The GBC mass for asymptotically hyperbolic manifolds (Q5892160)

The authors consider asymptotically hyperbolic manifolds \(\mathcal{M}^n\) of decay order \(\tau\) and define the Gauss-Bonnet-Chern mass \(m_k^{\mathbf H}\), in terms of (a modified) Gauss-Bonnet curvature, for positive integers \(k\) and suitable relations between \(n\), \(k\) and \(\tau\). It is shown that \(m_k^{\mathbf H}\) are geometric invariants. If \(\mathcal M^n\) is a graph with a horizon boundary, then a positive mass theorem for \(m_k^{\mathbf H}\) is proven and a Penrose-type inequality is established (with equality achieved by the anti-de Sitter Schwarzschild type metric). For the later, the authors use some new weighted Alexandrov-Fenchel inequalities (for any horospherical convex hypersurface) in the hyperbolic space, and an additional hypothesis involving a dominant energy-type condition.