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Transactions of the American Mathematical Society

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A $ C^m$ Whitney extension theorem for horizontal curves in the Heisenberg group


Authors: Andrea Pinamonti, Gareth Speight and Scott Zimmerman
Journal: Trans. Amer. Math. Soc. 371 (2019), 8971-8992
MSC (2010): Primary 53C17; Secondary 54C20
DOI: https://doi.org/10.1090/tran/7806
Published electronically: March 26, 2019
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Abstract: We characterize those mappings from a compact subset of $ \mathbb{R}$ into the Heisenberg group $ {\mathbb{H}}^{n}$, which can be extended to a $ C^{m}$ horizontal curve in $ \mathbb{H}^{n}$. The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.


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Additional Information

Andrea Pinamonti
Affiliation: Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy
Email: Andrea.Pinamonti@unitn.it

Gareth Speight
Affiliation: Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, Ohio 45221
Email: Gareth.Speight@uc.edu

Scott Zimmerman
Affiliation: Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, Connecticut 06269
Email: Scott.Zimmerman@uconn.edu

DOI: https://doi.org/10.1090/tran/7806
Keywords: Heisenberg group, horizontal curve, Whitney extension theorem
Received by editor(s): July 20, 2018
Received by editor(s) in revised form: January 12, 2019
Published electronically: March 26, 2019
Additional Notes: Part of this work was done while the first author was visiting the University of Cincinnati. This visit was partly supported by a Research Support Grant from the Taft Research Center at the University of Cincinnati.
The work of the second author was supported by a grant from the Simons Foundation (#576219).
Article copyright: © Copyright 2019 American Mathematical Society