Nonlinear equations related to the Levi form
In the geometric theory of functions of several complex variables, nonlinear equations related to the Levi form often occur. The most famous among them is perhaps the Monge-Ampère equation \(\text{det} (u_{\alpha \overline \beta}) = f\). Another equation which seems to play an important role in the complex analysis of CR-structures is \[ \text{det} \left( \begin{matrix} 0 & & u_1 &\ldots & u_n \\ u_{\overline 1} & & u_{\overline 11} &\ldots & u_{\overline 1n} \\ \vdots & & \vdots & & \vdots \\ u_{\overline n} &\ldots & u_{\overline n1} &\ldots & u_{\overline nn} \end{matrix} \right) = f. \tag{L} \] We call it the Levi equation. There is a great difference between these two equations. Indeed the former is elliptic in the family of strictly plurisubharmonic functions while the latter is elliptic degenerate in all reasonable classes of functions we may choose. It becomes clear that the traditional methods based on ``a priori estimates'' do not apply to (L) so that, in order to study the general problem, a preliminary analysis of the specific features of the solutions of (L) is necessary. This paper, which can be seen as a continuation of [the author, Ann. Mat. Pura Appl., IV. Ser. 152, 331-344 (1988; Zbl 0681.35017)], has just this motivation. In the first part, various aspects of the maximum principle are analysed. The second part is devoted to removable singularities.
scientific article; zbMATH DE number 60573
scientific article