Biholomorphic classification of the tubular tori in \({\mathbb{C}}^ 2\) (Q1073224)

Let \({\mathbb{C}}^ 2={\mathbb{R}}^ 2+i{\mathbb{R}}^ 2\), \(T(r_ 1,r_ 2)=\{x+iy\in {\mathbb{C}}^ 2; | x| =r_ 1, | y| =r_ 2\}\). The author studies the biholomorphic mapping of the manifold \(T(r_ 1,r_ 2)\) on the manifold of the same kind. One of the results is Theorem 1. If there exists a biholomorphic mapping of some neighbourhood of \(T(r_ 1,r_ 2)\) on some neighbourhood of \(T(p_ 1,p_ 2)\) which carries \(T(r_ 1,r_ 2)\) onto \(T(p_ 1,p_ 2)\) then \((r_ 1,r_ 2)=(tp_ 1,tp_ 2)\) or \((r_ 2,r_ 1)=(tp_ 1,tp_ 2)\) for some \(t>0\).