#### 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
##### Abstract

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