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In this article we develope a spinorial proof of the Shi-Tam theorem for thepositivity of the Brown-York mass without necessity of building non smoothinfinite asymptotically flat hypersurfaces in the Euclidean space and use thepositivity of the ADM mass proved by Schoen-Yau and Witten. This same compactapproach provides an optimal lower bound \cite{HMZ} for the first non nulleigenvalue of the Dirac operator of a mean convex boundary for a compact spinmanifold with non negative scalar curvature, an a rigidity result formean-convex bodies in flat spaces. The same machinery provides analogous, butnew, results of this type, as far as we know, in spherical contexts, includinga version of Min-Oo's conjecture.

Compact approach to the positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries in flat and spherical contexts