TUBES IN COMPLEX HYPERBOLIC MANIFOLDS

We prove a tubular neighborhood theorem for an embedded complex geodesic in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic chi of the embedded complex geodesic. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem. The first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. Finally, we prove a geometric combination theorem for two C-Fuchsian subgroups of PU(2 , 1). Using this combination theorem, we show that the optimal width size of a tube about an embedded complex geodesic is asymptotically bounded between 1 | chi| and 1 root |chi | .