RIGIDITY RESULTS FOR LICHNEROWICZ BAKRY-EMERY RICCI TENSORS

We investigate rigidity phenomena in the class of Riemannian manifolds with density. By a weighted manifold (or manifold with density) we mean a Riemannian manifold endowed with a weighted measure absolute continuous with respect to the Riemannian one. Associated to a weighted manifold there is also a natural divergence form second order diffusion operator: the f-Laplacian. Good choiches for the concept of curvature in this setting are those that reveal interplays with metric and topological properties of the space. We have focused on Lichnerowicz Bakry-Emery Ricci's tensors. When we impose the constancy of these tensors we are endowing the manifold of an additional structure, namely, a gradient Ricci soliton or a quasi-Einstein structure. The importance of Ricci solitons is due to Perelman's solution of Poincarè conjecture, while the importance of quasi-Einstein manifolds comes from the relationship they have with Einstein warped products. Actually in this thesis we also introduce an extension of the concept of gradient Ricci soliton, the Ricci almost soliton, allowing the soliton constant to be a generic smooth function on the weighted manifold. In view of the not necessarily constantness, one expects that a certain flexibility on the almost soliton structure is allowed and, consequently, the existence of almost solitons is easier to prove than in the classical situation. This feeling is confirmed by a number of diff erent examples of almost solitons. On the other hand we prove a rigidity result which indicates that almost solitons should reveal a reasonably broad generalization of the fruitful concept of classical soliton. Considering elliptic equations and inequalities (naturally involving the f-Laplacian) for various geometric quantities and making use of analytical techniques coming from stochastic analysis such as stochastic completeness, in the form of the weak Omori-Yau maximum principle, parabolicity and Lp-Liouville type results, we prove rigidity in the form of metric rigidity (scalar curvature estimates, classification results, gap theorems for some geometric quantities) and in the form of triviality of the additional structure. Moreover we obtain also topological rigidity for these tensors (with consequences e.g. on the structure of the fundamental group).