Year: 2026
Type: preprint
Abstract: We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.
Source: arXiv (Cornell University)
Author: Chen, Zhangchi
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216492850