Convexity and Teichmüller spaces

Resumo

We provide a negative answer to Royden’s problem whether any finite dimensional Teichmüller space of dimension greater than 1 is biholomorhically equivalent to a bounded convex domain in complex Euclidean space.

We prove that any Teichmüller space T(0, n) of punctured spheres with a sufficiently large number n ≥ n₀ > 4 of punctures cannot be mapped biholomorphically onto a bounded convex domain in ℂⁿ⁻³.