The mass of an asymptotically hyperbolic end and distance estimates

Abstract

Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\ge -n(n-1)$ 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 $R_g\ge (-1+\kappa )n(n-1)$ for any positive constant $\kappa$ less than one.

24 pages, 3 figures

References in corpus (5)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Boundary value problems for Dirac--type equations, with applications
• Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem
• The positive mass theorem and distance estimates in the spin setting
• Rigidity of non-compact static domains in hyperbolic space via positive mass theorems

Cited by in corpus (1)

• Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields