Projective structures and exact variational formula of monodromy group of the linear differential equations on compact Riemann surface

TL;DR

This paper explores the monodromy groups of linear differential equations on compact Riemann surfaces, establishing conditions for uniformization, analyzing the monodromy mapping, and deriving an exact variational formula for the monodromy group.

Contribution

It provides necessary and sufficient conditions for uniformization via linearly polymorphic functions and derives an exact variational formula for the monodromy group on Riemann surfaces.

Findings

Monodromy mapping has the path-lifting property over certain deformation spaces.

Derived an exact variational formula for the monodromy group.

Established conditions linking polymorphic functions to standard uniformization.

Abstract

In this paper we are investigated the monodromy group for linearly polymorphic functions on compact Riemann surface of genus $g \geq 2,$ in connection with standard uniformization of these surfaces by Kleinian groups, and are found a neccessary and sufficients conditions, that a linearly polymorphic function on compact Riemann surface gave a standard uniformization of this surface. We are investigated the monodromy mapping $p : {\bf T}_{g}Q \rightarrow \mathcal{M},$ where ${\bf T}_{g}Q$ is a vector bundle of holomorphic quadratic abelian differentials over the Teichmueller space of compact Riemann surfaces of genus $g,$ $\mathcal{M}$ is a space of monodromy groups for of genus $g.$ Here is proved that over any space, which consist from quasiconformal deformations by Koebe group of signature $\sigma = (h, s; i_{1},..., i_{m}),$ connected with standard uniformization compact Riemann surface of genus $g = |\sigma|,$ this mapping $p$ has the lifting of path property . Also we are received exact variational formula for monodromy group of the linear differential equation of the second order and the first variation for solution of the Schwartz equation on compact Riemann surface.