ON THE TYPE OF SEWING FUNCTIONS \MITH A SINGULARITY

When a Riemann surface is defined by identifying boundary arcs of plane re-gions, the process is called a conformal sewing. A permissible identifying homeo-morphism can be termed a sewing function, or a function with the sewing property (the terms are from [5]). In the case of sewing along real intervals, the question arises: when does a function with the sewing property on two adjacent intervals possess the global sewing property. This is a removability problem, with the com-mon endpoint taking the role of a singularity. Our assumptions are as follows: Suppose that g is an increasing homeomor-phism between two bounded open intervalsl lei the situation be normalized by the condition p(0) : 0. Suppose further that the function rp is locally quasisymmetric off the point zero. The latter assumption guarantees that 1) the restrictions of cp to both sides of zero admit sewing (of the lower and upper half-planes), and 2) the conformal structure of the resulting (doubly connected) Riemann surface.B is essentially unique. The surface.R has a parabolic or hyperbolic end at zero. Accordingly, th