Vol. 297, No. 1, 2018

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Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

Halyson Baltazar and Ernani Ribeiro, Jr.

Vol. 297 (2018), No. 1, 29–45

We provide a general Bochner type formula which enables us to prove some rigidity results for V-static spaces. In particular, we show that an n-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify V-static spaces with nonnegative sectional curvature.

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Riemannian functional, critical metrics, static space, scalar curvature
Mathematical Subject Classification 2010
Primary: 53C21, 53C25
Secondary: 53C24
Received: 12 December 2016
Revised: 11 January 2017
Accepted: 9 January 2018
Published: 7 October 2018
Halyson Baltazar
Departamento de Matemática
Universidade Federal do Piauí - UFPI
Campus Universitário Ministro Petrônio Portella
Ernani Ribeiro, Jr.
Departamento de Matemática
Universidade Federal do Ceará