Properties of associated quadratic differentials in certain extremal problems

Abstract

One investigates properties of extremal configurations in the problem of the maximum of the n-th diameter dn (E) (MR 88c:30028).

Some Geometric Properties of Extremal Decompositions

“…Therefore, it is of interest to describe geometric properties of the extremal decompositions in these problems in the general case. For instance, in [7] the author proved that the associated quadratic differential in Problem 3 has no multiple zeros. Geometrically, this means that in contrast with Problem 1, no point of the Riemman sphere can be a boundary point for more than three domains of the extremal decomposition for this problem.…”