Local integrability of Mizohata structures

Abstract:

In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank $n > 2$ (and co-rank $1$). These structures are locally generated in an appropriate coordinate system $({t_1}, \ldots ,{t_n},x)$ by flat perturbations of Mizohata vector fields ${M_j} = \frac {\partial } {{\partial {t_j}}} - i{t_j}\frac {\partial } {{\partial x}}$, $j = 1, \ldots ,n$. For this, we first prove the global integrability of small perturbations of the structure generated by $\frac {\partial } {{\partial \bar z}} + {\sigma _1}\frac {\partial } {{\partial z}}$, $\frac {\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac {\partial } {{\partial z}}$, $j = 2, \ldots ,n$, defined over a manifold ${\mathbf {C}} \times S$, where $S$ is simply connected.