Maximum Principles For Null Hypersurfaces And Null Splitting Theorems

Introduction The geometric maximum principle for smooth (spacelike) hypersurfaces, which is a consequence of Alexandrov's [1] strong maximum for second order quasilinear elliptic operators, is a basic tool in Riemannian and Lorentzian geometry. In [2], extending earlier work of Eschenburg [7], a version of the geometric maximum principle in the Lorentzian setting was obtained for rough (C 0 ) spacelike hypersurfaces which obey mean curvature inequalities in the sense of support hypersurfaces. In the present paper we establish an analogous result for null hypersurfaces (Theorem 3.4) and consider some applications. For the applications, it is important to have a version of the maximum principle for null hypersurfaces which does not require smoothness. The reason for this, which is described in more detail in Section 3, is that the null hypersurfaces which arise most naturally in spacetime geometry and general relativity, such as black hole