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

Abstract

Suppose that $\Sigma = \partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $(M, \langle , \rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle , \rangle {}_H = H^2 \langle , \rangle$ 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 $\Sigma$ admits an isometric and isospin immersion $F$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $M_0$ admitting a parallel spinor field, then$$\int_{\Sigma} H{ } d\Sigma \leq \int_{\Sigma} \frac{H^2_0}{H} { } d\Sigma$$where $H$ is the mean curvature of $\Sigma$ as the boundary of $M$ and $H_0$ stands for the mean curvature of the immersion $F$ of $\Sigma$ into $\mathbb{R}^{n+1}$. Equality holds if and only if $\Sigma$ is connected, $M$ is a Euclidean domain and the embedding of $\Sigma$ in $M$ and its immersion in $\mathbb{R}^{n+1}$ are congruent.