Compact static perfect fluid space-times and quasi-Einstein manifolds with boundary

Abstract

The purpose of this work is to study compact manifolds with boundary from the point of view of two distinct structures. In the first part, we investigate the geometry of static perfect fluid space-time on compact manifolds with boundary. We use the generalized Reilly’s formula to establish a geometric inequality for a static perfect fluid space-time involving the area of the boundary and its volume. Moreover, we obtain new boundary estimates for this space. One of the boundary estimates is obtained in terms of the Brown-York mass. In addition, we provide a new (simply connected) counterexample to the Cosmic no-hair conjecture for arbitrary dimension n ≥ 4. At the second part of this work, we turn our attention to the geometry of compact quasi-Einstein manifolds with boundary. We establish the possible values for the constant scalar curvature of a compact quasi-Einstein manifold with boundary. Moreover, we show that a 3-dimensional simply connected compact m-quasi-Einstein manifold with boundary and constant scalar curvature must be isometric, up to scaling, to either the standard hemisphere S3+, or the cylinder I × S2 with the product metric, where I is a closed interval. For dimension n = 4, we prove that a 4-dimensional simply connected compact m-quasi-Einstein manifold M 4 with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere S4+, or the cylinder I × S3 with the product metric, or the product space S2+ × S2 with the doubly warped product metric. Other results for dimension greater than or equal to 5 are also discussed.