From Period Coordinates to Teichmueller Distance

Geometry & Topology
Event time: 
Tuesday, November 7, 2017 - 4:15pm to 5:15pm
431 DL
Ian Frankel
Speaker affiliation: 
University of Chicago
Event description: 

A half-translation surface is a surface equipped with a flat Riemannian metric with finitely many cone-type singularities, and a system of charts to $\mathbb{R}^2$ valid away from the cone points such that all transition maps between charts are translations or rotations by $\pi$. The moduli space of such surfaces comes equipped with a family of locally defined Euclidean metrics, which can be pieced together to form a Euclidean metric for which the Teichm"uller geodesic flow has hyperbolic properties. We describe how to build quasiconformal maps between nearby half-translation surfaces surfaces in order to estimate the Teichm"uller distance between the underlying Riemann surface structures. In particular, we show that the projection from the moduli space of unit area half-translation surfaces (with the Euclidean metric) onto Teichm"uller space (with the Teichm"uller metric) is H"older on every compact set.