Rigidity of proper holomorphic self-mappings of the hexablock

Abstract: The hexablock (\mathbb{H}), introduced by Biswas-Pal-Tomar \cite{Hexablock}, is a Hartogs domain in (\mathbb{C}^4) fibered over the tetrablock (\mathbb{E}) in (\mathbb{C}^3), arising in the context of (\mu)-synthesis problems. In this paper, we prove that every proper holomorphic self-map of (\mathbb{H}) is necessarily an automorphism. Consequently, we resolve the conjecture (G(\mathbb{H}) = \mathrm{Aut}(\mathbb{H})) on the automorphism group structure, originally posed by Biswas-Pal-Tomar in \cite{Hexablock}.