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Harmonic diffeomorphisms and conformal distortion of Riemann surfaces. (English) Zbl 1024.53044
Author’s abstract: We study the change of conformal structure induced by harmonic diffeomorphisms between Riemann surfaces. The main result of this paper is to answer the following question raised by R. Schoen [Lect. Notes Pure Appl. Math. 143, 179-200 (1993; Zbl 0806.58013)]. Is it true that Riemann surfaces which are related by a surjective harmonic diffeomorphism are necessarily quasiconformally related? We show that there exists a pair of Riemann surfaces of infinite topological type, which are related by a surjective harmonic diffeomorphism but which are not quasiconformally related. Also we characterize when the above question has a positive answer in the case of Riemann surfaces of finite topological type.
MSC:
53C43Differential geometric aspects of harmonic maps
30F15Harmonic functions on Riemann surfaces
30F45Conformal metrics (hyperbolic, Poincaré, distance functions)
58E20Harmonic maps between infinite-dimensional spaces