Reciprocal lower bound on modulus of curve families in metric surfaces

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 ζi and ζj ; 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.