Each homotopically homogeneous tube in \(\mathbb C^2\) has an affine-homogeneous base (Q2773683)

A set of the form \(\Gamma + i{\mathbb R}^n\) in the \(n\)-dimensional complex affine space with \(\Gamma\) a hypersurface in \({\mathbb R}^n\) is called a tube. It is said that such a set is holomorphically homogeneous if any two points of this set have biholomorphically equivalent neighborhoods. In particular, spherical tubes which are by definition holomorphically equivalent to a sphere are holomorphically homogeneous. NEWLINENEWLINENEWLINEIn this paper it is proved that, for every nonspherical tube in \({\mathbb C}^2\), the set \(\Gamma\) is affine homogeneous, i.e. each two points of this set have affine equivalent neighborhoods.