A generalisation of a theorem of Fornaess-Sibony (Q1070401)

On any complex manifold M one may consider those tangent vectors \(\xi\) with \(F_ M(p,\xi)=0\), where \(F_ M\) denotes the infinitesimal Kobayashi metric. In general settings it is difficult to say very much about this because one can find examples of manifolds M such that \(\{\xi | F_ M(p,\xi)=0\}\) is a vector subspace of the tangent space whose dimension \((=:\) the corank of \(F_ M)\) depends on the point \(p\in M.\) One possibility is to assume M is exhausted by biholomorphic images of a fixed manifold \(\Omega\). If \(\Omega\) is hyperbolic and \(\Omega\) /Aut(\(\Omega)\) is compact, \textit{J. E. Fornaess} and \textit{N. Sibony} [Math. Ann. 255, 351-360 (1981; Zbl 0438.32012)] showed that the corank of \(F_ M\) is independent of p. Further, if this corank is one, then they proved that M is biholomorphic to a locally trivial holomorphic fiber bundle with fiber \({\mathbb{C}}\). In the present work the author considers manifolds M exhausted by biholomorphic images of a fixed strongly pseudoconvex \(C^ 2\) domain \(\Omega \subset \subset {\mathbb{C}}^ n\) and proves analogous results in this set-up. The proof uses the fact that such \(\Omega\) are taut and an argument of \textit{A. Kodama} [''On complex manifolds exhausted by biholomorphic images of a strictly pseudoconvex domain'', Preprint 1984] is needed to handle the case where a subsequence of maps converges to a constant map into the boundary of \(\Omega\). \(\{\) The reviewer has some results in the homogeneous setting (''On the Kobayashi pseudometric reduction of solv-manifolds'', to appear)\(\}\).