Small curvature surfaces in hyperbolic 3-manifolds
 

abstract
 

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S^3 for which the complement contains a closed embedded totally geodesic surface.  In this note, we show that one can get "as close as possible" to a counter-example.  Specifically, we construct a sequence of hyperbolic knots {K_n} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity.  We also construct a family of two-component links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera.  In addition, we prove that a closed embedded totally geodesic surface with sufficiently small principal curvatures is not only quasi-Fuchsian (a result of W. Thurston's), but it is also either acylindrical or else the boundary of a twisted I-bundle.