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

Overview & Synopsis

Suppose that Σ = ∂M is the n-dimensional boundary of a con-nected compact Riemannian spin manifold (M, ,) with non-negative 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 2 , is at least n/2 and equality holds if and only if there exists a non-trivial parallel spinor field on M . As a con-sequence, if Σ admits an isometric and isospin immersion F with mean curvature H 0 as a hypersurface into another spin Riemann-ian manifold M 0 admitting a parallel spinor field, then (1) Σ H dΣ ≤ Σ H 2 0 H dΣ and equality holds if and only if both immersions have the same shape operator. In this case, Σ has to be also connected. In the special case where M 0 = R n+1 , equality in (1) implies that M is a Euclidean domain and F is congruent to the embedding of Σ in M as its boundary. We also prove that Inequality (1) implies the Positive Mass Theorem (PMT).