An Alexander type Theorem for Reinhardt domains of C^2

The authors prove that any smooth bounded Reinhardt domain of $Bbb{C}^2$ whose boundary has constant Levi curvature is a ball. The main difficulty in obtaining such a conclusion is that the method of Aleksandrov cannot be applied, since the fully nonlinear PDE that expresses the fact that the Levi curvature is constant is not elliptic. To overcome this obstacle, the authors exploit the intrinsic additional symmetries of the boundary of any Reinhardt domain, which allow them to reduce the problem to a uniqueness result of global solutions of a singular second-order ODE.