Volume functional of compact $4$-manifolds with a prescribed boundary metric

Abstract

We prove that a critical metric of the volume functional on a four-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics on four-dimensional manifolds.