Uniformization with Infinitesimally Metric Measures

Kai Rajala, Martti Rasimus, Matthew Romney

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to R2. Given a measure μ on such a space, we introduce μ-quasiconformal mapsf: X→ R2, whose definition involves deforming lengths of curves by μ. We show that if μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a μ-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

Original languageEnglish
Pages (from-to)11445-11470
Number of pages26
JournalJournal of Geometric Analysis
Volume31
Issue number11
DOIs
StatePublished - Nov 2021

Keywords

  • Conformal modulus
  • Metric doubling measure
  • Quasiconformal mapping
  • Quasisymmetric mapping

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