On the existence of Kundt's metrics with compact sections of null   hypersurfaces
  It is shown that Kundt's metric for vacuum cannot be constructed when two-dimensional space-like sections of null hypersurfaces are compact, connected manifolds with no boundary unless they are tori or spheres, i.e. higher genus $\mathbf{g} \geq 2$ is excluded by vacuum Einstein equations. The so-called {\em basic equation} (resulting from Einstein equations) is examined. This is a non-linear PDE for unknown covector field and 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. 
Jacek Jezierski