Extremal Conformal Mappings and Poles of Quadratic Differentials

Abstract

The stability of the configuration of a non-linear electromagnetic field corresponding to a charged point source is studied. The static classical solution of the field equations, describing charged particles has been modified by a small perturbation obeying linearised field equations. Assuming time dependence of the perturbation in the form exp(-iwt) it has been shown that w 2 is non-negative and the classical static configuration of the field describing charged particles is stable. Detailed analysis of the linearised field equations has been performed for the Born-Infeld model of the non-linear electromagnetic field. There is only a continuous spectrum of w 2 with certain distinct values of o corresponding to the resonant states. The problem of the zero modes connected with translational invariance is also studied. Jackiw R 1977 Rev. Mod. Phys. 49 681 Nielsen H and Olesen P 1973 Nucl. Phys. B 61 45 Plebahski J 1970 Lectures on nonlinear electrodynamics, N O R D l T A lectures, Copenhagen Schwinger J 1951 Phys. Rev. 82 664

Approximation on the limit continuum of a degenerate Kleinian group

“…Then q>(a) = 0 for a E ~(A), and (24) yields the estimate IO(z)[ _< c(p(t>(z))) -k, where k = 1 or k = a + 1. In accordance with a theorem from [6], we have the following important relation:…”

Decompositions into nonoverlapping domains and extremal properties of univalent functions

“…The next result of [8] easily follows from inequality (2) and from the classical inequality for If'(z)i in the class E. The problem on the maximum of functional (4) is closely related with the problem on the minimum of minl,l-~lf'(z)[ in the class :7:(R,p). The solution of the latter problem is given by the following theorem.…”

“…The differential (2.6) is a real quadratic differential meromorphic in a triply connected domain; it represents the simplest generalization of the above-mentioned differential. In this and next subsections we describe the structure of orthogonal trajectories of the differential (2.6), which allows us to prove Lemma 2.1 below on the monotonicity of M (8).…”

“…As an example of an open problem, where the associated quadratic differential has five free poles, P. M. Tamrazov [8] …”

Quadratic differentials in problems of the product of inner radii of nonoverlapping domains

“…This principle has later been expressed in the form of a so-called "general coefficient theorem," which was formulated and proved by Jenkins [9]. The method of quadratic differentials and its applications were further developed by Tamrazov [10], who have also substantially complemented the Jenkins theorem indicated above.…”