| Abstrakt: |
The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $$m^{{\mathbb {H}}}_k$$ , 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 and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $${\mathbb {H}}^n$$ . In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $${\mathbb {H}}^n$$ for any horospherical convex hypersurface $$\Sigma $$ where $$\sigma _{j}$$ is the j-th mean curvature of $$\Sigma \subset {\mathbb {H}}^n$$ , $$V=\cosh r$$ and $$|\Sigma |$$ is the area of $$\Sigma $$ . As an application, we obtain an optimal Penrose type inequality for the new mass defined in the first part for asymptotically hyperbolic graphs with a horizon type boundary $$\Sigma $$ , provided that a dominant energy type condition $$\tilde{L}_k\ge 0$$ holds. Both inequalities are optimal. [ABSTRACT FROM AUTHOR] |
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