Lower bounds for eigenvalues of the Dirac-Witten operator

Abstract

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact (with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions, just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case, the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.