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It has become a standard fact that for any asymptotically Euclidean manifold one can define a mass which is a certain integral at infinity. A celebrated theorem of Schoen, Yau and Witten shows that if the manifold has nonnegative scalar curvature then the mass is positive and vanishes only for the Euclidean space. A similar statement holds for asymptotically hyperbolic manifolds. The analog of the positive mass theorem has been proven by Wang, Chrusciel and Herzlich for spin manifolds and by Andersson, Cai and Galloway under another restrictive assumption.
After reviewing these results, I will present two developments. The first one concerns the near equality case of the positive mass theorem: what can be said if the mass is close to zero. And the second one is a lower bound on the mass in terms of the geometry of the manifold known as the (conjectured) Penrose inequality which we prove in the context of manifolds which are graphs over the hyperbolic space.
A part of the work presented is joint with Mattias Dahl and Anna Sakovich.
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