Teichmüller geodesics and ends of hyperbolic 3-manifolds

Abstract

THE THEORY of the structure and deformations of hyperbolic 3-manifolds depends in an essential way on a good understanding of the geometry of incompressible maps of surfaces into these manifolds. Consider, for example, a hyperbolic 3-manifold N homeomorphic to S x R where S is a closed surface of genus g > 1. Thurston and Bonahon showed how to fill the convex hull of N with “pleated surfaces” homotopic to the obvious map S -+ S x {0} (see Sections 2.3, 2.4) and these surfaces have induced metrics which determine points in the Teichmiiller space r(S) of conformal (or hyperbolic) structures on S. It has been conjectured that the locus of these points is related in an approximate way to a geodesic in F(S), and this is known to be true for a class of examples arising from hyperbolic structures on surface bundles over a circle (see [S]). This kind of information has implications concerning the geometry of N, which will be discussed more fully in a forthcoming paper ([25]). From a more differential-geometric point of view, one can consider, for any metric 0 on S, a mapf, : S -+ N of least “energy” (see Section 3) in the above homotopy class. This least energy is then some function 8(c), andf, is a harmonic map. One can then ask about the locus of points [o] in 3(S) where 8 is bounded above by a given constant, and its relationship to the above set of metrics induced from pleated surfaces. We give here a proof of the following result in this direction, where the crucial restrictive hypothesis we need to make is a positive lower bound on the injectivity radius inj,(x) for all x E N.