Year: 2026
Type: preprint
Abstract: We prove that for every integer $n\geq 2$ and $8π(n-1)<ρ<8πn$, the singular Liouville equation $Δu+\e^u=ρδ_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2π\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2π\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for $ρ\leq 8π$. We develop a unified approach for all $ρ\in (0,+\infty)\setminus 8π\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lamé equation.
Source: arXiv (Cornell University)
Authors: Chen, Zhijie, Zhang, Shihong
Institution: Tsinghua University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3327967141, G6507539299, G8888635134
OpenAlex ID: W7204946606