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The GBC mass for asymptotically hyperbolic manifolds. (La masse de Gauss-Bonnet-Chern sur des variétés asymptotiquement hyperboliques.) (English. French summary) Zbl 1285.53026

Authors’ abstract: By using the Gauss-Bonnet curvature, we introduce a higher-order mass, the Gauss-Bonnet-Chern mass, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs. Then, we prove the weighted Alexandrov-Fenchel inequalities in the hyperbolic space \(\mathbb{H}^n\) for any horoscopical convex hypersurface \(\Sigma\). As an application, we obtain an optimal Penrose-type inequality for this new mass for asymptotically hyperbolic graphs with a horizon type boundary \(\Sigma\), provided that a dominant energy condition \(\widetilde{L}_k\geq 0\) holds.
Reviewer: Ioan Pop (Iaşi)

MSC:

53C20 Global Riemannian geometry, including pinching
57R20 Characteristic classes and numbers in differential topology
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