Dirac Operator on Embedded Hypersurfaces
  New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces. 
Oussama Hijazi, Sebastian Montiel and Xiao Zhang