Let \({\mathcal {F}}\) be a smooth Riemann surface foliation on \(M \setminus E\) , where M is a complex manifold and the singular set \(E \subset M\) is an analytic set of codimension at least two. Fix a hermitian metric on M and assume that all leaves of \({\mathcal {F}}\) are hyperbolic. Verjovsky’s modulus of uniformization \(\eta \) is a positive real function defined on \(M \setminus E\) defined in terms of the family of holomorphic maps from the unit disc \({\mathbb {D}}\) into the leaves of \({\mathcal {F}}\) and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of \(\eta \) on \(M {\setminus } E\) . The main question that is addressed here is its continuity at points of E. To do this, we adapt Whitney’s \(C_4\) -tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of \({\mathcal {F}}\) at points of E. This leads to the definition of a foliation that is of transversal type at points of E. It is shown that the map \(\eta \) associated to such foliations is continuous at E provided that it is continuous on \(M \setminus E\) and \({\mathcal {F}}\) is of transversal type. We also present observations on the locus of discontinuity of \(\eta \) . Finally, for a domain \(U \subset M\) , we consider \({\mathcal {F}}_U\) , the restriction of \({\mathcal {F}}\) to U and the corresponding positive function \(\eta _U\) . Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto–Martins on the variation \(U \mapsto \eta _U\) .
