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Volume functional of compact $4$-manifolds with a prescribed boundary metric (1709.07839v2)
Published 22 Sep 2017 in math.DG
Abstract: 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.