Unique extremality of affine maps on plane domains

TL;DR

The paper proves that affine maps are uniquely extremal quasiconformal maps on the complement of any well distributed set $E\subset\mathbb{C}$. It introduces Reich sequences and reduces the problem to complements of quasilattices by showing any $\tfrac{1}{8}$-well distributed set contains a quasilattice, then constructs a meromorphic partition of unity on $\mathbb{C}\setminus L$ via Bergman projections and local quadratic differentials. A Reich sequence $\phi_n(z)=\sum_{k,l} P_{k,l}(z)/( |z_{k,l}|/n+1)^4$ is built from these partitions, and a sequence of estimates ensures $\phi_n$ satisfies Reich’s three conditions, yielding unique extremality. This generalizes prior lattice results (e.g., the integer lattice) to broader well distributed sets and provides a conceptual framework connecting Bergman kernels, partitions of unity, and extremal quasiconformal theory in planar domains.

Abstract

We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman projections, and meromorphic partitions of unity.

Theorems & Definitions (25)

• Remark 1
• Definition 1
• Conjecture 1
• Theorem 1
• Remark 2
• Theorem 2
• Remark 3
• Definition 2
• Lemma 1
• proof