Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates

Abstract

The present thesis is divided in three different parts. The aim of the first part is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive. The aim of the second part is to study the critical metrics of the total scalar curvature funcional on compact manifolds with constant scalar curvature and unit volume, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. We prove that the Conjecture is true for CPE metrics under a suitable integral condition and we also prove that it suffices the metric to be conformal to an Einstein metric. In the third part we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension one C 2-foliations of open subsets Ω of Riemannian manifolds M and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of Ω.