Harmonic mappings and disc bundles over compact Kähler manifolds (Q1081000)

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.