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
**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* ⊂ *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
Mod Γ(*ζ*_{1},*ζ*_{3};*Q*)
Mod Γ(*ζ*_{2},*ζ*_{4};*Q*)
≥ 1/κ for κ = 2000^{2} · (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 **R**^{2}.

**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

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