Teichmüller geometry and dynamics
A geodesic lamination, lifted to the hyperbolic plane
The Teichmüller space parametrizes marked conformal structures on a Riemann surface, or equivalently, isotopy classes of hyperbolic metrics. In the modern theory, how to evolve one metric structure most efficiently to another has been vital to understanding connections between any number of problems in dynamics and geometry, and their relation to the study of 3-dimensional spaces, or manifolds. Brock’s work focuses on understanding and classifying Weil-Petersson geodesics, straight lines in a negatively curved Riemannian metric on Teichmüller space. The idea of an ending lamination arises here as well, as a parameter for infinite length geodesics – they are naturally associated, but the extent to which they control or parameterize non recurrent geodesics is still under exploration.
Related Papers
Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II (arXiv)
(With Chris Leininger, Babak Modami, and Kasra Rafi). Crelle's Journal, DOI: 10.1515/crelle-2017-0024.
Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence and flows.
(With Howard Masur and Yair Minsky). Geom. & Funct. Anal. 19 (2010) pp. 1229-1257.
Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity
(With Howard Masur). Geometry & Topology, 12 (2008) 2453-2495.