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

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 Rad\'o and Kneser. R. Shoen and S.T. 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.