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 $m^{{\mathbb {H}}}_k$, 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 $\Sigma$

$$\begin{aligned} \int _{\Sigma } V \sigma _{2k-1} d\mu \ge C_{n-1}^{2k-1} \omega _{n-1}{\left( \left( \frac{|\Sigma |}{\omega _{n-1}} \right) ^{\frac{n}{k(n-1)}}+\left( \frac{|\Sigma |}{\omega _{n-1}} \right) ^{\frac{n-2k}{k(n-1)}} \right) }^{k}, \end{aligned}$$

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

$$\begin{aligned} m_{k}^{{\mathbb {H}}} \ge \frac{1}{2^k}{\left( \left( \frac{|\Sigma |}{\omega _{n-1}} \right) ^{\frac{n}{k(n-1)}}+\left( \frac{|\Sigma |}{\omega _{n-1}} \right) ^{\frac{n-2k}{k(n-1)}} \right) }^{k}, \end{aligned}$$

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.