Calculus of Variations and Geometric Measure Theory

K. Rajala - M. Romney

Reciprocal lower bound on modulus of curve families in metric surfaces

created by romney on 15 Jan 2019


Accepted Paper

Inserted: 15 jan 2019
Last Updated: 15 jan 2019

Journal: Ann. Acad. Sci. Fenn. Math.
Year: 2018

ArXiv: 1801.08223 PDF


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 $ζ_1, ζ_2, ζ_3, ζ_4$ and let $Γ(ζ_i, ζ_j; Q)$ denote the family of curves in $Q$ connecting $ζ_i$ and $ζ_j$; then $\text{mod } Γ(ζ_1, ζ_3; Q) \text{mod } Γ(ζ_2, ζ_4; Q) \geq 1/κ$ for $κ= 2000^2\cdot (4/π)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in $\mathbb{R}^2$.

Tags: GeoMeG