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

TL;DR

This paper establishes a holographic principle linking parallel spinor fields to geometric inequalities, showing how boundary eigenvalues relate to the existence of parallel spinors and deriving an inequality that implies the Positive Mass Theorem.

Contribution

It introduces a new inequality involving mean curvature and spinor fields, connecting boundary spectral properties to the existence of parallel spinors and geometric rigidity.

Findings

Eigenvalue of boundary Dirac operator is at least n/2 with equality characterizing parallel spinors.

Derived a new inequality relating mean curvatures of immersed hypersurfaces.

Proved that the inequality implies the Positive Mass Theorem.

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 parallel spinor field on $ M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $\phi$ 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, $\Sigma$ has to be also connected. In the special case where $M_0=\R^{n+1}$, equality in (\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $\phi$ is congruent to the embedding of $\Sigma$ 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 $\phi$ 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}) .