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