Riemannian geometry of Hartogs domains

Let DF = {(z0,z) ∈ Cn | |z0|2 < b,∥z∥2 < F(|z0|2)} be a strongly pseudoconvex Hartogs domain endowed with the K ̈ahler metric gF associated to the K ̈ahler form ωF =−i∂∂log(F(|z0|2)−∥z∥2). 2 This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if DF admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then DF is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of DF do not self-intersect, we find necessary and sufficient conditions on F for DF to be geodesically complete and we prove that DF is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric gB and the metric gF in a bounded Hartogs domain and we prove that if gB is a multiple of gF , namely gB = λgF , for some λ ∈ R+, then DF is holomorphically isometric to an open subset of the complex hyperbolic space.