Tubes in complex hyperbolic manifolds

Abstract

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 χ \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 \mathbb {C} -Fuchsian subgroups of PU ⁡ ( 2 , 1 ) \operatorname {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 | χ | \frac {1}{|\chi |} and 1 | χ | \frac {1}{\sqrt {|\chi |}} .