On the scarcity of non-totally geodesic complete spacelike hypersurfaces of constant mean curvature in a Lie group with bi-invariant Lorentzian metric

Abstract

The results of this paper can be viewed as giving a sort of heuristic explanation of why it is so hard to give examples of non-totally geodesic, complete, spacelike, constant mean curvature hypersurfaces Mn of a Lorentzian group Gn+1. More precisely, let N be a timelike unit vector field on M and suppose that the Ricci curvature of G in the direction of N is greater than or equal to −H2n, where H is the mean curvature of M with respect to N. If M is compact and transversal to a timelike element of the Lie algebra of G, then we show that it is a lateral class of a Lie subgroup of G and, as such, totally geodesic in G. If M is noncompact and parabolic, then we get the same result, provided M has bounded hyperbolic Gauss map. We also discuss some related examples and, along the way, give a simple proof of the parabolicity of a Riemannian product of a compact and a parabolic Riemannian manifold.