A holographic principle for the existence of imaginary Killing spinors

Suppose that $\mathrm{\Sigma }=\mathrm{\partial }\mathrm{\Omega }$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $({\mathrm{\Omega }}^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\text{textgreater}0$. If $\mathrm{\Sigma }$ admits an isometric and isospin immersion $F$ into the hyperbolic space ${\mathbb{H}}^{n+1}\mathrm{\_}-k^2$, we define a quasi-local mass and prove its positivity as well as the associated rigidity statement. The proof is based on a holographic principle for the existence of an imaginary Killing spinor. For $n=2$, we also show that its limit, for coordinate spheres in an Asymptotically Hyperbolic (AH) manifold, is the mass of the (AH) manifold.