Volume functional of compact 4-manifolds with a prescribed boundary metric (Q2024633)

The paper refers to critical metrics of the volume functional. It is shown that such metrics 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 \(\mathbb{S}^4\). Also, an integral curvature estimate is given involving the Yamabe constant for critical metrics of the volume functional, hence obtaining a rigidity result for such critical metrics.