A holographic principle for the existence of imaginary Killing spinors

Oussama Hijazi (1) , Sebastian Montiel (2) , Simon Raulot (3)

1 IECL - Institut Élie Cartan de Lorraine
2 UGR - Universidad de Granada = University of Granada
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

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 $k>0$. 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.

Mots clés: Positive Mass Theorem; Imaginary Killing spinors; Asymptotically Hyperbolic manifolds; Dirac Operator; Rigidity; Manifolds with Boundary

Domaines: Géométrie différentielle [math.DG]; Mathématiques générales [math.GM]

Oussama Hijazi, Sebastian Montiel, Simon Raulot. A holographic principle for the existence of imaginary Killing spinors. Journal of Geometry and Physics, 2015, 91, pp.12-28. ⟨10.1016/j.geomphys.2015.01.012⟩. ⟨hal-01116656⟩