Métricas críticas do funcional volume e não-existência de múltiplos buracos negros em espaço-tempo estático - Baltazar, Halyson Irene (2017)
This work is divided in two parts. In the first one we prove a Böchner type formula for critical metrics of the volume functional on compact manifolds with fixed metric on boundary (such critical metrics are called Miao-Tam critical metrics). As an application, we derive an integral formula that will be crucial to deduce a generalization of a result obtained by Miao and Tam in 2011 for the Einstein case. More precisely, we prove that a Miao-Tam critical metric with parallel Ricci curvature must be isometric to a geodesic ball in a simply connected space form Rn, Sn or Hn. Furthermore, in dimension 3, we prove that critical metrics with non-negative sectional curvature are precisely geodesic balls of R3 or S3. Moreover, we generalize a result due to Kim and Shin (2016), replacing the harmonic Weyl tensor condition by the second order divergence free Weyl tensor condition (i.e., div2W = 0), which is weaker that the former. To be precise, we shall show that a 4-dimensional Miao-Tam critical metric, with boundary isometric to a standard sphere S3 and satisfying div2W = 0 is isometric to a geodesic ball in a simply connected space form R4, S4 or H4. At the same time, we get some rigidity results for positive static triples. In the second part, we study static vacuum space-times, which can be seen as a special case of the V-static metrics for complete Riemannian manifolds with null scalar curvature. In this case, we focus our attention on four dimensions. We prove that there are no multiple black holes on static vacuum space-times with half harmonic Weyl tensor (i.e., divW+ = 0).