Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus
Year: 2026
Type: preprint
Abstract: We study the following curvature equation with two singular sources on a flat torus $$ Δu+e^u=8π(δ_p+δ_{-p})\quad\text{on } E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ), $$ where $δ_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case $\wp'(2p)\neq 0$. A novel phenomenon for $\wp'(2p)\neq 0$ is that the Hill discriminant of the associated generalized Lamé equation has an essential singularity. We develop new ideas to determine the intersection of two conditional stability sets completely, which is quite surprising since it seems impossible to determine the exact structure of these two sets due to the essential singularity. This leads to a sharp criteria for the existence and non-existence of solutions when $E_τ$ is a rectangular torus and $\wp(p)\in \mathbb{R}$.
Source: arXiv (Cornell University)
Authors: Chen, Zhijie, Song, Zhen
Institutions: Northeastern University, Tsinghua University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6507539299, G8888635134
OpenAlex ID: W7214019411