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Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus

  • Zhijie Chen
  • Zhen Song
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Abstract

We study the following curvature equation with two singular sources on a flat torus Δu+eu=8π(δp+δ−p) on Eτ:=ℂ/(ℤ+ℤτ), where δp denotes the Dirac measure at p. The case ℘'(2p)=0 was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case ℘'(2p)≠ 0. A novel phenomenon for ℘'(2p)≠ 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 Eτ is a rectangular torus and ℘(p)∈ ℝ.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus}},
  author = {Zhijie Chen and Zhen Song},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23531},
  eprint = {2609.23531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23531},
}
RIS
TY  - UNPB
TI  - Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus
AU  - Zhijie Chen
AU  - Zhen Song
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23531
UR  - https://arxiv.org/abs/2609.23531
AB  - We study the following curvature equation with two singular sources on a flat torus Δu+e^u=8π(δ_p+δ_−p) on E_τ:=ℂ/(ℤ+ℤτ), where δ_p denotes the Dirac measure at p. The case ℘'(2p)=0 was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case ℘'(2p)≠ 0. A novel phenomenon for ℘'(2p)≠ 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 ℘(p)∈ ℝ.
ER  -