Rigidity of proper holomorphic mappings between certain nonequidimensional unbounded non-hyperbolic domains (Q6071775)

This is a paper about the rigidity of proper holomorphic maps between Fock-Bargmann-Hartogs domains, which are defined by \[ D_{n,m}(\mu):=\big\{(z,w)\in\mathbb C^n\times\mathbb C^m:\|w\|^2<e^{-\mu\|z\|^2}\big\}, \] where \(\mu>0\). The author first proves that every rational proper holomorphic map from \(D_{n,m}(\mu)\) to \(D_{n,M}(\mu)\) is equivalent to the standard linear embedding if \(m\geq 3\) and \(M<2m-1\). If one considers the projection from \(D_{n,m}(\mu)\) to \(\mathbb C^n\) given by \((z,w)\mapsto z\), then the fiber over each fixed \(z_0\in\mathbb C^n\) is a Euclidean complex ball of dimension \(m\). So one of the key ingredients of the proof is to show that such a proper holomorphic map preserves these fibers. This is achieved by the author with some basic tools from Nevanlinna theory. He can then apply the well-known rigidity result for proper holomorphic maps from \(\mathbb B^m\) to \(\mathbb B^M\) when \(M<2m-1\) and deduce the desired result. The other result of the paper is concerned with a proper holomorphic map from \(D_{n,m}(\mu)\) to \(D_{n,m+1}(\mu)\) (\(m\geq 3\)) that is only assumed to be \(\mathcal C^2\) on the boundary. The author proves that such a map is again equivalent to the standard embedding. The hard part of his proof is to show that such a map extends holomorphically across a dense open subset of the boundary, and this is achieved by a long and hard computation involving taking derivatives with respect to the CR vector fields and solving a system of linear equations. After establishing such extension, the rest of the argument is parallel to that of the rational case.