  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 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.
