Real hyperbolic hyperplane complements in the complex hyperbolic plane

Abstract

This paper studies Riemannian manifolds of the form $M \setminus S$, where $M^4$ is a complete four dimensional Riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane $\mathbb{C} \mathbb{H}^2$, and $S$ is a compact totally geodesic codimension two submanifold whose induced Riemannian metric is modeled on the real hyperbolic plane $\mathbb{H}^2$. In this paper we write the metric on $\mathbb{C} \mathbb{H}^2$ in polar coordinates about $S$, compute formulas for the components of the curvature tensor in terms of arbitrary warping functions (Theorem 7.1), and prove that there exist warping functions that yield a complete finite volume Riemannian metric on $M \setminus S$ whose sectional curvature is bounded above by a negative constant (Theorem 1.1(1)). The cases of $M \setminus S$ modeled on $\mathbb{H}^n \setminus \mathbb{H}^{n-2}$ and $\mathbb{C} \mathbb{H}^n \setminus \mathbb{C} \mathbb{H}^{n-1}$ were studied by Belegradek in [Bel12] and [Bel11], respectively. One may consider this work as "part 3" to this sequence of papers.