An analog of the Löwner equation for mappings of strips

Abstract

The Lowner equation was first stated in the famous work [1]. It was connected with an attempt to solve the coefficient problem in the class of univalent in a disk functions. This equation analytically describes the deletion of a cut in terms of conformal mappings with the Riemann normalization. Later this equation was generalized in connection with the solution of various extremal problems. Proceeding to boundary normalizations, several analogs of the Lowner equation were obtained for a half-plane (with the hydrodynamic normalization) and for a strip (with two fixed boundary points [2]). These generalizations are aimed at the variation approach in the corresponding classes of conformal mappings, therefore they contain no construction, erasing a cut. On the other hand, new applications of the Lowner equation such as the SLE (Shramm(stochastic)–Lowner evolution, see, e. g., [3]), are based on such a construction. In this paper, we obtain a complete analog of the Lowner equation for conformal mappings of a strip with finite angular derivatives at infinite points. Unlike the classical case, where the time parameter equals the conformal radius of the variable domain, here the time is connected with the characteristic which expresses the difference between the angular derivatives at infinite points. This is caused by the necessity to enforce the boundary value conditions. The specific character of the case under consideration is also proved by the fact that certain approaches connected with the application of the Riemann–Schwartz symmetry principle ([4], § 3.3, pp. 84–87) are unsuccessful, because they lead to the mappings of infinite-connected domains. Let Π = {z : 0 < Im z < π}, let f be a conformal mapping of the strip Π into itself. The limits c±(f) = lim Re z→±∞ (z − f(z)),