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A note on harmonic maps between surfaces. (English) Zbl 0585.58011
The author studies the relation between harmonic maps between surfaces and holomorphic quadratic differentials. If \(\Sigma_ 1\) and \(\Sigma_ 2\) are surfaces with conformal metrics \(\sigma^ 2dzd\bar z\) and \(\rho^ 2dud\bar u\), resp., and \(u: \Sigma\) \({}_ 1\to \Sigma_ 2\) is harmonic, then \(\phi:=\rho^ 2u_ z\bar u_ zdz^ 2\) is a holomorphic quadratic differential on \(\Sigma_ 1\). It has been an open problem to which extent the converse is true, i.e. whether a map with holomorphic \(\phi\) is harmonic. A variational procedure is invented that produces a map with holomorphic \(\phi\) in every homotopy class of maps between closed surfaces. Some other related results are also discussed in detail.
Reviewer: T.Rassias

MSC:
58E20 Harmonic maps, etc.
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