Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori

@misc{chen2026uniqueness,
  title = {{Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori}},
  author = {Zhijie Chen and Shihong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29991},
  eprint = {2609.29991},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29991},
}

TY  - UNPB
TI  - Uniqueness and nonuniqueness for mean field equations with multiple singularities on flat tori
AU  - Zhijie Chen
AU  - Shihong Zhang
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29991
UR  - https://arxiv.org/abs/2609.29991
AB  - We study mean field equations with multiple positive singularities on flat tori. We establish a uniform uniqueness criterion in terms of the total singular mass and the scale-invariant spectral quantity λ_1(𝕋)|𝕋|, where λ_1(𝕋) denotes the first positive eigenvalue of the Laplacian. Combining nodal-set analysis for Jacobi fields with a cylindrical Alexandrov--Bol type inequality and sharp small-capacity asymptotics, we obtain the optimal leading coefficient 8/π^2 in the total-mass uniqueness bound as λ_1(𝕋)|𝕋|→0. Sharpness is demonstrated by examples with two symmetric singularities on degenerating rectangular tori that admit at least three distinct solutions. For the case of two singularities with total mass 2ρ, we further derive uniqueness and multiplicity results near ρ=4π from the critical-point structure of the associated symmetrized Green function. In particular, when its only critical points are the four nondegenerate two-torsion points, the equation admits a unique solution for ρ just below 4π and exactly three solutions for ρ just above 4π.
ER  -