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Entropy, minimal surfaces and negatively curved manifolds

Published online by Cambridge University Press:  04 July 2016

ANDREW SANDERS
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA email andysan@uic.edu
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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.

Type
Research Article
Copyright
© Cambridge University Press, 2016 

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