Аннотация:
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.
Язык доклада: английский
Список литературы
S. Nasyrov, “Uniformization of simply-connected ramified coverings of the sphere by rational functions”, Lobachevskii J. Math., 39:2 (2018), 252–258
S. R. Nasyrov, “Uniformization of one-parameter families of complex tori”, Izv. Vyssh. Uchebn. Zaved. Mat., 2017, no. 8, 42–52
S. R. Nasyrov, “Families of elliptic functions and uniformization of complex tori with a unique point over infinity”, Probl. Anal. Issues Anal., 7(25):2 (2018), 98–111