Alternating Beltrami equation and conformal multifolds

A. N. Kondrashov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2015), pp. 6-24

Résumé

The problem of the study of alternating Beltrami equation was posed by L.I. Volkovyskiǐ [5]. In [8] we proved that solutions of the alternating Beltrami equation of a certain structure ($(A,B)$-multifolds) are composition of conformal multifold and suitable homeomorphism. Thus, lines of change of orientation cannot be arbitrary, and only mapped by the specified homeomorphism in analytical arcs. Therefore, understanding of the structure of conformal multifolds is the key to understanding the structure of $(A, B)$-multifolds. The main results of this work. I. The theorem on removability of conformal multifolds cuts. This theorem is about the possibility of extending by continuity from the domain $D_{\Gamma_0} = D\setminus\bigcup_{\gamma\in\Gamma_0}|\gamma|$ to the whole domain $D$. Here $\Gamma_0$ is family of arcs which belong to the set change of type. Theorem 3. Suppose that conditions are hold. (A1) Functions $f_k(z)$ $(k = 1,2)$ are analytical ( antianalytical ) extended from each white ( black ) domain $D_i$ to a domain $\Omega\supset[D]$ and these extensions $f^i_k(z)$ $(i=1,\ldots,N)$, are homeomorphisms of $\Omega$. (A2) $\bigcap_{i=1}^Nf^i_1(\Omega)\supset[f_1(D)]$. Then the conformal multifold $f_2(z)$ in $D_{\Gamma_0}$ is also conformal multifold in $D$. II. Description of a process of constructing conformal multifolds on analytical arcs of change type.

Détail

Keywords: alternating Beltrami equation, conformal multifold, black-white cut of domain, multidomain, continuous extending.