Quadratic differentials and function theory on Riemann surfaces

TL;DR

This paper characterizes when measured foliations on Riemann surfaces are realized by finite-area quadratic differentials, extending classical results to more general surfaces and exploring invariance properties of certain classes of Riemann surfaces.

Contribution

It extends the Hubbard-Masur realization theorem to arbitrary Fuchsian groups and bounded geometry surfaces, and investigates invariance of classes of surfaces supporting harmonic functions.

Findings

Measured foliations with finite Dirichlet integral correspond to quadratic differentials.

The realization theorem is extended to infinite Riemann surfaces with bounded geometry.

The class of surfaces supporting only constant harmonic functions is invariant under quasiconformal maps.

Abstract

A finite-area holomorphic quadratic differentials on an arbitrary Riemann surface $X=\mathbb{H}/\Gamma$ is uniquely determined by its horizontal measured foliation. By extending our prior result for $\Gamma$ of the first kind to arbitrary Fuchsian group $\Gamma$, we obtain that a measured foliation $\mathcal{F}$ is realized by the horizontal foliation of a finite-area holomorphic quadratic differential on $X$ if and only if $\mathcal{F}$ has finite Dirichlet integral. We determine the image of this correspondence when the infinite Riemann surface has bounded geometry -- an extension of the realization result of Hubbard and Masur for compact surfaces. A corollary is that a planar surface $X$ with bounded pants decomposition and with (at most) countably many ends is parabolic, i.e., does not support Green's function, in notation $X\in O_G$ where $G$ is Green's function. The class of harmonic functions with finite Dirichlet integral is denoted by $HD$. We give a geometric proof that the class $O_{HD}$ of the Riemann surfaces (that do not support non-constant $HD$-functions) is invariant under quasiconformal maps. Lyons proved that the $O_{HB}$ class (surfaces that do not support non-constant bounded harmonic functions) is not invariant under quasiconformal maps, and it is well-known that the $O_G$ class is invariant. Therefore, the noninvariant class $O_{HB}$ is between two invariant classes: $O_G\subset O_{HB}\subset O_{HD}$.