The Relation Between the Geodesic Flow and Finite-Area Holomorphic Quadratic Differentials on Infinite-Genus Riemann Surfaces

Explore the intricate relationship between geodesic flows and finite-area holomorphic quadratic differentials on infinite-genus Riemann surfaces in this advanced mathematical physics lecture. Examine the Hopf-Tsuji-Sullivan theorem, which establishes that geodesic flow on infinite Riemann surfaces is ergodic if and only if the Poincaré series diverges and Brownian motion is recurrent. Learn how infinite Riemann surfaces are constructed by gluing infinitely many hyperbolic pairs of pants, with the resulting surface determined by corresponding Fenchel-Nielsen parameters. Discover various sufficient conditions on these parameters that guarantee surface membership in the ergodic class, based on joint research with Hakobyan and Basmajian. Understand the key theorem proving that geodesic flow is ergodic if and only if almost every horizontal leaf of every finite-area holomorphic quadratic differential is recurrent. Apply this criterion to determine when geodesic flow fails to be ergodic using Fenchel-Nielsen parameters, providing concrete tools for analyzing the dynamical behavior of these complex geometric structures.