Real hyperbolic hyperplane complements in the complex hyperbolic plane

TL;DR

This paper constructs complete finite volume Riemannian metrics with negative curvature on complex hyperbolic manifolds minus certain submanifolds, extending previous work on hyperbolic space complements.

Contribution

It provides explicit formulas for curvature components and demonstrates the existence of warping functions that produce negatively curved, complete, finite volume metrics on complex hyperbolic hyperplane complements.

Findings

Curvature tensor formulas in polar coordinates

Existence of warping functions for negative curvature

Construction of complete finite volume metrics

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.