Extremal properties of quadratic differentials with strip-shaped domains in the structure of the trajectories

Abstract

One considers the modulus problem for a family ~ of homotopic classes {Hi}, in the extended complex plane ~ , of the following types. The classes H i consist of closed Jordan curves, homotopic to appropriate nondegenerate contours or point curves, and also of arcs with endpoints in (distinct or coinciding) distinguished points in ~.One establishes the relation of the indicated extremal metric problem and the problem on the extremal decomposition of ~ in the family ~=~)~ of systems of mutually nonoverlapping domains {Di}, associated with the family ~ of the classes ~I 9 The results of this paper complement a previous theorem of the author [Moduli of families of curves and quadratic differentials.Trudy Mat. Inst.

Existence of quadratic differentials with prescribed properties

“…One of the possible proofs is given in [8]. It follows the approach used earlier in the proof of Theorem 1 in [5] and is based on the following considerations. Let 5 = (81,... ,St) be a system of small positive numbers.…”

“…Let c~, (~(1), and h be arbitrary systems of positive numbers. Let 7:) be a family of all systems of nonoverlapping domains {Di}, {D}l)}, {D~ 2)} in C\ {A U B (2) } associated with the family 7/, where the domains 0 (2) satisfy condition (*) and their inner angular characteristics at the boundary elements bk, k = k'(s),k"(s) (see (5), (6) …”

“…The further extension of the indicated principle to more general problems on extremal decomposition [4,5] is connected with the notion of reduced module with respect to the vertices for a biangle with nonzero inner angles at its vertices. Differentials Q(z)dz 2 defining extremal configurations in problems of such type have poles of the second order at these vertices and their trajectories in the vicinities of the poles are asymptotically similar to rectilinear rays.…”

The maximum of the conformal radius in the families of domains satifying additional conditions

“…Let M(D, a) denote the reduced module of the simply connected domain D with respect to the point a E D, and let M(D, ~, "~) denote the reduced module of the bigon D with respect to its distinguished boundary elements ~ and ~" with supports at the points u and v (see [1,2]). For a doubly-connected domain D with boundary continua P and F' let HD(P, a) denote the harmonic measure of the continuum P with respect to the domain D at the point a E D. Let re(D) denote the conformal module of the doubly-connected domain D, i.e., rn(D) = r2/rl if D is con_formally equivalent to the circular ring KT(rl, r2) = {z : rl "( [z[ < r2}.…”

The boundary distortion and extremal problems in certain classes of univalent functions

“…Throughout the paper we use the following notation: the conformal radius of a simply connected domain D. If D is a (topological) digon D(e~, e2) or a triangle D(e0, el, e2) with the vertices ek E OD, k = 0, 1, 2, then re(D; el, c2) and re(D; e0 ]el, e~) denote the corresponding reduced moduli of these configurations (for precise definitions, see [7][8][9]). Let w(z, E, D) denote the harmonic measure of a set E C OD with respect to D at a point z E D; mes E denotes the Lebesgue 1-measure of E. w 1.…”