  An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient
hyperbolic manifold contains a closed incompressible minimal surface with
principal curvatures contained in (-1,1). We show that the domain of
discontinuity of an almost-Fuchsian group contains many balls of a fixed
spherical radius in the visual boundary of hyperbolic 3-space. This yields a
necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which
involves only conformal geometry. As an application, we prove that there are no
doubly-degenerate geometric limits of almost-Fuchsian groups.
