Some problems of conformal mappings of spherical domains (Q2461523)

Let \(\Omega\) be a simply connected domain on the sphere, and let \(f\) be a conformal mapping of \(\Omega\) onto a domain \(D\) in the plane. One of the most used characteristics of the local properties of \(f\) is the scale function \(m\) defined as the ratio between the elementary arc lengths along a planar curve and the respective spherical curve. For the measure of distortion over the entire domain \(\Omega\) the quantity \[ \delta(\Omega) = \frac{\sup_\Omega{m}}{\inf_\Omega{m}} \] is used, and it is called the distortion coefficient. In this paper the authors consider conformal mappings with a scale function depending only on latitude, and for this class of mappings they solve the problem of restoration of conformal mappings. They also consider the problem of minimization of the distortion coefficient, and solve this problem for domains of the form of a spherical disk. Finally, they compare the distortion coefficient for different mappings of spherical domains with the distortion of orthogonal mappings.