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

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 ${\lambda }_1(\mathbb{T})\mathrm{\mid }\mathbb{T}\mathrm{\mid }$, where ${\lambda }_1(\mathbb{T})$ 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 $8\mathrm{/}{\pi }^2$ in the total-mass uniqueness bound as ${\lambda }_1(\mathbb{T})\mathrm{\mid }\mathbb{T}\mathrm{\mid }\to 0$. 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 $2\rho$, we further derive uniqueness and multiplicity results near $\rho =4\pi$ 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 $\rho$ just below $4\pi$ and exactly three solutions for $\rho$ just above $4\pi$.