О регулярных тканях, определенных плюригармоническими функциями

Abstract

As is known, the function of two variables z = f (x, y) on the plane of the variables (x, y) in the neighborhood of a regular point defines a three-web formed by the foliations

x = const, y = const, and f (x, y) = const.

A three-web is called regular if it is equivalent (locally diffeomorphic) to a three-web formed by three families of parallel lines.

In this case, the equation of the three-web has the form $z=f\left(\alpha(x)+\beta(y)\right)$.

In one of the works of the authors of this article, all regular three-webs, determined by some well-known partial differential equations, in particular, determined by harmonic functions, were found. In the present paper, the results are generalized for pluriharmonic functions of the form

u=f(x_1, ..., x_r, y_1, ... , y_r).

First, a function of this type on a manifold of dimension 2r determines a (2r + 1)-web formed by foliations of codimension 1:

x_i = const, 

y_i = const,    (i = 1 , 2, ...., r), 

u = const.

A (2r + 1)-web is called regular if in some local coordinates its equation can be written as

\[ u=f\bigl( \varphi_1(x_1)+\ldots + \varphi_r(x_r)+\psi_1(y_1)+ \ldots +\psi_r( y_r) \bigr).\]

In the present paper, we find all pluriharmonic functions defining regular (2r + 1)-webs (Theorem 1).

On the other hand, each pluriharmonic function u = f (x_1, ...., x_r, y_1, ....., y_r) defines a three-web                  W (r, r, 2r-1) on a 2r-dimensional manifold formed by two r-dimensional foliations x_i = const and y_i = const and one foliation u = const of codimension 1. This three-web is called regular if its equation in some local coordinates can be written as

$$

u=f\left(\varphi(x_1, x_2,\ldots, x_r)+\psi(y_1, y_2,\ldots, y_r)\right).

$$

In this work found all pluriharmonic functions defining regular W(r,r,2r-1)-webs (Theorem 2)