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

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.

To appear in The Journal of Geometric Analysis

References in corpus (2)

• Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary
• Fourth order curvature flows and geometric applications