On existence of quadratic differentials with prescribed properties

G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 13, Tome 226 (1996), pp. 120-137

Résumé

The paper continues the author's studies of the question on the existence of quadratic differentials $Q(z)dz^2$ having given structure of trajectories and poles of high orders. It is shown that such differentials can be considered as the limits of sequences of quadratic differentials that have poles of second order with trajectories asymptotically similar to logarithmic spirals and realize extremal configurations in suitable families of nonoverlapping domains. It is established that there exist differentials $Q(z)dz^2$ of indicated form having given initial terms of the Laurent expansions in the vicinities of the poles of $Q(z)dz^2$ of order not smaller than three. Some discrepancies in an earlier paper are corrected. Bibl. 9 titles.