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

Suppose that $\mathrm{\Sigma }=\mathrm{\partial }M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $(M,⟨,⟩)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\mathrm{\Sigma }$ 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 $n\mathrm{/}2$ and equality holds if and only if there exists a parallel spinor field on $M$. As a consequence, if $\mathrm{\Sigma }$ admits an isometric and isospin immersion $\varphi$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $M_0$ admitting a parallel spinor field, then \begin{equation} \label{HoloIneq} \int_\Sigma H,d\Sigma\le \int_\Sigma \frac{H^2_0}{H}, d\Sigma \end{equation} and equality holds if and only if both immersions have the same shape operator. In this case, $\mathrm{\Sigma }$ has to be also connected. In the special case where $M_0={\mathbb{R}}^{n+1}$, equality in (\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $\varphi$ is congruent to the embedding of $\mathrm{\Sigma }$ in $M$ as its boundary. We also prove that Inequality (\ref{HoloIneq}) implies the Positive Mass Theorem (PMT). Note that, using the PMT and the additional assumption that $\varphi$ is a strictly convex embedding into the Euclidean space, Shi and Tam \cite{ST1} proved the integral inequality \begin{equation}\label{shi-tam-Ineq} \int_\Sigma H,d\Sigma\le \int_\Sigma H_0, d\Sigma, \end{equation} which is stronger than (\ref{HoloIneq}) .