Riemannian geometry of Hartogs domains

TL;DR

This paper investigates the Riemannian geometry of Hartogs domains with Kähler metrics, characterizing when they are hyperbolic, complete, and related to complex hyperbolic space, with results on geodesics and metric comparisons.

Contribution

It provides new geometric characterizations of Hartogs domains, including conditions for hyperbolicity, geodesic behavior, and metric equivalences with the Bergman metric.

Findings

If a non-special geodesic through the origin is a straight line, the domain is isometric to complex hyperbolic space.

All geodesics through the origin do not self-intersect.

Conditions for geodesic completeness and local irreducibility of the domain.

Abstract

Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$ associated to the \K form $\omega_F = -\frac{i}{2} \partial \bar{\partial} \log (F(|z_0|^2) - \|z\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ 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 $D_F$ do not self-intersect, we find necessary and sufficient conditions on $F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally irreducible as a Riemannian manifold. Finally, we compare the Bergman metric $g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if $g_B$ is a multiple of $g_F$, namely $g_B=\lambda g_F$, for some $\lambda\in \R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space.