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 $$\mathrm{\Delta }u+e^u=8\pi \sum _{k=0}^3{n}_k{\delta }_{\frac{{\omega }_k}{2}}$$ where $n_k\in \mathbb{N}$ and ${\delta }_a$ 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.