Poincar\'e metric of holomorphic foliations with non-degenerate singularities

TL;DR

This paper studies the Poincaré metric of Brody hyperbolic holomorphic foliations with non-degenerate singularities, proving its transversally Hölder continuity with a logarithmic slope near singularities.

Contribution

It establishes the regularity of the leafwise Poincaré metric in the presence of non-degenerate singularities, revealing new geometric properties of such foliations.

Findings

Leafwise Poincaré metric is transversally Hölder continuous.

The metric exhibits a logarithmic slope near singularities.

Results apply to Brody hyperbolic foliations on compact complex manifolds.

Abstract

Consider a Brody hyperbolic foliation $\mathscr{F}$ with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincar\'{e} metric is transversally H\"{o}lder continuous with a logarithmic slope towards the singular set of $\mathscr{F}$.