On the Hausdorff dimension of CAT($\kappa$) surfaces

TL;DR

This paper establishes that closed CAT(κ) surfaces have Hausdorff dimension 2, provides bounds on measures of small metric balls, and explores implications for geodesic flow dynamics and entropy rigidity.

Contribution

It proves the Hausdorff dimension result for CAT(κ) surfaces and links measure uniformity to geodesic flow dynamics and entropy rigidity.

Findings

Hausdorff dimension of CAT(κ) surfaces is 2

Uniform bounds on Hausdorff measure of small metric balls

Connection between measure uniformity and geodesic flow dynamics

Abstract

We prove that a closed surface with a CAT($\kappa$) 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.