The moduli space of Riemann surfaces is Kahler hyperbolic
  Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures.   From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant curvature -1 in the Teichm\"uller (=Kobayashi) metric.   When $r=\dim_{\cx} \cM_{g,n}$ is greater than one, however, $\cM_{g,n}$ carries no complete metric of bounded negative curvature. Instead, Dehn twists give chains of subgroups $\zed^r \subset \pi_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$.   In this paper we introduce a new K\"ahler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank. 
Curtis T. McMullen