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.
@article{bwmeta1.element.doi-10_1515_agms-2016-0010,
author = {David Constantine and Jean-Fran\c cois Lafont},
title = {On the Hausdorff Dimension of CAT($\kappa$) Surfaces},
journal = {Analysis and Geometry in Metric Spaces},
volume = {4},
year = {2016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0010}
}
David Constantine; Jean-François Lafont. On the Hausdorff Dimension of CAT(κ) Surfaces. Analysis and Geometry in Metric Spaces, Tome 4 (2016) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0010/