2021/08/09 by G. P. Balakumar, Diganta Borah, Balakumar, G. P. +5
Mathematics · #32F45 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32F45
paper · pdf · doi:10.48550/arxiv.2108.03951
arxiv created 2021/08/09 · arxiv updated 2021/08/10
Let M be a complex manifold which admits an exhaustion by open subsets Mj each of which is biholomorphic to a fixed domain Ω⊂ \mathbb Cn. The main question addressed here is to describe M in terms of Ω. Building on work of Fornaess--Sibony, we study two cases namely, M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when Ω is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in \mathbb C2, or (iv) a simply connected domain in \mathbb C2 with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when Ω is the minimal ball or the symmetrized polydisc in \mathbb Cn can also be handled. When the Kobayashi metric on M has corank one and Ω is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre \mathbb C over a holomorphic retract of Ω or that of a limiting domain associated with it. Finally, when Ω= Δ× \mathbb Bn-1, the product of the unit disc Δ⊂ \mathbb C and the unit ball \mathbb Bn-1 ⊂ \mathbb Cn-1, a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and Ω= Δ× \mathbb Bn-1, it is shown that M is biholomorphic to Ω. Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to Z × \mathbb C, where Z ⊂ Ω= Δ× \mathbb Bn-1 is a holomorphic retract.