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

Martin, Gaven

doi:10.1090/S0002-9947-2015-06444-7

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