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

DOI: 10.2140/pjm.2018.297.29

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.

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