The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the
conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean
oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.
