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- W4367367279 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 {it 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 $U mapsto eta_U$." @default.
- W4367367279 created "2023-04-30" @default.
- W4367367279 creator A5022173409 @default.
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- W4367367279 date "2023-04-27" @default.
- W4367367279 modified "2023-10-14" @default.
- W4367367279 title "Regularity of the leafwise Poincare metric on singular holomorphic foliations" @default.
- W4367367279 doi "https://doi.org/10.48550/arxiv.2304.14206" @default.
- W4367367279 hasPublicationYear "2023" @default.
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