Real hyperbolic hyperplane complements in the complex hyperbolic plane

This paper studies Riemannian manifolds of the form M∖S, where M4 is a complete four dimensional Riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane CH2, and S is a compact totally geodesic codimension two submanifold whose induced Riemannian metric is modeled on the real hyperbolic plane H2. In this paper we write the metric on CH2 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∖S whose sectional curvature is bounded above by a negative constant (Theorem 1.1(1)). The cases of M∖S modeled on Hn∖Hn−2 and CHn∖CHn−1 were studied by Belegradek in [4] and [3], respectively. One may consider this work as “part 3” to this sequence of papers.