Gennadiĭ Mikhaĭlovich Goluzin and geometric function theory

Abstract

Abstract. G. M. Goluzin crucially influenced the development and extension of geometric function theory. His results received world-wide recognition, and his monograph "Geometric theory of functions of a complex variable" has been a reference book for several generations of analysts.

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

“…The explicit evaluation of B(z) is a very hard problem studied, so far, only in the case q = 3. For details on Chebotarev's problem, and some other related problems in the theory of functions, as well, the reader may be referred to [16], [17], [35].…”

“…In a series of articles from 1946 through 1951, G. M. Goluzin created his own method, different from the earlier approach introduced by M. Schiffer; namely, the so-called method of internal variations. Applying his method, G. M. Goluzin solved Chebotarev's problem in terms of quadratic differentials (see [17]).…”

“…Theorem of Goluzin ((see [17])). Under conditions (10), the family of compacta K {∞},f possesses a unique minimizing element F with respect to the measure δ ∞ (or, equivalently, to the field V −δ∞ = 0).…”

Вариация Равновесной Энергии И $S$-Свойство Стационарного Компакта

“…Так как u ≡ 0 на F (1) ∪ E (3) , то u гармонически продолжается с одного экземпляра R на другой с заменой знака. Продолженная функция гармонична на R 3 всюду кроме точек z = ∞ (2) , ∞ (3) , ∞ (4) , ∞ (5) , где она имеет логарифмические особенности: 3 log |z| при z → ∞ (2) , ∞ (3) и −3 log |z| при z → ∞ (4) , ∞ (5) . Следовательно, u(z) = Re W (z),…”

“…dW (z) = dW (z; ∞ (2) , ∞ (3) ; ∞ (4) , ∞ (5) ) -(единственный) абелев дифференциал на R 3 с чисто мнимыми периодами и особенностями вида 1/z в точках z = ∞ (2) , ∞ (3) и вида −1/z в точках z = ∞ (4) , ∞ (5) . Компакт F соответствует нулевой линии уровня функции Re W (z; ∞ (2) , ∞ (3) ; ∞ (4) , ∞ (5) ): F = {z ∈ C : Re W (z; ∞ (2) , ∞ (3) ; ∞ (4) , ∞ (5) ) = 0} \ E.…”