Campos conformes e métricas críticas em variedades compactas com bordo. - Viana, Emanuel Mendonça (2019)
This work is divided into two parts and it aims to study conformal vector fields and critical metrics on compact manifold with smooth boundary. The first of these parts is related to compact Riemannian manifold (M n , g) with smooth boundary under the existence of nontrivial conformal gradient vector field. With appropriate controls on the Ricci’s curvature, we show that M is isometric to a hemisphere of the sphere, where we use the stiffness results of Reilly (1977 e 1980). Next, considering the case in which the manifold is Einstein with the existence of nonzero conformal gradient vector field, we prove that its scalar curvature is positive and it must be isometric to a hemisphere of Sn . Finally, we conclude that part by showing that a suitable control on the energy of a conformal vector field implies that M is isometric to a hemisphere S+n. In the second part, we study compact Riemannian manifolds (M n , g) that admit a non-constant solution to the system of equations −Δf g + Hessf − fRic = µRic + λg, where Ric is the Ricci tensor of g where as µ and λ are two real parameters. More precisely, under assumption that (M n , g) has zero radial Weyl curvature, this means that the interior product of ∇f with the Weyl tensor W is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.