Liouville-type theorems for foliations with complex leaves

In this paper we discuss some structure problems regarding the foliation of Levi flat manifolds, i.e. those CR manifolds which are foliated by complex leaves or, equivalently, whose Levi form vanishes. Levi flat manifolds have a particular significance among CR manifolds, since due to their degenerate nature they often behave as “limit case”, or they are an obstruction in extension problems of CR objects (see, for example, [3]). The geometry of Levi flat manifolds is a source of many interesting problems (see [2], [1]). Here, we address the following general question: assuming that a Levi flat submanifold S ⊂ C is bounded in some directions, what can be said about its foliation? We will show that, in some circumstances, it is possible to conclude that the foliation (or even, more generally, a complex leaf of a foliated manifold) is “trivial”, i.e. is made up by complex planes. A first result in this direction is the following