Regularity of the Leafwise Poincaré Metric on Singular Holomorphic Foliations.

Let F be a smooth Riemann surface foliation on M \ E , where M is a complex manifold and the singular set E ⊂ M is an analytic set of codimension at least two. Fix a hermitian metric on M and assume that all leaves of F are hyperbolic. Verjovsky's modulus of uniformization η is a positive real function defined on M \ E defined in terms of the family of holomorphic maps from the unit disc D into the leaves of 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 η on M \ 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 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 η associated to such foliations is continuous at E provided that it is continuous on M \ E and F is of transversal type. We also present observations on the locus of discontinuity of η . Finally, for a domain U ⊂ M , we consider F U , the restriction of F to U and the corresponding positive function η U . Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto–Martins on the variation U ↦ η U . [ABSTRACT FROM AUTHOR]