TY - JOUR
T1 - Uniformization with Infinitesimally Metric Measures
AU - Rajala, Kai
AU - Rasimus, Martti
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2021, The Author(s).
PY - 2021/11
Y1 - 2021/11
N2 - 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.
AB - 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.
KW - Conformal modulus
KW - Metric doubling measure
KW - Quasiconformal mapping
KW - Quasisymmetric mapping
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U2 - 10.1007/s12220-021-00689-y
DO - 10.1007/s12220-021-00689-y
M3 - Article
AN - SCOPUS:85106515855
SN - 1050-6926
VL - 31
SP - 11445
EP - 11470
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
IS - 11
ER -