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 \mathrmCAT(-1) Riemannian manifolds.

SEEK ID: https://publications.h-its.org/publications/959

DOI: 10.1017/etds.2016.23

Research Groups: Groups and Geometry

Publication type: Journal

Journal: Ergodic theory and dynamical systems

Citation: Ergod. Th. Dynam. Sys. 38(1):336-370

Date Published: 1st Feb 2018

URL: https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/entropy-minimal-surfaces-and-negatively-curved-manifolds/61E3D3677BEBD30669AE5F084E05C40E

Registered Mode: imported from a bibtex file

Author: ANDREW SANDERS

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Citation
SANDERS, A. N. D. R. E. W. (2016). Entropy, minimal surfaces and negatively curved manifolds. In Ergodic Theory and Dynamical Systems (Vol. 38, Issue 1, pp. 336–370). Cambridge University Press (CUP). https://doi.org/10.1017/etds.2016.23
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Created: 30th Jan 2020 at 09:39

Last updated: 5th Mar 2024 at 21:24

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