---
abstract: 'We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\zeta_1, \zeta_2, \zeta_3, \zeta_4$ and let $\Gamma(\zeta_i, \zeta_j; Q)$ denote the family of curves in $Q$ connecting $\zeta_i$ and $\zeta_j$; then $\operatorname{mod}\Gamma(\zeta_1, \zeta_3; Q) \operatorname{mod}\Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question in [@Raj:16] concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in $\mathbb{R}^2$.'
address: 'Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland.'
author:
- Kai Rajala
- Matthew Romney
bibliography:
- 'ReciprocalLowerBoundBiblio.bib'
title: Reciprocal lower bound on modulus of curve families in metric surfaces
---

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Introduction
============

The classical uniformization theorem states that any simply connected Riemann surface can be mapped onto either the Euclidean plane $\mathbb{R}^2$, the sphere $\mathbb{S}^2$, or the unit disk $\mathbb{D}$ by a conformal mapping. For obtaining similar results in the setting of metric spaces, the class of conformal mappings is too restrictive 