This preprint was accepted February, 1997.
Contact: A. Yu. Solynin
The paper deals with problems on an extremal partition of a finite Riemann surface into configurations of a special topological form. The foundations of this theory, closely relative to the theory of quadratic differentials, were established by J. A. Jenkins. A lot of important results were also proved by K. Strebel, H. Renelt, G. V. Kuz'mina, and others. The present paper includes a detailed exposition of this theory for the problems associated with quadratic differentials that have poles at most second order with circular and radial structure of trajectories. In particular, we begin by proving the existence and uniqueness theorems for the extremal configurations then develop the theory of differentiability for the weighted sum of extremal moduli. Bibliography--71.