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Transactions of the American Mathematical Society

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Harmonic degree 1 maps are diffeomorphisms: Lewy's theorem for curved metrics


Author: Gaven J. Martin
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
Published electronically: January 29, 2015
MathSciNet review: 3413878
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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
Email: g.j.martin@massey.ac.nz

DOI: https://doi.org/10.1090/S0002-9947-2015-06444-7
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
Article copyright: © Copyright 2015 American Mathematical Society