Green functions for the Dirac operator under local boundary conditions and applications

Simon Raulot (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

In 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.

Mots clés: Manifolds with boundary; Conformally invariant operators; Dirac operator; Green function; Local boundary conditions

Domaines: Géométrie différentielle [math.DG]

Simon Raulot. Green functions for the Dirac operator under local boundary conditions and applications. Annals of Global Analysis and Geometry, 2011, 39 (4), pp.337-359. ⟨10.1007/s10455-010-9236-y⟩. ⟨hal-00135368⟩