Sobre métricas críticas do funcional curvatura escalar total

Abstract

This dissertation aims to explain the critical metrics of the total scalar curvature functional (CPE) and to detail the main results obtained in the articles entitled "A note on critical point metrics of the total scalar curvature" due to Leandro Benedito [Math. Anal. Appl. Vol. 424, 1544-1548 (2015)] and "Remarks on critical point metrics of the total scalar curvature" due to Francisco B. Filho [Math. Arch. Vol 104, 463-470 (2015)]. In the first, it has been proved that if a given function in terms of the potential function of a CPE is constant then the manifold is Einstein. Already in the second, it has been shown that under some suitable integral conditions of the canonical sphere, the manifold is isometric to a standard sphere of some ray and its potential function is a first autofunction of the Laplacian.