The mass of an asymptotically hyperbolic end and distance estimates

Let (M, g) be a completely connected n-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies Rg ≥ −n(n − 1), and let E⊂M be an asymptotically hyperbolic end. We prove that the mass functional of the end E 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 Rg ≥ (−1 + κ)n(n − 1) for any positive constant κ less than one.