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We prove that a critical metric of the volume functional on a $4$-dimensionalcompact manifold with boundary satisfying a second-order vanishing condition onthe Weyl tensor must be isometric to a geodesic ball in a simply connectedspace form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, weprovide an integral curvature estimate involving the Yamabe constant forcritical metrics of the volume functional, which allows us to get a rigidityresult for such critical metrics.

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