Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
Year: 2026
Type: preprint
Abstract: Let $X$ be a genus $g\geq 0$ compact Riemann surface that admits an antiholomorphic involution $ι$ with non empty fixed-point set, and let $D = \{p_1,\dots,p_k\}\subset \text{Fix}(ι)$. 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 $\mathbb{C}$ to the interior of an ideal polygon in $\mathbb{H}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map \[ h:X\backslash D \to \mathbb{H}^2 \] 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 $\mathbb{H}^2$ factors of horizontal catenoids in $\mathbb{H}^2 \times \mathbb{R}$.
Source: arXiv (Cornell University)
Authors: Li, Qiongling, Liu, Jinsong, Xu, Yihui
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213528933