Behavior of the riemann mapping function under complex analytic deformations of the domain

Abstract

Suppose that the boundary of a simply connected domain depends holomorphically on a complex parameter. Then the domain will depend on that parameter: one can ask how its normalized Riemann mapping function will depend on the parameter. Previously known results have required smoothness conditions on the boundary and have restricted the parameter to real values. In this paper we show that the Improved Lambda Lemma can be used to remove these requirements and restrictions.

AMS (MOS):

• 30C35

^*This research was partially supported by the National Science Foundation.

^*This research was partially supported by the National Science Foundation.

Notes

^*This research was partially supported by the National Science Foundation.