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Harmonic maps in Kähler geometry. (English) Zbl 0573.53040

Harmonic mappings and minimal immersions, Lect. 1st 1984 Sess. C.I.M.E., Montecatini/Italy 1984, Lect. Notes Math. 1161, 193-205 (1985).
[For the entire collection see Zbl 0566.00013.]
A Riemannian manifold Y has Hermitian negative curvature at a point if \(R_{jklm} \omega^{il} \omega^{km}\leq 0\) for every positive definite Hermitian matrix \((\omega^{jl})\); strongly so if equality holds only for \((\omega^{jl})\) of rank \(\leq 1\). The author shows that if Y has constant negative curvature, then it is strongly Hermitian negative. Furthermore if \(f: M\to Y\) is a harmonic map of a compact Kähler manifold M into a Riemannian manifold (Y,g’) with Hermitian negative curvature, then (1) \((f^*g')^{2,0}\) is holomorphic; (2) if Y has strongly Hermitian negative curvature, then every such harmonic map f has rank\(\leq 2\). In particular, there is no minimal immersion of a compact Kähler manifold M of \(\dim_{{\mathbb{C}}} M>1\) into a manifold of constant negative curvature.
Reviewer: J.Eells

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E20 Harmonic maps, etc.

Citations:

Zbl 0566.00013