Quadratic differentials and function theory on Riemann surfaces

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: $O_G\subset O_{HB}\subset O_{HD}$.