Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 681-692

RECIPROCAL LOWER BOUND ON MODULUS OF CURVE FAMILIES IN METRIC SURFACES

Kai Rajala and Matthew Romney

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; kai.i.rajala 'at' jyu.fi

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; matthew.d.romney 'at' jyu.fi

Abstract. We prove that any metric space X homeomorphic to R² with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let Q ⊂ X be a topological quadrilateral with boundary edges (in cyclic order) denoted by ζ₁, ζ₂, ζ₃, ζ₄ and let Γ(ζ_i, ζ_j;Q) denote the family of curves in Q connecting ζ_i and ζ_j; then Mod Γ(ζ₁,ζ₃;Q) Mod Γ(ζ₂,ζ₄;Q) ≥ 1/κ for κ = 2000² · (4/π)². This answers a question in [6] concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in R².

2010 Mathematics Subject Classification: Primary 30L10; Secondary 30C65, 28A75.

Key words: Quasiconformal mapping, uniformization, conformal modulus, coarea inequality.

Reference to this article: K. Rajala and M. Romney: Reciprocal lower bound on modulus of curve families in metric surfaces. Ann. Acad. Sci. Fenn. Math. 44 (2019), 681-692.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4442