  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 in
fact quasi-Fuchsian, and we construct a Canary-Storm type compactification for
the weakly almost Fuchsian space. We apply these results 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 give examples of quasi-Fuchsian
manifolds which admit unique minimal surfaces (resp. stable minimal surfaces)
without being almost Fuchsian (resp. weakly almost Fuchsian). We also show that
for every $g$ there is an $\epsilon$ depending only on $g$ such that if a
closed hyperbolic 3-manifold fibers over the circle with fiber a surface of
genus $g$, then any embedded minimal surface isotopic to the fiber has the
maximum principal curvature $\lambda_0$ larger than $1+\epsilon$.
