Holomorphic maps on projective spaces and continuations of Fatou maps (Q2519157)

A holomorphic map \(\varphi\) from a complex analytic space \(R\) into a complex projective space \(\mathbb{P}^{n}\) is said to be a Fatou map for \(f\) if \(\{f^{j}\circ \varphi\}_{j\geq 0}\) is a normal family. This definition was given by \textit{J. W. Robertson} [Int. J. Math. Math. Sci. 2003, No. 19, 1233--1240 (2003; Zbl 1016.37022)] and \textit{T. Ueda} [RIMS Kokyuroku 988, 188--191 (1997; Zbl 0925.32009)] independently. In this paper, the author proves the following two theorems: Theorem 1. Let \(S\) be a Riemann surface and let \(E\) be a closed polar set in \(S\). If \(\varphi :S-E \rightarrow \mathbb{P}^{n}\) is a Fatou map, then \(\varphi\) can be extended to a Fatou map \(\check{\varphi}:S\rightarrow \mathbb{P}^{n}\). Theorem 2. Let \(S\) be a Riemann surface and let \(E\) be a closed polar set in \(S\). If \(\varphi :S \rightarrow \mathbb{P}^{n}\) is a holomorphic map and if \(\varphi(S-E)\) is contained in the Fatou set \(\Omega\), then \(\varphi(S)\) is also contained in \(\Omega\). Theorem 1 may be viewed as an analogue of the Nishino's theorems concerning continuation of a holomorphic map from \(S-E\) into a compact Riemann surface of genus \(\geq 2\) or, more generally, into a complex manifold whose universal cover is a bounded domain of certain type [ Susuki, 1988]. The author's proof is an adaptation of Suzuki's idea. As an application of these theorems, the author makes a remark concerning a result due to Fornæss and Sibony. The author proves that in case \((2)\) of the Fornæss and Sibony's theorem, the manifold \(\Sigma\) is not biholomorphic to the punctured disk.