One-parametric families of ramified coverings of the sphere and uniformization

We consider smooth one-parametric families of ramified coverings of the sphere by compact Riemann surfaces of a fixed genus $\rho$. In the case $\rho=0$, these coverings are realized by rational functions and for $\rho=1$ the functions are elliptic. The main problem is to describe trajectories of critical points and poles of the uniformizing functions if trajectories of critical values are known.
In the simply-connected case ($\rho=0$), the solution to the problem is given in [1].
In the case of complex tori ($\rho=1$), the problem is more complicated, since, besides of trajectories of critical points and poles, we also need to describe change of modules of the tori. Earlier, the author investigated the problem under the condition that the uniformizing elliptic functions have a unique pole. The case of simple branch-points lying over finite points of the Riemann sphere is described in [2]; the case of arbitrary multiplicities is studied in [3]. Here we give a solution for the general case, when the uniformizing functions of the family can have a few poles of arbitrary multiplicity.
The desired trajectories of critical points and poles are described with the help of a system of ODEs. Solving the Cauchy problem for the system we can approximately find the uniformizing functions for one-parametric families of ramified coverings.
We also give applications our method to some problems of geometric function theory and potential theory.
This work was supported by the Russian Foundation for Basic Research and the government of the Republic of Tatarstan, project No.18-41-160003.