Mappings with finite length distortion and Riemann surfaces

Abstract

We prove a series of criteria in terms of dilatations for the continuous and homeomorphic extension of the map pings with finite length distortion between domains on Riemann surfaces to the boundary. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be a very simple condition on the integrability of the dilatations in the first power. Moreover, the domain of the mapping is assumed to be locally connected on the boundary and its range has a weakly flat boundary. The criteria for the continuous extension of the direct map pings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at the corresponding boundary point. A stronger (but simpler) one is that the mean value of the dilatation over infinitesimal disks centered at the corresponding boundary point is finite. The domain is again assumed to be locally connected on the boundary and its range has a strongly accessible boundary. We give also many other criteria for the continuous extension of the direct mappings to the boundary. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains are obtained.