Ricci solitons - The equation point of view

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. Some simple proofs of 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.Comment: Revised Version - Corrected some mistakes and inaccuracie

Some properties of the distance function and a conjecture of De

ABSTRACT. In the paper [2] Ennio De Giorgi conjectured that any compact n– dimensional regular submanifold M of Rn+m, moving by the gradient of the functional 1 + |∇ M k η M | 2 dHn, where ηM is the square of the distance function from the submanifold M and Hn is the n–dimensional Hausdorff measure in Rn+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

Ricci solitons: the equation point of view

ABSTRACT. 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