Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π

It is known from [17] that the solvability of the mean field equation ∆u + e^u = 8nπδ₀ with n ∈ N_{≥ 1} on a flat torus E_τ essentially depends on the geometry of E_τ. A conjecture is the non-existence of solutions for this equation if E_τ is a rectangular torus, which was proved for n = 1 in [17]. For any n ∈ N_≥2, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for n = 2 (i.e. at critical parameter 16π).