TY - JOUR
T1 - Quasiconformal mappings on the Grushin plane
AU - Gartland, Chris
AU - Jung, Derek
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
AB - We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
KW - Conformal modulus
KW - Grushin plane
KW - Quasiconformal mapping
KW - Sub-Riemannian geometry
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U2 - 10.1007/s00209-017-1851-x
DO - 10.1007/s00209-017-1851-x
M3 - Article
AN - SCOPUS:85012231816
SN - 0025-5874
VL - 287
SP - 915
EP - 928
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 3-4
ER -