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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:
 5.8e+21 Harmonic maps, etc.
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##### References:
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