Extremal length and duality

TL;DR

This paper explores advanced duality properties of extremal length, extending classical conformal invariance concepts to diverse settings including metric surfaces, higher-dimensional quasiconformal maps, and multiply connected domains.

Contribution

It develops a unified abstract theory of extremal length focusing on duality and applies it to new contexts like metric surface uniformization and higher-dimensional quasiconformal analysis.

Findings

Demonstrates the flexibility of extremal length theory in various geometric settings

Provides new duality results for extremal length in higher dimensions

Extends extremal length concepts to multiply connected domains and transboundary contexts

Abstract

Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In ``Extremal length and functional completion'', Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings: Extremal length and uniformization of metric surfaces, Extremal length of families of surfaces and quasiconformal maps between $n$-dimensional spaces, and Schramm's transboundary extremal length and conformal maps between multiply connected plane domains.