Asymptotically hyperbolic manifolds with small mass

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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.