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 Hn for any horospherical convex hypersurface Σ. As an application, we obtain an optimal Penrose-type inequality for this new mass for asymptotically hyperbolic graphs with a horizon type boundary Σ, provided that a dominant energy condition L˜k⩾0 holds.
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