2023/08/15 by Ting-Jung Kuo, Kuo, Ting-Jung · 1 citation
Mathematics · Physics and Astronomy · #53A10 #58J10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2308.07620
openalex publication_date 2023/08/15 · openalex created_date 2023/08/17 · openalex updated_date 2026/07/28
In this paper, we investigate the following curvature equation: Δu+eu=8π(δ0+δ_\fracωk2) in Eτ, τ∈ ℍ (0.1) Here Eτ represents a flat torus and \fracωk2 is one of the half periods of Eτ. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: Δu+eu=8πδ0 in Eτ, τ∈ ℍ. This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all τ. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: Δu+eu=8πn(δ0+δ_\fracωk2) in Eτ, n∈ ℕ. As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.