The hyperbolic positive energy theorem

Abstract

We show that the causal-future-directed character of the energy-momentum vector of $n$-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, $n\ge 3$, can be traced back to that of asymptotically Euclidean general-relativistic initial data sets satisfying the dominant energy condition.

Figures (2)

• Figure 3.1: A small spherical cap $D^1_\varepsilon\subset {\mathbb S}^{n-1}$ (middle picture) is mapped to the upper-half sphere by the conformal transformation $\Lambda_\varepsilon^{1 }$. The final metric coincides with the hyperbolic one in a region which includes the upper half-ball. We assume the topology of conformal infinity to be spherical, but the interior does not have to be topologically trivial. Thus, the topology inside the final upper half-sphere is that of a half-ball, with the boundary of the lower-half sphere (including the equatorial hyperplane) bounding the remaining topology of $M_1$.
• Figure 3.2: A small spherical cap $D^2_\varepsilon\subset {\mathbb S}^{n-1}$ (left picture) is mapped to the lower-half sphere by the conformal transformation $\Lambda_\varepsilon^{2 }$. The final manifold $(M,g_\varepsilon)$ is obtained by gluing together the manifold from the right Figure \ref{['F8XI18.1']} with the manifold from the middle figure here. The metric $g_\varepsilon$ coincides with $g_{1,\varepsilon}$ on the "lower half" of the final manifold and with $g_{2,\varepsilon}$ on the "upper half", except for two transition strips near the equatorial hyperplane, and is exactly hyperbolic in an overlapping strip enclosing the equatorial hyperplane shown in the right figure.