Static perfect fluid space-time on compact manifolds

Abstract

The purpose of this article is to investigate the geometry of static perfect fluid space-time metrics on compact manifolds with boundary. In the first part, we provide a boundary estimate for static perfect fluid space-time. In the second part, we establish a unified Böchner type formula for a large class of spaces that include the static perfect fluid space-time, critical metrics of the volume functional, static spaces and CPE metrics. Moreover, as a consequence of such a formula we obtain a gap result for a compact static perfect fluid space-time.