Let \(({\cal L},{\mathfrak g})\) be a line bundle over a closed Riemann surface (Σ, g), \(\Gamma({\cal L})\) be the set of all smooth sections, and \({\cal D}:\Gamma ({\cal L}) \to T^{\ast}\Sigma \otimes \Gamma ({\cal L})\) be a connection independent of the bundle metric \({\mathfrak g}\) , where T*Σ is the cotangent bundle. Suppose that there exists a global unit frame ζ on \(\Gamma ({\cal L})\) . Precisely for any \(\sigma \in \Gamma ({\cal L})\) , 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 \({{\mathcal J}_\rho}:{W^{1,2}}(\Sigma, \, {\cal L}) \to \mathbb{R}\) by \({\cal J}_{\rho}(\sigma)={1 \over 2} \int_\Sigma {\vert{\cal D}\sigma \vert}^{2}dv_{g}+{{\rho} \over {\vert \Sigma \vert}} \int_\Sigma \langle\sigma, \, \zeta \rangle dv_{g}-\rho\log\int_\Sigma {he^{\langle \sigma, \, \zeta \rangle}} dv_{g},\) where \({W^{1,2}}(\Sigma, \, {\cal L})\) is a completion of \(\Gamma ({\cal L})\) 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 \({\mathfrak g}\) . The Euler-Lagrange equations of \({{\cal J}_\rho}\) are called mean field type equations. Write \({{\cal H}_0} = \{ \sigma \in {W^{1,2}}(\Sigma ,\,{\cal L}):{\cal D}\sigma = 0\}\) and \({\cal H}_1 =\left\{\sigma \in {W^{1,2}}(\Sigma, \, {\cal L}): \int_\Sigma {\langle \sigma, \, \tau \rangle d{v_g} = 0}, \, {\forall_\tau } \in {{\cal H}_0}\right\}.\) Based on the variational method, we prove that \({\cal J_\rho}\) has a constraint critical point on the space \({{\cal H}_1}\) for any ρ < 8π; Based on blow-up analysis, we calculate the exact value of \(\mathop {\inf}\limits_{\sigma \in {{\cal H}_1}} {{\cal J}_{8\pi}} (\sigma)\) , provided that it is not achieved by any \(\sigma \in {\cal H}_{1}\) ; If we further assume \({\cal D}\zeta=0, \, {\cal J}_{\rho}:W^{1,2}(\Sigma, {\cal L}) \to {\mathbb R}\) is reduced to a functional related to the classical mean field equation.
