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We prove the rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality p0 = p 2 1 + · · · + p 2 n holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds [18,23,2,6] or under special asymptotics [1].

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We present a scalar curvature splitting result patterned to some extent after a Ricci curvature splitting result of Croke and Kleiner. The proof is an application of results on marginally outer trapped surfaces. Using a local version of this result (and a variation thereof), we obtain a splitting result for manifolds with boundary that admit a solution to an Obata type equation. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning aspects of asymptotically locally hyperbolic manifolds.

No abstract