No abstract

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L p -Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L p conditions on the relevant quantities.

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For 1 < p < ∞, these are inequalities of the formvalid a priori for all smooth functions u with compact support, and constants C 1 ≥ 0, C 2 > 0. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón-Zygmund inequality (like new denseness results for second order L p -Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.

Abstract. We prove that the stochastic completeness of a Riemannian manifold (M, , ) is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.