  We introduce a new elliptic operator on null hypersurfaces of
four-dimensional Lorentzian manifolds. This operator depends on the first and
second fundamental forms of the sections of a foliation of the null
hypersurface and its novelty originates from its covariant transformation under
change of foliation. It thus provides at any point an elliptic structure
intimately connected with the geometry of the null hypersurface, independent of
the choice of a specific section through that point. No analytic or algebraic
symmetries or other conditions are imposed on the metric. The spectral
properties of this elliptic operator are relevant to the evolution of the wave
equation, and in particular, the existence of conservation laws along null
hypersurfaces.
