A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type

Suppose that Σ = ∂M is the n-dimensional boundary of a connected compact Riemannian spin manifold (M, 〈 , 〉) with nonnegative scalar curvature, and that the (inward) mean curvature H of Σ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric 〈 , 〉H = H 〈 , 〉 is at least n/2 and equality holds if and only if there exists a non-trivial parallel spinor field on M . As a consequence, if Σ admits an isometric and isospin immersion F with mean curvature H0 as a hypersurface into another spin Riemannian manifold M0 admitting a parallel spinor field, then