Mean field type equations on line bundle over a closed Riemann surface

Abstract

Let $(\mathcal{L},\mathfrak{g})$ be a line bundle over a closed Riemann surface $(Σ,g)$, $Γ(\mathcal{L})$ be the set of all smooth sections, and $\mathcal{D}:Γ(\mathcal{L})\rightarrow T^\astΣ\otimes Γ(\mathcal{L})$ be a connection independent of the bundle metric $\mathfrak{g}$, where $T^\astΣ$ is the cotangent bundle. Suppose that there exists a global unit frame $ζ$ on $Γ({\mathcal{L}})$. Precisely for any $σ\inΓ(\mathcal{L})$, there exists a unique smooth function $u:Σ\rightarrow\mathbb{R}$ such that $σ=uζ$ with $|ζ|\equiv 1$ on $Σ$. For any real number $ρ$, we define a functional $\mathcal{J}_ρ:W^{1,2}(Σ,\mathcal{L})\rightarrow\mathbb{R}$ by $$\mathcal{J}_ρ(σ)=\frac{1}{2}\int_Σ|\mathcal{D} σ|^2dv_g+\fracρ {|Σ|}\int_Σ\langleσ,ζ\rangle dv_g-ρ\log\int_Σh e^{\langleσ,ζ\rangle}dv_g,$$ where $W^{1,2}(Σ,\mathcal{L})$ is a completion of $Γ(\mathcal{L})$ under the usual Sobolev norm, $|Σ|$ is the area of $(Σ,g)$, $h:Σ\rightarrow\mathbb{R}$ is a strictly positive smooth function and $\langle\cdot,\cdot\rangle$ is the inner product induced by $\mathfrak{g}$. The Euler-Lagrange equations of $\mathcal{J}_ρ$ are called mean field type equations. Write $\mathcal{H}_0=\{σ\in W^{1,2}(Σ,\mathcal{L}):\mathcal{D}σ=0\}$ and $$\mathcal{H}_1=\left\{σ\in W^{1,2}(Σ,\mathcal{L}):\int_Σ\langleσ,τ\rangle dv_g=0,\,\,\forall τ\in \mathcal{H}_0\right\}.$$ Based on the variational method, we prove that $\mathcal{J}_ρ$ has a constraint critical point on the space $\mathcal{H}_1$ for any $ρ<8π$; Based on blow-up analysis, we calculate the exact value of $\inf_{σ\in\mathcal{H}_1}\mathcal{J}_{8π}(σ)$, provided that it is not achieved by any $σ\in\mathcal{H}_1$;