*Published Paper*

**Inserted:** 9 jun 2003

**Last Updated:** 5 jun 2013

**Journal:** J. Geom. Anal.

**Volume:** 14

**Number:** 2

**Pages:** 267-279

**Year:** 2004

**Abstract:**

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

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