Compact approach to the positivity of Brown-York mass]{Compact approach
to the positivity of Brown-York mass and rigidity of manifolds with
mean-convex boundaries in flat and spherical contexts
In this article we develope a spinorial proof of the Shi-Tam theorem for the
positivity of the Brown-York mass without necessity of building non smooth
infinite asymptotically flat hypersurfaces in the Euclidean space and use the
positivity of the ADM mass proved by Schoen-Yau and Witten. This same compact
approach provides an optimal lower bound \cite{HMZ} for the first non null
eigenvalue of the Dirac operator of a mean convex boundary for a compact spin
manifold with non negative scalar curvature, an a rigidity result for
mean-convex bodies in flat spaces. The same machinery provides analogous, but
new, results of this type, as far as we know, in spherical contexts, including
a version of Min-Oo's conjecture