Convexity and Teichmüller spaces (Q2357040)

It is well known that the Teichmüller spaces with their canonical complex structure are pseudoconvex. Moreover, all finite-dimensional Teichmüller spaces are Runge domains, hence polynomially convex. An old open question in this theory is whether any finite- or infinite-dimensional Teichmüller space \(T(\Gamma)\) (of dimension greater than one) is biholomorphically equivalent to a convex domain in a complex Banach space \(X\) (of the same dimension as \(T(\Gamma)\)). It was posed for finite-dimensional spaces by Royden and for the universal Teichmüller space by Sullivan. The author provides a negative answer to Royden's problem whether any finite-dimensional Teichmüller space of dimension greater than 1 is biholomorphically equivalent to a bounded convex domain in complex Euclidean space. He proves that any Teichmüller space \(T(0,n)\) of punctured spheres with a sufficiently large number \(n \geq n_{0} > 4\) of punctures cannot be mapped biholomorphically onto a bounded convex domain in \(\mathbb C^{n}.\)