Holomorphic realizations of pairs of foliations on Riemann surfaces
Abstract
Let X be a hyperbolic Riemann surface and let μ and ν be laminations on X homotopic to measured foliations with finite Dirichlet integral. We prove that μ and ν are filling if and only if there exists a homeomorphism to another Riemann surface f:X→ Y and an integrable holomorphic quadratic differential q on Y, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of q respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual ℝ-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two ℝ-trees. We construct examples such that f:X→ Y is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.
Related
- Opers on transversely holomorphic foliations
- Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
- Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
- Oka realizations, fundamental groups, and holomorphic cofibrations
- Singular Levi-flats and holomorphic foliations on projective spaces
BibTeX
@misc{sagman2026holomorphic,
title = {{Holomorphic realizations of pairs of foliations on Riemann surfaces}},
author = {Nathaniel Sagman and Dragomir Saric},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12217},
eprint = {2609.12217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12217},
note = {CC0},
}
RIS
TY - UNPB
TI - Holomorphic realizations of pairs of foliations on Riemann surfaces
AU - Nathaniel Sagman
AU - Dragomir Saric
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12217
UR - https://arxiv.org/abs/2609.12217
AB - Let X be a hyperbolic Riemann surface and let μ and ν be laminations on X homotopic to measured foliations with finite Dirichlet integral. We prove that μ and ν are filling if and only if there exists a homeomorphism to another Riemann surface f:X→ Y and an integrable holomorphic quadratic differential q on Y, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of q respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual ℝ-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two ℝ-trees. We construct examples such that f:X→ Y is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.
ER -