Polygons and non-starlikeness of Teichmüller spaces

The problem of starlikenness of Teichmüller spaces in Bers’ embedding has been raised in 1974 and is solved (negatively) for Teichmüller spaces of sufficiently large dimensions.

The original proof given by the author relies on the existence of conformally rigid domains established by Thurston. Later the author found another proof of non-starlikeness of universal Teichmüller space based on geometric features of rectilinear polygons.

This paper provides a complete solution of the problem for Teichmüller spaces T(g, 0) of closed Riemann surfaces of genus g ≥ 2.