A Homogeneous Function Constant along the Leaves of a Foliation

TL;DR

This paper constructs a smooth, positive, homogeneous function constant along leaves of a complex foliation compatible with spherical foliation, aiding in pseudoconvex domain modifications in complex analysis.

Contribution

It introduces a method to build functions constant along foliation leaves that are compatible with spherical foliations in complex spaces.

Findings

Constructed a smooth, positive, homogeneous function constant along foliation leaves.

Applied the function to facilitate bumping out pseudoconvex domains in complex three-space.

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 $\mathbb{C}^3$.