A boundary value problem for the nonlinear Dirac equation on compact spin manifold

The positive energy theorem is a significant subject in general relativity theory. In Witten's proof of this theorem, the solution of a free Dirac equation which is a spinor filed plays an important role. In order to prove the positive energy theorem for black holes, Gibbons, Hawking, Horowitz and Perry imposed a local boundary condition on the apparent horizon of the black hole. Then the Dirac equation under this boundary condition forms an elliptic boundary value problem. In fact, this kind of local boundary condition can be generally defined by a Chirality operator on the Dirac bundle over a spin manifold. In this paper, by establishing a proper analysis setting and developing variational arguments, we study a nonlinear Dirac equation on a compact spin manifold (M, g) which satisfies the local boundary condition with respect to a Chirality operator.