

Seminar on Complex Analysis (Gonchar Seminar)
April 25, 2016 17:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)






Families of rational functions uniformizing Riemann surfaces over the sphere
S. R. Nasyrov^{} ^{} Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University

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Abstract:
We deduce differential equation for a smooth oneparametric family of rational functions which maps the Riemann sphere onto a given family of ramified coverings of the sphere. As a corollary, we find a system of ODE for the critical points and the poles of these rational functions; the righthand sides of the equations also depend on the corresponding critical values. The results are used in the problem of uniformization of given compact simplyconnected Riemann surfaces over the sphere. The case of genus one (complex tori) is considered as well.

