On the existence of Kundt's metrics and degenerate (or extremal) Killing horizons

It is shown that Kundt's metric for vacuum cannot be constructed when two-dimensional spacelike sections of null hypersurfaces are compact, connected manifolds with no boundary unless they are tori or spheres, i.e. higher genus g ≥ 2 is excluded by vacuum Einstein equations. The so-called basic equation (resulting from Einstein equations) is examined. This is a nonlinear PDE for an unknown covector field and an unknown Riemannian structure on the two-dimensional manifold. It implies several important results derived in this paper. It arises not only for Kundt's class but also for degenerate Killing horizons and vacuum degenerate isolated horizons.