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arXiv · 1603.05456

Critical metrics of the volume functional on compact three-manifolds with smooth boundary

R. Batista·R. Diógenes·M. Ranieri·E. Ribeiro Jr
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Abstract

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manifolds. In addition, we obtain a B\"ochner type formula which enables us to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in $\Bbb{S}^3.$

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R. Batista, R. Diógenes, M. Ranieri, E. Ribeiro Jr. 2016-03-17. Critical metrics of the volume functional on compact three-manifolds with smooth boundary. https://arxiv.org/abs/1603.05456

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