Uniformization with Infinitesimally Metric Measures
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AbstractWe consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to $${{\mathbb {R}}}^2$$ R 2 . Given a measure $$\mu $$ μ on such a space, we introduce $$\mu $$ μ -quasiconformal maps$$f:X \rightarrow {{\mathbb {R}}}^2$$ f : X → R 2 , whose definition involves deforming lengths of curves by $$\mu $$ μ . We show that if $$\mu $$ 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 $$\mu $$ μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.
Keyword(s):
2018 ◽
Vol 11
(4)
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pp. 387-404
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Keyword(s):
2015 ◽
Vol 2015
(1)
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