My current research is related to the study of the pressure metric on higher rank Teichmüller spaces. This Riemannian metric is a generalization of the Weil-Petersson metric on the classical Teichmüller space arising from thermodynamic formalism. In contrast to the traditional means of studying the Teichmüller’s metric using extremal quasiconformal maps or the Weil-Petersson metric by quadratic differentials on Teichmüller space, for the pressure metric we work dynamically, for example by studying families of reparametrizations of geodesic flows on hyperbolic surfaces. The pressure metric is then defined in terms of the variance of the first variation of reparametrization functions.
My thesis work is about the properties of the pressure metric. I show the existence of natural geodesic coordinates of the pressure metric near the Fuchsian locus for a model case of a higher Teichmuller spaces. This work turns out to be a combination of both thermodynamic formalism and Higgs bundle theory.
I am also interested in the curvature properties of the pressure metric which are properties of the pressure metric and behaviors of the pressure metric as one leaves compacta in higher Teichmuller spaces.
Geodesic Coordinates for the Pressure Metric at the Fuchsian Locus (In preparation)