The mass of an asymptotically hyperbolic end and distance estimates

Abstract

Let ( M, g) be a completely connected n-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies R g ≥ − n( n − 1), and let [Formula: see text] be an asymptotically hyperbolic end. We prove that the mass functional of the end [Formula: see text] is timelike future-directed or zero. Moreover, it vanishes if and only if ( M, g) is isometrically diffeomorphic to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with a compact boundary, and we prove that the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a function defined by using distance estimates. For applications, the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by −( n − 1) or the scalar curvature satisfies R g ≥ (−1 + κ) n( n − 1) for any positive constant κ less than one.