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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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[1] Bedford, E., Taylor, B. A., The Dirichlet problem for the complex Monge-Ampere equation, Invent. Math., 37 (1976), 1-44. · Zbl 0315.31007 · doi:10.1007/BF01418826 · eudml:142425
[2] Diederich, K., Ohsawa, T., A Levi problem on two-dimensional complex manifolds. Math. Ann., 261 (1982), 255-261. · Zbl 0502.32010 · doi:10.1007/BF01456222 · eudml:182878
[3] Diederich, K., Fornaess, J. E., A smooth pseudoconvex domain without pseudocon- vex exhaustion, Manuscripta math., 39 (1982), 119-123. · Zbl 0524.32010 · doi:10.1007/BF01312449 · eudml:154876
[4] Eells, J., Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. · Zbl 0122.40102 · doi:10.2307/2373037
[5] Grauert, H., On Levi’s problem and the embedding of real-analytic manifolds, Ann. Math., 68 (1958), 460-472. · Zbl 0108.07804 · doi:10.2307/1970257
[6] _ ., Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Zeitsckrift, 81 (1963), 377-391.
[7] Hamilton, R. S., Harmonic maps of manifolds with boundary, LN in Mathematics, vol. 471. Berlin-Heidelberg-New York 1975. · Zbl 0308.35003
[8] Narasimhan, R., The Levi problem in the theory of functions of several complex variables, ICM Stockholm 1962. · Zbl 0116.06103
[9] Ohsawa, T., Cohomology vanishing theorems on weakly 1 -complete manifolds, Publ. RIMS Kyoto, 19 (1983), 1181-1201. · Zbl 0537.32014 · doi:10.2977/prims/1195182026
[10] _ ., On the domain of existence for P-pluriharmonic functions, Publ. RIMS Kyoto, 20 (1984), 1021-1024. · Zbl 0571.31004 · doi:10.2977/prims/1195180878
[11] Oka, K., Domains pseudoconvexes, Tohoku Math. J., 49 (1942), 15-52. · Zbl 0060.24006
[12] Sacks, J., Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Ann. Math., 113 (1981), 1-24. · Zbl 0462.58014 · doi:10.2307/1971131
[13] Siu, Y. T., The complex analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. Math., 112 (1980), 73-111. · Zbl 0517.53058 · doi:10.2307/1971321
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