published Paper

Inserted: 9 jun 2003
Last Updated: 5 jun 2013

Journal: J. Geom. Anal.
Volume: 14
Number: 2
Pages: 267-279
Year: 2004


Ennio De Giorgi conjectured that any compact $n$-dimensional regular submanifold $M$ of ${\mathbf R}^{n+m}$, moving by the gradient of the functional \[ \int_M 1+\vert\nabla^{k}\eta^M\vert^2\,d{\mathcal {H}}^n\,, \] where $\eta^M$ is the square of the distance function from the submanifold $M$ and ${\mathcal H}^n$ is the $n$-dimensional Hausdorff measure in ${\mathbf R}^{n+m}$, does not develop singularities in finite time provided $k$ is large enough, depending on the dimension $n$.

We prove this conjecture by means of the analysis of the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of independent interest.

Keywords: Gradient Flow, distance function, second fundamental form