A holographic principle for the existence of imaginary Killing spinors

TL;DR

This paper introduces a holographic principle linking imaginary Killing spinors to quasi-local mass positivity in hyperbolic manifolds, with implications for asymptotically hyperbolic spaces.

Contribution

It establishes a new holographic principle connecting imaginary Killing spinors to mass positivity and rigidity in hyperbolic manifolds.

Findings

Defined a quasi-local mass with positivity and rigidity properties.

Proved the limit of the quasi-local mass matches the manifold's mass in asymptotically hyperbolic spaces.

Extended results to the case of 2-dimensional boundary surfaces.

Abstract

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge -n(n+1)k^2$, for some $k\textgreater{}0$. If $\Sigma$ admits an isometric and isospin immersion $F$ into the hyperbolic space ${\mathbb{H}^{n+1}\_{-k^2}}$, 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.