Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations

2025/03/10 by Chen, Zhijie, Fu, Erjuan, Lin, Chang-Shou
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.06935

Abstract

Let E_τ:=ℂ/(ℤ+ℤτ) with Imτ>0 be a flat torus and G(z;τ) be the Green function on E_τ with the singularity at 0. Consider the multiple Green function G_n on (E_τ)ⁿ: \[ G_n(z₁,⋯,z_n;τ):=∑_i0\ such that G_n(⋅;τ) has degenerate critical points for any τ on the union of these curves. In this paper, we prove that there is a measure zero subset O_n⊂ \mathbb H (containing these curves) such that for any τ∈ \mathbb H\setminusO_n, all critical points of G_n(⋅;τ) are non-degenerate. Applications to counting the exact number of solutions of the curvature equation Δu+e^u=ρδ₀ on E_τ will be given.

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