2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125238In this paper, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the $\MIT$ bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in particular, solve the Yamabe problem on manifolds with boundary in some cases. Finally, using the $\MIT$ Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.23 pagesDifferential Geometry53A30, 53C27 (Primary), 58J50, 58C40 (Secondary)text