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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Harmonic degree 1 maps are diffeomorphisms: Lewy’s theorem for curved metrics
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by Gaven J. Martin PDF
Trans. Amer. Math. Soc. 368 (2016), 647-658 Request permission

Abstract:

In 1936, H. Lewy showed that the Jacobian determinant of a harmonic homeomorphism between planar domains does not vanish and thus the map is a diffeomorphism. This built on the earlier existence results of Rado and Kneser. Schoen and Yau generalised this result to degree $1$ harmonic mappings between closed Riemann surfaces. Here we give a new approach that establishes all these results in complete generality.
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Additional Information
  • Gaven J. Martin
  • Affiliation: NZ Institute for Advanced Study, Massey University, Auckland, New Zealand — and — Department of Mathematics, Magdelen College, University of Oxford, Oxford OX1 4AU, United Kingdom
  • MR Author ID: 120465
  • Email: g.j.martin@massey.ac.nz
  • Received by editor(s): December 1, 2013
  • Published electronically: January 29, 2015
  • Additional Notes: Research supported in part by grants from the N.Z. Marsden Fund
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 647-658
  • MSC (2010): Primary 30C62, 58E20
  • DOI: https://doi.org/10.1090/S0002-9947-2015-06444-7
  • MathSciNet review: 3413878