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Analysis and Geometry in Metric Spaces

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On the Hausdorff Dimension of CAT(κ) Surfaces

David Constantine
  • Department of Mathematics, Ohio State University, Columbus, Ohio 43210
/ Jean-François Lafont
  • Wesleyan University, Mathematics and Computer Science Department, Middletown, CT 06459
Published Online: 2016-09-20 | DOI: https://doi.org/10.1515/agms-2016-0010


We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

Keywords: metric geometry; Hausdorff dimension; CAT(k) surface; topological entropy


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Received: 2015-03-02

Accepted: 2016-07-04

Published Online: 2016-09-20

Citation Information: Analysis and Geometry in Metric Spaces. Volume 4, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2016-0010, September 2016

© 2016 David Constantine and Jean-François Lafont . This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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