Dynamics Seminar

An infinite Riemann surface is closest to a compact Riemann surface if it does not support a Green’s function. This condition is equivalent to the ergodicity of the geodesic flow. We give another characterization of this class of Riemann surfaces using the space of finite-area holomorphic quadratic differentials and prove that holomorphic quadratic differentials with single cylinders are dense among all holomorphic quadratic differentials.

Then, we show that a larger class of Riemann surfaces without non-constant harmonic functions with finite Dirichlet integral is quasiconformally invariant. The proof uses the trajectory structure of finite-area holomorphic quadratic differentials.