The GBC mass for asymptotically hyperbolic manifolds

Abstract

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 $m^{\H}_k$, 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 $\H^n$. In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $\H^n$ for any horospherical convex hypersurface $\Sigma$. 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 condition ${\tilde{L}}_k\ge 0$ holds. Both inequalities are optimal.

41 pages

References in corpus (7)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• The Higher Dimensional Positive Mass Theorem II
• A geometric inequality on hypersurface in hyperbolic space
• A generalized mass involving higher order symmetric function of the curvature tensor
• Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space
• A new mass for asymptotically flat manifolds
• Rigidity for nearly umbilical hypersurfaces in space forms