Reciprocal lower bound on modulus of curve families in metric surfaces

Kai Rajala, Matthew Romney

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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 (ζi, ζj ;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.

Original languageEnglish
Pages (from-to)681-692
Number of pages12
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume44
Issue number2
DOIs
StatePublished - 2019

Keywords

  • Coarea inequality
  • Conformal modulus
  • Quasiconformal mapping
  • Uniformization

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