  In 2004, Taubes introduced the space of minimal hyperbolic germs with
elements consisting of the first and second fundamental form of an equivariant
immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further
study of this space by studying the behavior of a dynamically defined function
which records the entropy of the geodesic flow on the associated Riemannian
surface. We provide a useful estimate on this function which, in particular,
yields a new proof of Bowen's theorem on the rigidity of the Hausdorff
dimension of the limit set of quasi-Fuchsian groups. These follow from new
lower bounds on the Hausdorff dimension of the limit set which allow us to give
a quantitative version of Bowen's rigidity theorem. To demonstrate the strength
of the techniques, these results are generalized to convex-cocompact surface
groups acting on $n$-dimensional CAT$(-1)$ Riemannian manifolds.
