We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.

In the paper [2] Ennio De Giorgi conjectured that any compact ndimensional regular submanifold M of R n+m , moving by the gradient of the functionalwhere η M is the square of the distance function from the submanifold M and H n 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. Proposition 2.2 allows us to write A k in terms of the tensors p k,s and the projections on the tangent and normal spaces (hence contracting with the scalar product of R n+m ), so we get the following corollary.Corollary 2.4. For every k ≥ 3 the symmetric tensor A k can be expressed as a polynomial tensor in B and its covariant derivatives, contracted with the scalar product of R n+m .