A few remarks on the Poincaré metric on a singular holomorphic foliation

Let F be a Riemann surface foliation on M \ E, where M is a complex manifold and E subset of M is a closed set. Assume that F is hyperbolic, i.e. all the leaves of the foliation F are hyperbolic Riemann surfaces. Fix a Hermitian metric g on M. We consider the Verjovsky modulus of uniformization map eta, which measures the largest possible derivative in the class of holomorphic maps from the unit disc into the leaves of F. Various results are known to ensure the continuity of the map eta along the transverse directions, with suitable conditions on M, F and E. For a domain U subset of M, let FU be the holomorphic foliation given by the restriction of F to the domain U, i.e. F|U. We consider the modulus of uniformization map eta U corresponding to the foliation FU and study its variation when the corresponding domain U varies in the Caratheodory kernel sense, motivated by the work of Lins Neto and Canille Martins. (c) 2024 Elsevier Inc. All rights reserved.