A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary (Q2788769)

The aim of this paper is to provide a positive mass theorem in the context of asymptotically hyperbolic (AH) manifolds with inner boundary.NEWLINENEWLINENEWLINEThe authors recall first the positive Mass Conjecture and the Penrose Inequality Conjecture for Assymptotically Flat (AF) manifolds, which is the time-symmetric case of the full Penrose Inequality.NEWLINENEWLINENEWLINEIn [Commun. Math. Phys. 188, No. 1, 121--133 (1997; Zbl 0886.53032)] \textit{M. Herzlich} proved a Penrose-type inequality for spin manifolds of arbitrary dimension. In the present paper, the authors extend Herzlich's proof from the AF context to AH spin manifolds with compact inner boundary, improving a result from [\textit{P. T. Chruściel} and \textit{M. Herzlich}, Pac. J. Math. 212, No. 2, 231--264 (2004; Zbl 1056.53025)]. More precisely, in the last mentioned work it was proved that the energy-momentum vector of a complete spin AH \(n\)-dimensional manifold with a compact inner boundary of mean curvature \(H\leq 1\) and with scalar curvature greater than \(-n(n-1)\) is time-like future-directed. In the present paper this result is generalized in such a way that the energy-momentum vector is time-like future-directed or zero. The authors treat first the 3-dimensional case. By assuming the compact inner boundary \(\Sigma\) to be homeomorphic to a 2-sphere of mean curvature \(H\leq \sqrt{\frac{4\pi}{\mathrm{Area}_g\big(\sum\big)}+1}\), they obtain that the energy-momentum vector is time-like future-directed or zero, and moreover, when it is zero, the manifold is isometric to the complement of a geodesic ball in the hyperbolic space \(\mathbb H^3\). In the case of complete AH manifolds of dimension \(n\geq 4\), the same causality of the energy momentum as above is obtained by controlling the mean curvature by an expression depending on the Yamabe invariant of \(\Sigma\) for the induced metric. In this situation, if the energy momentum vanishes, then there exists an imaginary Killing spinor field on the manifold and \(\Sigma\) is a totally umbilical hypersurface with constant mean curvature, endowed with a real Killing spinor.