Title: Volume functional of compact 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 $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or
$\mathbb{S}^{4}.$ 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.