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A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type
Suppose that is the -dimensional boundary of a
connected compact Riemannian spin manifold with
non-negative scalar curvature, and that the (inward) mean curvature of
is positive. We show that the first eigenvalue of the Dirac operator
of the boundary corresponding to the conformal metric
is at least and equality
holds if and only if there exists a parallel spinor field on . As a
consequence, if admits an isometric and isospin immersion with
mean curvature as a hypersurface into another spin Riemannian manifold 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, has to be also connected. In the special case where
, equality in (\ref{HoloIneq}) implies that is an Euclidean
domain and is congruent to the embedding of in 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
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}) .Comment: arXiv admin note: text overlap with arXiv:1502.0408
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