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Harmonic mappings and disc bundles over compact Kähler manifolds. (English) Zbl 0601.32023
Theorem 1. Let X be a compact Kähler manifold and $$\Omega\to^{\pi}X$$ a locally trivial holomorphic disc bundle. Then $$\Omega$$ is weakly 1- complete. The proof uses the harmonic sections with respect to the Kähler metric $$ds^ 2_ X$$ on X and the Kähler metric $$ds^ 2$$ induced on $$\Omega$$ by $$ds^ 2_ X$$ and the Poincaré metric $$ds^ 2_ h$$ on $$\Delta$$.
The authors provide the following existence theorem for harmonic sections. Theorem 2. Let $$\Omega\to^{\pi}X$$ be a locally trivial holomorphic disc bundle over the compact Kähler manifold X. Suppose that the corresponding $$P^ 1$$-bundle $${\hat \Omega}\to^{{\hat \pi}}X$$ does not allow a flat section in $$\partial \Omega$$. Then there exists a harmonic section $$s: X\to \Omega.$$
Then Theorem 1 is generalized to Theorem 3. Let X be a compact complex manifold which is bimeromorphically equivalent to a compact Kähler manifold. Then any locally trivial holomorphic disc bundle $$\Omega\to^{\pi}X$$ is weakly 1-complete. An extendibility result for harmonic maps is also given.
Reviewer: V.Oproiu

MSC:
 32F99 Geometric convexity in several complex variables 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32J99 Compact analytic spaces 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 58C05 Real-valued functions on manifolds
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References:
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