Infinite circle patterns in the Weil-Petersson class

TL;DR

This paper explores infinite circle patterns in the Euclidean plane linked to Weil-Petersson quasicircles, revealing their geometric structure, associated Riemannian metrics, and connections to Teichmüller theory.

Contribution

It introduces a new infinite-dimensional Hilbert manifold of circle patterns, relates its metric to hyperbolic volume, and connects boundary extensions to Weil-Petersson class homeomorphisms.

Findings

The space of such circle patterns forms a Hilbert manifold homeomorphic to a Sobolev space.

The Riemannian metric is derived from the Hessian of a hyperbolic volume functional.

Boundary extensions of patterns belong to the Weil-Petersson class of the universal Teichmüller space.

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 Teichm\"uller space. Our results shed light on Jordan domains packed by infinite circle patterns of hyperbolic type, a subject highlighted by He and Schramm.