Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
Abstract
Let X be a genus g≥ 0 compact Riemann surface that admits an antiholomorphic involution ι with non empty fixed-point set, and let D = {p1,…,pk}⊂ Fix(ι). At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from ℂ to the interior of an ideal polygon in ℍ2, whose Hopf differential is the given polynomial. We construct a harmonic map h:Xbackslash D → ℍ2 whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of h tends to an ideal polygon near each end. The resulting h covers harmonic maps obtained by taking the ℍ2 factors of horizontal catenoids in ℍ2 × ℝ.
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BibTeX
@misc{li2026harmonic,
title = {{Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics}},
author = {Qiongling Li and Jinsong Liu and Yihui Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18297},
eprint = {2609.18297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18297},
}
RIS
TY - UNPB
TI - Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
AU - Qiongling Li
AU - Jinsong Liu
AU - Yihui Xu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.18297
UR - https://arxiv.org/abs/2609.18297
AB - Let X be a genus g≥ 0 compact Riemann surface that admits an antiholomorphic involution ι with non empty fixed-point set, and let D = {p_1,…,p_k}⊂ Fix(ι). At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from ℂ to the interior of an ideal polygon in ℍ^2, whose Hopf differential is the given polynomial. We construct a harmonic map h:Xbackslash D → ℍ^2 whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of h tends to an ideal polygon near each end. The resulting h covers harmonic maps obtained by taking the ℍ^2 factors of horizontal catenoids in ℍ^2 × ℝ.
ER -