Entropy, minimal surfaces and negatively curved manifolds

Abstract

Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr.7 (2004), 69–100 (electronic)] 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 $\text{CAT}\,(-1)$ Riemannian manifolds.