A couple of real hyperbolic disc bundles over surfaces

Applying the techniques developed in [1], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [6], such bundles M -> S should satisfy the inequality vertical bar eM/chi S vertical bar <= 1, where eM stands for the Euler number of the bundle and chi S, for the Euler characteristic of the surface. In this paper, we construct new examples that provide a maximal value of vertical bar eM/chi S vertical bar = 3/5 among all known examples. The former 5 maximum, belonging to Feng Luo [10], was vertical bar eM/chi S vertical bar = 1/2.