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

We prove that a critical metric of the volume functional on a $4$-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.