Métricas críticas do funcional volume, volume mínimo e curvatura mínima em variedades de dimensão quatro

Abstract

This aim of this is to study the critical metrics of the volume functional, minimal volume and minimal curvature on four-dimensional compact manifolds. In the first part, we investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form R4, H4 or S4. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that M has divergence-free Bach tensor. In the second part we investigate the geometric invariants: minimal volume and minimal curvature. In 1982, Gromov introduced the concept of minimal volume for a smooth manifold as the greatest lower bound of the total volumes of Mn with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value by 1. While the minimal curvature, introduced by G. Yun in 1966, is the smallest pinching of the sectional curvature among metrics of volume 1. In both cases we give below estimates to minimal volume and minimal curvature on 4-dimensional compact manifolds involving some differential and topological invariants. Among these ones, we get some sharp estimates. Moreover, we deduce characterizations for the equality case in some estimates.