[论文解读] Quasiconformal Normalization of Random Meromorphic Functions

Michael Iofin|arXiv (Cornell University)|Mar 16, 2026

Meromorphic and Entire Functions被引用 0

一句话总结

这篇论文构建了从分布在球面上的曲面出发的随机拟共形正规化的随机有理函数，几乎必然的抛物型并给出 Nevanlinna 特性增长界限。

ABSTRACT

We study the conformal type of surfaces spread over the sphere via random quasiconformal maps. Constructing a random Beltrami coefficient on the complex plane, we obtain a locally quasiconformal homeomorphism with prescribed dilatation that is almost surely surjective and, with high probability, approximately linear. This yields a normalization for random meromorphic functions associated to surfaces spread over the sphere, from which we prove that the surfaces are almost surely parabolic and obtain bounds on the growth order of their Nevanlinna characteristic.

研究动机与目标

• Motivate the conformal type problem for surfaces spread over the sphere and its connection to random meromorphic functions.
• Develop a probabilistic construction of Beltrami coefficients on partitions of the plane with bounded geometry.
• Establish surjectivity and rough quasiconformality of the associated random maps to infer normalization results.
• Deduce almost sure parabolicity of the random surfaces and bound the growth (Nevanlinna) characteristics of the corresponding meromorphic functions.

提出的方法

• Partition the plane into a countable collection of Jordan regions with bounded geometry.
• Define a random Beltrami coefficient by independently selecting per-region dilatations from prescribed distributions.
• Use truncated Beltrami differentials and a limiting argument to construct a locally quasiconformal map w^{μ} with Beltrami coefficient μ.
• Prove probabilistic boundedness and, under periodicity, approximate linearity on large disks (rough quasiconformality).
• Apply percolation-type estimates to compare chemical and Euclidean distances, enabling modulus control and surjectivity results.
• Derive consequences for associated meromorphic functions, including parabolic type and bounds on the Nevanlinna characteristic.]
• research_questions46001: [
• Can a random Beltrami coefficient on a partition with bounded geometry yield a surjective quasiconformal map almost surely?
• Under periodicity, can the random map approximate a linear map on large scales with high probability?
• What is the conformal type (parabolic/hyperbolic) of surfaces spread over the sphere under the random construction?
• What bounds can be obtained for the growth of Nevanlinna characteristics of meromorphic functions arising from these random surfaces?

实验结果

研究问题

• RQ1 Can a random Beltrami coefficient on a partition with bounded geometry yield a surjective quasiconformal map almost surely?
• RQ2 Under periodicity, can the random map approximate a linear map on large scales with high probability?
• RQ3 What is the conformal type (parabolic/hyperbolic) of surfaces spread over the sphere under the random construction?
• RQ4 What bounds can be obtained for the growth of Nevanlinna characteristics of meromorphic functions arising from these random surfaces?

主要发现

• The random Beltrami map w^{μ} is surjective on ℂ with probability 1 under probabilistically bounded dilatations.
• Under periodicity, w^{μ} is close to a linear map on large disks with high probability, giving a rough quasiconformal control.
• Almost all surfaces in the chosen probabilistic model are parabolic.
• The associated meromorphic functions have order at least 2 and lower order at most 2 with high probability.