The mass of an asymptotically hyperbolic end and distance estimates

TL;DR

This paper establishes conditions under which the mass of asymptotically hyperbolic ends of Riemannian spin manifolds is future-directed timelike or zero, characterizing hyperbolic space through mass vanishing.

Contribution

It proves the positivity and causal nature of the mass functional for asymptotically hyperbolic manifolds, extending previous results and providing new geometric bounds involving boundary mean curvature.

Findings

Mass is timelike future-directed or zero for asymptotically hyperbolic ends.

Mass vanishes if and only if the manifold is hyperbolic space.

Boundary mean curvature bounds imply positivity of mass.

Abstract

Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with compact boundary, we prove the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a function defined using distance estimates. As an application, 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 $R_g\geq (-1+\kappa)n(n-1)$ for any positive constant $\kappa$ less than one.