Liouville-type theorems for foliations with complex leaves

TL;DR

This paper investigates the structure of foliations with complex leaves in certain submanifolds of complex space, proving that under specific boundedness conditions, the leaves are necessarily complex planes.

Contribution

It establishes Liouville-type theorems for foliations with complex leaves, showing that boundedness conditions imply leaves are complex planes, which is a novel structural result.

Findings

Leaves are complex planes under boundedness conditions

Foliation structure is constrained by geometric conditions

Results apply to Levi flat submanifolds in C^n

Abstract

We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes.