Rigidez e estimativas de volume de métricas tipo Einstein

Abstract

The purpose of this work is to study like-Einstein metrics, namely, Ricci solitons, almost Ricci solitons and quasi-Einstein metrics. First, we deduce two compactness theorem for gradient Ricci solitons satisfying certain special conditions. In the sequel we prove some integral formulae which allow us to prove that every compact almost Ricci solitons with constant scalar curvature must be gradient type. Moreover, we prove that every compact locally conformally at gradient Ricci soliton must be isometric to standard sphere under an integral condition. Finally, we study the growth of the geodesic balls of steady quasi-Einstein metrics. Moreover, we use Einstein quasi-metric theory to prove a triviality theorem and then to produce a certain class of Einstein warped products under a suitable hypothesis in the fiber.