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arXiv · 1709.07839

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

H. Baltazar·R. Diógenes·E. Ribeiro Jr
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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.

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H. Baltazar, R. Diógenes, E. Ribeiro Jr. 2017-09-22. Volume functional of compact $4$-manifolds with a prescribed boundary metric. https://arxiv.org/abs/1709.07839

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