Domains of discontinuity for almost-Fuchsian groups

An almost-Fuchsian group Γ > I s o m + ( H 3 ) Γ >Isom⁺(H^3) is a quasi-Fuchsian group such that the quotient hyperbolic manifold H 3 / Γ H^3/Γ contains a closed incompressible minimal surface with principal curvatures contained in ( − 1 , 1 ) . (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius R > 0 R>0 in C ∪ { ∞ } = ∂ ∞ ( H 3 ) . C∪ \{∞ \} =∂ ∞(H^3). 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.