Sharp nonexistence results for curvature equations with four singular sources on rectangular tori
Journal
American Journal of Mathematics
Journal Volume
142
Journal Issue
4
Pages
1269-1300
Date Issued
2020
Author(s)
Chen, Z.
Abstract
In this paper, we prove that there are no solutions for the curvature equation Δu + eu = 8πnδ0 on Eτ, n∈ N, where Eτ is a flat rectangular torus and δ0 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 8πn in the RHS is replaced by 2πρ with 0 <ρ/Ɇ 4N. Geometrically, our result implies that a rectangular torus Eτ admits a metric with curvature +1 acquiring a conic singularity at the lattice points with angle 2πα if and only if α 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.
Type
journal article