Real hyperbolic hyperplane complements in the complex hyperbolic plane

Abstract

This paper studies Riemannian manifolds of the form M \S, where M 4 is a complete four dimensional Riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane CH 2 , and S is a compact totally geodesic codimension two submanifold whose induced Riemannian metric is modeled on the real hyperbolic plane H 2 . In this paper we write the metric on CH 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 \ S whose sectional curvature is bounded above by a negative constant (Theorem 1.1(1)). The cases of M \ S modeled on H n \ H n−2 and CH n \ CH n−1 were studied by Belegradek in [Bel12] and [Bel11], respectively. One may consider this work as "part 3" to this sequence of papers.