Reciprocal lower bound on modulus of curve families in metric surfaces

TL;DR

This paper establishes a reciprocal lower bound on the modulus of curve families in metric surfaces homeomorphic to , advancing understanding of conditions for quasiconformal parametrizations in such spaces.

Contribution

It proves a specific reciprocal lower bound on modulus of curve families in metric surfaces homeomorphic to  with finite Hausdorff measure, addressing minimal conditions for quasiconformal mappings.

Findings

Established a quantitative lower bound on the product of moduli of curve families in metric surfaces.

Provided conditions under which metric spaces admit quasiconformal parametrizations.

Answer to a key question on minimal hypotheses for quasiconformal mappings in metric surfaces.

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 $\text{mod} \Gamma(\zeta_1, \zeta_3; Q) \text{mod} \Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in $\mathbb{R}^2$.