2022/06/03 by Jie Yang, Yang, Jie, Yunyan Yang +1
Mathematics · #58J05 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2206.01437
openalex publication_date 2022/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (L,\mathfrakg) be a line bundle over a closed Riemann surface (Σ,g), Γ(L) be the set of all smooth sections, and D:Γ(L)→ T^∗Σ⊗ Γ(L) be a connection independent of the bundle metric \mathfrakg, where T^∗Σ is the cotangent bundle. Suppose that there exists a global unit frame ζ on Γ(L). Precisely for any σ∈Γ(L), there exists a unique smooth function u:Σ→ℝ such that σ=uζ with |ζ|≡ 1 on Σ. For any real number ρ, we define a functional Jρ:W1,2(Σ,L)→ℝ by Jρ(σ)=(1)/(2)∫Σ|D σ|2dvg+\fracρ |Σ|∫Σ⟨σ,ζ⟩ dvg-ρlog∫Σh e⟨σ,ζ⟩dvg, where W1,2(Σ,L) is a completion of Γ(L) under the usual Sobolev norm, |Σ| is the area of (Σ,g), h:Σ→ℝ is a strictly positive smooth function and ⟨⋅,⋅⟩ is the inner product induced by \mathfrakg. The Euler-Lagrange equations of Jρ are called mean field type equations. Write H0=\σ∈ W1,2(Σ,L):Dσ=0\ and H1=\σ∈ W1,2(Σ,L):∫Σ⟨σ,τ⟩ dvg=0, ∀ τ∈ H0\. Based on the variational method, we prove that Jρ has a constraint critical point on the space H1 for any ρ<8π; Based on blow-up analysis, we calculate the exact value of infσ\inH1J8π(σ), provided that it is not achieved by any σ\inH1;