Variedades quasi-Einstein completas e métricas críticas do funcional volume em variedades compactas com bordo

Abstract

The purpose of this work is to study quasi-Einstein manifolds and Miao-Tam critical metrics. In the first part, we will study the structure at infinity of a complete non-compact quasi-Einstein manifold. In particular, we show that if M is the basis of a warped product Ricci-flat then M is connected at infinity. When M is a quasi-Einstein manifold with λ < 0 there are examples showing that such a result is not true. In this case, we show that M is f -non-parabolic and, under a certain hypothesis on the scalar curvature, M has only one f -non-parabolic end. Furthermore, we obtain two estimates for the volume of the geodesic balls of M. Next, we show that a Bach-flat non-compact quasi-Einstein manifold with λ= 0 and positive Ricci curvature must be isometric to a warped product metric g = dt2+ψ2(t)gL, where gL is an Einstein metric. In the second part, we will study the critical metrics of the functional volume restricted to the set of metrics with constant scalar curvature and boundary prescribed metric on a compact manifold. We obtain a sharp upper bound for the area of the boundary of a Miao-Tam critical metric (M3;g) with non-negative scalar curvature. Moreover, we show that the equality holds if and only if (M3;g) is isometric to a geodesic ball in simply connected space form R3 or S3. Finally, we get a type-Bochner formula for a 3-dimensional Miao-Tam critical metric, which allows us to get the same rigid result provided that/ Ric/ ≤R6 .