Sharp nonexistence results for curvature equations with four singular sources on rectangular tori

2017/09/13 by Zhijie Chen, Chang‐Shou Lin, Chen, Zhijie +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1709.04287

openalex publication_date 2017/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that there are no solutions for the curvature equation Δu+e^u=8πnδ₀ on E_τ, n∈ℕ, where E_τ is a flat rectangular torus and δ₀ is the Dirac measure at the lattice points. This confirms a conjecture in \citeCLW2 and also improves a result of Eremenko and Gabrielov \citeEG. The nonexistence is a delicate problem because the equation always has solutions if 8πn in the RHS is replaced by 2πρ with 0

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• Green functions, Hitchin's formula and curvature equations on tori
• Even Cone Spherical Metrics: Blow-Up at a Cone Singularity
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