Uniformization by Rectangular Domains and Quadratic Differentials
Vidanage, Bellana Vidanalage Nadeesha Chathuranga
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In the first part of this dissertation we consider uniformizing a given domain in Riemann sphere onto a domain bounded by a rectangles (which is a rectangluar domain) via an extremal problems. We work with the class of functions ( ) which has the expansion f(z) = z + a1 z +· · · near 1 and univalent on a finitely connected domain , 1 2 C. One of the classical results due to G. Grötzsch, the function f0 maximizing S0(f, ) = 2 <a1 over the class ( ) maps onto a domain in C bounded by slits parallel to the real axis. Recently, M. Bonk found a similar extremal problem, which maximizer f1 2 ( ) maps onto a domain on C, whose complementary components are squares. In this dissertation , we present a parametric family of extremal problems on the class ( ) with maximizers fm, 0 < m < 1, mapping onto domains on C, whose complementary components are rectangles with horizontal and vertical sides and with module m. In the second part of the dissertation we focus on conformal mappings and its relation with the quadratic differentials. Our main goal is to give a general existing theorem for conformal mappings on a finitely connected domain 2 C onto a domain bounded by slits on trajectories of a quadratic differential Q(z)dz2.