An Explicit Variational Formula for the Monodromy Group

We study the monodromy groups of linearly polymorphic functions on compact Riemann surfaces of genus $g\ge 2$ in connection with standard uniformizations of these surfaces by Kleinian groups. We find necessary and sufficient conditions under which a linearly polymorphic function on a compact Riemann surface gives a standard uniformization of this surface. We study the monodromy mapping $p\colon\mathbf T_gQ\to\mathcal M$, where $\mathbf T_gQ$ is the vector bundle of holomorphic quadratic abelian differentials over the Teichmüller space of compact Riemann surfaces of genus $g$ and $\mathcal M$ is the space of monodromy groups for genus $g$. We prove that $p$ possesses the path lifting property over each space of quasiconformal deformations of the Koebe group of signature $\sigma=(h,s;i_1,\dots,i_m)$ connected with the standard uniformization of a compact Riemann surface of genus $g=|\sigma|$. Moreover, we obtain an explicit variational formula for the monodromy group of a second-order linear differential equation and the first variation for a solution to a Schwartz equation on a compact Riemann surface.