  An almost Fuchsian manifold is a hyperbolic 3-manifold of the type $S\times
\mathbb{R}$ which admits a closed minimal surface (homeomorphic to $S$) with
the maximum principal curvature $\lambda_0 <1$, while a weakly almost Fuchsian
manifold is of the same type but it admits a closed minimal surface with
$\lambda_0 <= 1$. We first prove that any weakly almost Fuchsian manifold is
geometrically finite, and we construct a Canary-Storm type compactification for
the weakly almost Fuchsian space. We use this to prove uniform upper bounds on
the volume of the convex core and Hausdorff dimension for the limit set of
weakly almost Fuchsian manifolds, and to prove a gap theorem for the principal
curvatures of minimal surfaces in hyperbolic 3-manifolds that fiber over the
circle. We also give examples of quasi-Fuchsian manifolds which admit unique
stable minimal surfaces without being weakly almost Fuchsian.
