Quasiregular maps on Carnot groups |
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Authors: | Juha Heinonen Ilkka Holopainen |
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Institution: | (1) Department of Mathematics, University of Michigan, 48109 Ann Arbor, MI, USA;(2) Department of Mathematics, University of Helsinki, Yliopistonkatu 5, P.O. Box 4, FIN-00014, Finland |
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Abstract: | In this paper we initiate the study of quasiregular maps in a sub-Riemannian geometry of general Carnot groups. We suggest
an analytic definition for quasiregularity and then show that nonconstant quasiregular maps are open and discrete maps on
Carnot groups which are two-step nilpotent and of Heisenberg type; we further establish, under the same assumption, that the
branch set of a nonconstant quasiregular map has Haar measure zero and, consequently, that quasiregular maps are almost everywhere
differentiable in the sense of Pansu. Our method is that of nonlinear potential theory. We have aimed at an exposition accessible
to readers of varied background.
Dedicated to Seppo Rickman on his sixtieth birthday
J.H. was partially supported by NSF, the Academy of Finland, and the A. P. Sloan Foundation. I.H. was partially supported
by the EU HCM contract no. CHRX-CT92-0071. |
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Keywords: | Math Subject Classification" target="_blank">Math Subject Classification 30C65 31C45 58G03 |
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