Some properties of the distance function and a conjecture of De Giorgi

In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ ^n+m ,moving by the gradient of the functional where η^M is the square of the distance function from the submanifold M and Hⁿ is the n-dimensional Hausdorff measure in 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.