Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinite circle patterns in the Weil-Petersson class

Wai Yeung Lam

Abstract

Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an infinite-dimensional Hilbert manifold homeomorphic to the Sobolev space of half-differentiable functions on the unit circle. The Hilbert manifold is equipped with a Riemannian metric induced from the Hessian of a hyperbolic volume functional. We relate this Riemannian metric to the symplectic form on the Sobolev space of half-differentiable functions via an analogue of the Hilbert transform. Every such circle pattern induces a quasiconformal homeomorphism from the unit disk to itself, whose boundary extension belongs to the Weil-Petersson class of the universal Teichmüller space. Our results shed light on Jordan domains packed by infinite circle patterns of hyperbolic type, a subject highlighted by He and Schramm.

Infinite circle patterns in the Weil-Petersson class

Abstract

Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an infinite-dimensional Hilbert manifold homeomorphic to the Sobolev space of half-differentiable functions on the unit circle. The Hilbert manifold is equipped with a Riemannian metric induced from the Hessian of a hyperbolic volume functional. We relate this Riemannian metric to the symplectic form on the Sobolev space of half-differentiable functions via an analogue of the Hilbert transform. Every such circle pattern induces a quasiconformal homeomorphism from the unit disk to itself, whose boundary extension belongs to the Weil-Petersson class of the universal Teichmüller space. Our results shed light on Jordan domains packed by infinite circle patterns of hyperbolic type, a subject highlighted by He and Schramm.
Paper Structure (27 sections, 55 theorems, 260 equations, 6 figures)

This paper contains 27 sections, 55 theorems, 260 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Further related work
  3. Background
  4. Classical Dirichlet-finite functions and half-differentiable functions
  5. Weil-Petersson class in the universal Teichmüller space
  6. Discrete harmonic functions on graphs
  7. Dirichlet-finite functions on graphs and equivalent edge weights
  8. Conjugate discrete harmonic functions
  9. Circle patterns in the Euclidean plane
  10. Intersection angles of circle patterns
  11. Euclidean radii of circle patterns
  12. Infinitesimal deformations of circle patterns
  13. Volume of hyperbolic polyhedra associated with edges
  14. Geometric edge weights equivalent to combinatorial edge weights
  15. Hilbert manifold $P(\Theta,R^{\dagger})$
  16. ...and 12 more sections

Key Result

Theorem 1.3

Let $(V,E,F)$ be a cell decomposition of a topological open disk $\Omega$ and $\Theta : E^* \to (\epsilon_0,\pi - \epsilon_0)$ satisfying Definition def:infintheta. Then there exists a function $R^{\dagger}\in \tilde{P}(\Theta)$ such that the developing map $\mathop{\mathrm{dev}}\nolimits$ is an iso

Figures (6)

  • Figure 1: A uniformized circle pattern with constant intersection angle $\Theta\equiv \pi/2$ filling the unit disk. It is induced from a circle packing together with its dual circle packing.
  • Figure 2: Circle patterns in the deformation space $P(\Theta,R^{\dagger})$ with constant intersection angle $\Theta \equiv \pi/2$. Each arises from a superposition of a circle packing and its dual circle packing. The circle pattern on the bottom right is not embedded in $\mathbb{R}^2$.
  • Figure 3: A circle pattern corresponding to $u \in P(\Theta,R^{\dagger})$ whose boundary value $u_{\partial \mathbf{D}} \in H^{\frac{1}{2}}(\partial \mathbb{D})$ is infinite at one point on $z=1$. Specifically, the boundary value is $u_{\partial \mathbb{D}}(z) = \mathop{\mathrm{Re}}\nolimits \left( \sum_{n=2}^{\infty} \frac{z^n}{n \log n} \right)$.
  • Figure 4: Every dual edge $\phi \psi$ is associated with a kite formed by the two circumcenters of the faces $\phi, \psi$ and the two intersection points of their circumcircles. The side lengths of the kite are the radii $R_{\phi}, R_{\psi}$, and the intersection angle between the circles is $\Theta_{\phi\psi}$. We denote the half-angles at the circumcenters by $\alpha_{\phi \psi}$ and $\alpha_{\psi \phi}$.
  • Figure 5: The hyperbolic polyhedron associated with an edge is obtained by gluing two hyperbolic tetrahedra. The signed hyperbolic volume of the polyhedron is given by the sum of the Lobachevsky functions of the half-angles at the circumcenters.
  • ...and 1 more figures

Theorems & Definitions (118)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Discrete uniformization HeSchramm1993Ge2025
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 108 more