Quasiconformal curves and quasiconformal maps in metric spaces

In this article, we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> Ω → {{R}}ᵐ from some domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> Ω ⊂ {{R}}ⁿ to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> {{R}}ᵐ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>m</m:mi> </m:math> n≤ m , belong in a suitable Sobolev class and satisfy a certain distortion inequality for some smooth, closed and non-vanishing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n -form in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> {{R}}ᵐ . These mappings can be seen as quasiconformal mappings between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> </m:math> Ω and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> f(Ω ) . We prove that a quasiconformal curve always satisfies the analytic definition of quasiconformal mappings and the lower half of the modulus inequality. Moreover, we give a sufficient condition for a quasiconformal curve to satisfy the metric definition of quasiconformal mappings. We also show that a quasiconformal map from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> </m:math> Ω to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> </m:math> f(Ω )⊂ {{R}}ᵐ is a quasiconformal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> ω curve for some form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> ω under suitable assumptions. Finally, we show that same is true when