The classification of proper holomorphic mappings between special Hartogs triangles of different dimensions (Q950685)

The proper holomorphic mapping theory dates from the 1950s, and there are many good results on it. The classification of proper holomorphic mappings is an important and difficult problem, especially between bounded domains of different dimensions. By a classical result of \textit{H. Alexander} [Math. Ann. 209, 249--256 (1974; Zbl 0272.32006)], every proper holomorphic self-mapping of \(B^{n}\) with \(n\geq 2\) is equivalent to the identity mapping. In 2001, \textit{X. Huang} and \textit{S. Ji} [Invent. Math. 145, No. 2, 219--250 (2001; Zbl 1038.32018)] proved that there are only two equivalence classed in \(\text{Rat}(B^{n},B^{N})\) for \(n>2\). In 2006, \textit{X. Huang, D. Xu} and \textit{S. Ji} [Math. Res. Lett. 13, No. 4, 515--529 (2006; Zbl 1158.32007)] found a new gap phenomenon for proper holomorphic mappings from \(B^{n}\) to \(B^{N}\) when \(N\leq 3n-4\). In this paper, the authors consider the special Hartogs triangles defined as follows: \[ \begin{aligned} \Omega(n_{1},m_{1})&= \bigg\{(z,w)\in\mathbb C^{n_{1}+m_{1}}: 0<\sum_{i=1}^{n_{1}}|z_{i}|^{2}<\sum_{j=1}^{m_{1}}|w_{j}|^{2}<1 \bigg\}\\ \Omega(n_{2},m_{2})&= \bigg\{(z',w')\in\mathbb C^{n_{2}+m_{2}}: 0<\sum_{i=1}^{n_{2}}|z_{i}'|^{2}<\sum_{j=1}^{m_{2}}|w_{j}'|^{2}<1\bigg\}. \end{aligned} \] They discuss proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem. The main idea is from [\textit{Z. H. Chen}, and \textit{Y. C. Liu}, Chin. J. Contemp. Math. 24, No.~3, 215--220 (2003; Zbl 1069.32500)].