Reciprocal lower bound on modulus of curve families 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 $\text{mod}\Gamma ({\zeta }_1,{\zeta }_3;Q)\text{mod}\Gamma ({\zeta }_2,{\zeta }_4;Q)\ge 1/\kappa$ 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 ${\mathbb{R}}^2$.

arXiv comment12 pages, to appear in Ann. Acad. Sci. Fenn. Math