On the existence of bounded holomorphic functions on complete Kähler manifolds (Q1073218)

Sufficient conditions are given for the existence of bounded (non- constant) holomorphic functions on a complete Kähler manifold \(M^ n\), \(n\geq 2\). Modified exponential coordinates and asymptotic conditions on the curvature are used to determine the asymptotic behavior of the metric and the J operator. This leads to a smooth CR structure on \(\partial M\) (diffeomorphic to \(\partial {\mathbb{B}}^{2n})\) for which CR functions on \(\partial M\) extend to holomorphic functions on M. For \(n\geq 3\), an imbedding theorem of \textit{L. Boutet de Monvel} [Sémin. Goulaouic-Lion- Schwartz 1974-1975, Exposé IX, 13 p. (1975; Zbl 0317.58003)] provides smooth non-constant CR functions on \(\partial M\). Thus, bounded holomorphic functions exist for \(M^ n\), \(n\geq 3\), a complete Kähler manifold with non-positive sectional curvature under some condition for suitable exponential coordinates.