Desigualdade de Caffarelli-Kohn-Nirenberg e solitons de Yamabe gradiente

Abstract

This thesis deals with two distinct problems. Namely, we study [(P1)] Rigidity of metric spaces that support CKN inequality; [(P2)] Gradient Yamabe solitons on top of warped product manifolds B x f F. For the first problem, we prove that the metric measure spaces that support the CKN inequality have n-dimensional volume growth, that is, there exists a universal constant C 0gt; 0 such that, m(B x (ρ)) ≥ C 0 ρ n , ∀x ∈ M, ρ gt; 0. As application, some rigidity theorems are obtained in the following spaces: Riemannian manifolds, Finsler manifolds and Alexandrov spaces. For the second problem, taking a gradient Yamabe soliton (B x f F, g, h, ρ), we obtain triviality results for h and f by means of some hypotheses on the base B. Furthermore, under a hypothesis involving the Ricci curvature of the base Ric gB , we prove estimates for h, f and for scalar curvature scal g , in addition, by means of a warping gradient estimates, we present a beautiful obstruction in the construction of gradient Yamabe solitons on warped product manifolds. Finally, by making use of invariant solution techniques, we classify all steady gradient Yamabe solitons with a conformally flat base that is invariant by the action of a codimension 1 translation group.