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Quasiconformal Normalization of Random Meromorphic Functions

Michael Iofin

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.

Quasiconformal Normalization of Random Meromorphic Functions

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.
Paper Structure (10 sections, 14 theorems, 47 equations, 5 figures)

This paper contains 10 sections, 14 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Random Differentials on Partitions of the Plane
  4. Percolation
  5. Rough Quasiconformality
  6. Rescaling
  7. Random Riemann Surfaces
  8. Quasiconformal Deformation of an Elliptic Function
  9. Directions for Further Research
  10. Acknowledgements

Key Result

Theorem 1

If $\mu$ is a probabilistically bounded Beltrami differential on a partition $\mathcal{P}$ with bounded geometry, then the map $w^\mu$ is surjective onto $\mathbb{C}$ with probability 1.

Figures (5)

  • Figure 1: Curve $\gamma$ connecting opposite boundary components of $Q(N, 2N)$
  • Figure 2: Labeling the vertices and dividing $\mathbb{R} \cup \{ \infty \}$ into intervals.
  • Figure 3: The cells of the rectangular grid divided into triangular Jordan regions.
  • Figure 4: The action of $\phi$ on one hemisphere
  • Figure 5: $\mathcal{R}$ is the Riemann surface of $f^{-1}$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition
  • Proposition 5: Ahlfors
  • Proposition 6: Conformal Invariants
  • Proposition 7: Conformal Invariants
  • Proposition 8
  • proof
  • ...and 22 more