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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

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
Full Text: DOI
[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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