Asymptotically hyperbolic manifolds with small mass

Abstract

For asymptotically hyperbolic manifolds of dimension $n$ with scalar curvature at least equal to $-n(n-1)$ the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.

References in corpus (5)

• On the Riemannian Penrose inequality in dimensions less than 8
• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold
• A Level Set Analysis of the Witten Spinor with Applications to Curvature Estimates
• De l'équation de prescription de courbure scalaire aux équations de contrainte en relativité générale sur une variété asymptotiquement hyperbolique

Cited by in corpus (6)

• On the asymptotic behavior of static perfect fluids
• Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry
• On the stability of the positive mass theorem for asymptotically hyperbolic graphs
• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow
• Sobolev stability of the PMT and RPI using IMCF
• Hilbert manifold structure for asymptotically hyperbolic relativistic initial data