Riemann surface foliations with non-discrete singular set

Abstract

Let $F$ be a singular Riemann surface foliation on a complex manifold M, such that the singular set $E\subset M$ is non-discrete. We study the behavior of the foliation near the singular set E, particularly focusing on singular points that admit invariant submanifolds (locally) passing through them. Our primary focus is on the singular points that are removable singularities for some proper subfoliation. The aim of this note is two-fold. First, we classify singular points based on the dimension of their invariant submanifold and, consequently, establish that for hyperbolic foliations $F$, the presence of such singularities ensures the continuity of the leafwise Poincaré metric on $M\setminus E$. Secondly, we look into the behavior of the leafwise Poincaré metric near the singular set. We prove that if a foliation is of transversal type in a sufficiently large set, then the modulus of uniformization map η is continuous on M.