Rigidez de métricas críticas para funcionais riemannianos.

Abstract

The aim of this work is to study metrics that are critical points for some Riemannian functionals. In the first part, we investigate critical metrics for functionals which are quadratic in the curvature on closed Riemannian manifolds. It is known that space form metrics are critical points for these functionals, denoted by F t,s (g). Moreover, when s = 0, always Einstein metrics are critical to F t (g). We proved that under some conditions the converse is true. For instance, among others results, we prove that if n ≥ 5 and g is a Bach-flat critical metric to F −n/4(n−1) , with second elementary symmetric function of the Schouten tensor σ 2 (A) > 0, then g should be Einstein. Furthermore, we show that a locally conformally flat critical metric with some additional conditions are space form metrics. In the second part, we study the critical metrics to volume functional on compact Riemannian manifolds with connected smooth boundary. We call such critical points of Miao-Tam critical metrics due to the variational study making by Miao and Tam (2009). In this work, we show that the geodesics balls in space forms Rn , Sn and Hn have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary be an Einstein manifold. In the same spirit, we also extend a rigidity theorem due to Boucher et al. (1984) and Shen (1997) to n-dimensional static metrics with positive constant scalar curvature, which give us another way to get a partial answer to the Cosmic no-hair conjecture already obtained by Chrusciel (2003).