On Leafwise Meromorphic Functions with Prescribed Poles

Abstract

Let $\mathcal{F}$ be a complex foliation by Riemann surfaces defined by a trivial (in the differentiable sense) fibration $\pi :M\longrightarrow B$ but for which the complex structure on each fibre $\pi ^{-1}(t)$ may depend on t. Let $\sigma :B\longrightarrow M$ be a section of $\pi$ contained in a $\mathcal{F}$-relatively compact subset of M. We prove: for any $\mathcal{F}$-relatively compact open set U containing $\Sigma =\sigma (B)$ and any integer $s\ge 0$, there exists a function $U\longrightarrow {\mathbb {C}}$ of class $C^s$ nonconstant on any leaf of $(U,\mathcal{F})$, meromorphic along the leaves and whose set of poles is exactly $\Sigma$.