On totally geodesic boundaries of hyperbolic $3$-manifolds

Abstract

By a hyperbolic manifold, we will mean a Riemannian manifold with constant sectional curvature —1. In this paper, we study complete oriented hyperbolic 3-manifolds each of which has a totally geodesic boundary. A totally geodesic boundary of such a 3-manifold becomes a hyperbolic surface. Let g be an integer greater than or equal to 2 and let Mg be the Riemann moduli space consisting of all isometry classes of connected closed hyperbolic surfaces of genus g. Let Sg be the subset of Mg consisting of those hyperbolic surfaces which are boundary components of compact oriented hyperbolic 3manifolds with totally geodesic boundary. As pointed out by Soma, there is an argument making use of a theorem of Brooks [1] which shows that Sg is dense in Mg. However it seems that almost nothing is known about the characterization of elements of Sg. For example, it is by no means easy to construct elements of Sg except for obvious ones. In this paper, we will obtain certain concrete examples of elements of Sg by taking a complete hyperbolic 3-manifold with one torus cusp and totally geodesic boundary and then analyzing the effect of the hyperbolic Dehn surgery on the toral end. More precisely, we consider the following. Namely, take any complete hyperbolic 3-manifold P with one torus cusp and connected totally geodesic boundary Σg of genus g. Denote P(p, q) the 3-manifold obtained by performing Dehn surgery on P of type (p, q) on the toral end, where (p, q) is a coprime pair of integers near oo in R2\J{oo\. Then, using the results of Thurston and Mostow, we see that P(p, q) admits a complete hyperbolic structure with connected totally geodesic boundary Σg(p, q) of genus g. For each g, we show that there are infinitely many mutually non-isometric P(p, q)'s by considering the lengths of the adjoining closed short geodesies (see [3]). Now in such situations, it seems that the following phenomenon may quite often happen: if {p, q) is close to oo in R2U{°°], then the original surface Σg and the resulting surface Σg(p, q) are different in the Teichmϋller space which are very close to each other, so they are also different in the moduli space Mg. However at present, there are no general theory about this. Moreover, as far as the author