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 Σ Σ

An Alexandrov–Fenchel-type inequality for hypersurfaces in the sphere

“…They found a formula for the mass of a graph and (2) was the only thing left to show in order to establish the Penrose inequality in this context. Similar inequalities, with the mean curvature replaced by the k-mean curvature σ k , for odd k, were obtained in [11], where they also proved versions of the Penrose inequality for graphs in the context of the hyperbolic Gauss-Bonnet-Chern mass. Consider now the unit n-sphere S n , n ≥ 3, to be the ambient space.…”

“…where prime means derivative with respect to r. Notice that, if N = (S n−1 , h) and η(r) = sin(r), so that (M, g) is (part of) the round unit sphere, than the function (10) coincides with (3). We assume throughout this section that (M, g) has scalar curvature R g = n(n − 1) (11) and that ρ satisfies the identity…”

A Penrose type inequality for graphs over Reissner-Nordström-anti-deSitter manifold

“…Hence from (6.10) and above, when s → ∞, By integration by parts , we get (see [19] A direct computation (see [1]) gives (6.14) div g (T r ) = −Bdiv g (T r−1 ) − i Rg(ξ, T r−1 (∂ i ))∂ i T where {∂ i } is a tangent frame of M, Rg(·, ·) is the Riemannian curvature tensor of (Q,g). In particular, for r = 1 we have (6.15) div g T 1 , ∂ ∂t where Rg = −n(n + 1)κ 2 − (n − 2)(n − 3)q 2 λ 2−2n is the scalar curvature of the manifold (Q,g).…”

On the stability of the positive mass theorem for asymptotically hyperbolic graphs

“…This lower bound will suffice for our purposes, although it is not optimal. We remark that in [16], Ge, Wang and Wu defined a higher order mass invariant for asymptotically hyperbolic graphs and obtained corresponding Penrose type inequalities. Now, we would like to impose some conditions on the "shape" of the graph.…”