Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations

We study the curvature equation with multiple singular sources on a torus \[Δu+eᵘ=8π∑ₖ₌₀³nₖδ_{{ωₖ}{2}}% +4π( δₚ+δ₋ₚ) on \;E_τ:=C/( Z+Zτ),\] where nₖ∈ N and δₐ denotes the Dirac measure at a. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point p, and we give a sharp criterion of p in terms of Painlevé VI equations.