2024/07/23 by Dragomir Šarić, Saric, Dragomir · 3 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #Meromorphic and Entire Functions #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2407.16333
openalex publication_date 2024/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A finite-area holomorphic quadratic differentials on an arbitrary Riemann surface X=ℍ/Γ is uniquely determined by its horizontal measured foliation. By extending our prior result for Γ of the first kind to arbitrary Fuchsian group Γ, we obtain that a measured foliation F is realized by the horizontal foliation of a finite-area holomorphic quadratic differential on X if and only if 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∈ OG 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 OHD of the Riemann surfaces (that do not support non-constant HD-functions) is invariant under quasiconformal maps. Lyons proved that the OHB class (surfaces that do not support non-constant bounded harmonic functions) is not invariant under quasiconformal maps, and it is well-known that the OG class is invariant. Therefore, the noninvariant class OHB is between two invariant classes: OG⊂ OHB⊂ OHD.