Sharp Nonexistence Results for Curvature Equations with Four Singular Sources on Rectangular Tori

In this paper, we prove that there are no solutions for the curvature equation $$ \Delta u+e^u=8\pi n\delta_0\ {\rm on}\ E_{\tau},\quad n\in\Bbb{N}, $$ where $E_{\tau}$ is a flat rectangular torus and $\delta_0$ is the Dirac measure at the lattice points. This confirms a conjecture of Lin and Wang and also improves a result of Eremenko and Gabrielov. The nonexistence is a delicate problem because the equation always has solutions if $8\pi n$ in the RHS is replaced by $2\pi\rho$ with $0<\rho\notin 4\Bbb{N}$. Geometrically, our result implies that a rectangular torus $E_{\tau}$ admits a metric with curvature $+1$ acquiring a conic singularity at the lattice points with angle $2\pi\alpha$ if and only if $\alpha$ is not an odd integer. Unexpectedly, our proof of the nonexistence result is to apply the spectral theory of finite-gap potential, or equivalently the algebro-geometric solutions of stationary KdV hierarchy equations. Indeed, our proof can also yield a sharp nonexistence result for the curvature equation with singular sources at three half periods and the lattice points.