A Homogeneous Function Constant along the Leaves of a Foliation

Abstract

Given a smooth foliation by complex curves (locally around a point $x\in {\mathbb{C}}^2\setminus \{0\}$) which is "compatible" with the foliation by spheres centered at the origin, we construct a smooth real-valued function $g$ in a neighborhood of said point, which is positive, homogeneous and constant along the leaves. A corollary we obtain from this is relevant to the problem of "bumping out" certain pseudoconvex domains in ${\mathbb{C}}^3$.

13 pages