Regularity of the Leafwise Poincaré Metric on Singular Holomorphic Foliations

Abstract

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 $U \mapsto \eta _U$.