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 $(\Sigma ,g)$, $\Gamma (\mathcal{L})$ be the set of all smooth sections, and $\mathcal{D}:\Gamma (\mathcal{L})\to T^{\ast }\Sigma \otimes \Gamma (\mathcal{L})$ be a connection independent of the bundle metric $\mathfrak{g}$, where $T^{\ast }\Sigma$ is the cotangent bundle. Suppose that there exists a global unit frame $\zeta$ on $\Gamma (\mathcal{L})$. Precisely for any $\sigma \in \Gamma (\mathcal{L})$, there exists a unique smooth function $u:\Sigma \to \mathbb{R}$ such that $\sigma =u\zeta$ with $|\zeta |\equiv 1$ on $\Sigma$. For any real number $\rho$, we define a functional ${\mathcal{J}}_{\rho }:W^{1,2}(\Sigma ,\mathcal{L})\to \mathbb{R}$ by $${\mathcal{J}}_{\rho }(\sigma )=\frac{1}{2}{\int }_{\Sigma }|\mathcal{D}\sigma {|}^{2}d{v}_g+\frac{\rho }{|\Sigma |}{\int }_{\Sigma }⟨\sigma ,\zeta ⟩d{v}_g-\rho \log {\int }_{\Sigma }h{e}^{⟨\sigma ,\zeta ⟩}d{v}_g,$$ where $W^{1,2}(\Sigma ,\mathcal{L})$ is a completion of $\Gamma (\mathcal{L})$ under the usual Sobolev norm, $|\Sigma |$ is the area of $(\Sigma ,g)$, $h:\Sigma \to \mathbb{R}$ is a strictly positive smooth function and $⟨\cdot ,\cdot ⟩$ is the inner product induced by $\mathfrak{g}$. The Euler-Lagrange equations of ${\mathcal{J}}_{\rho }$ are called mean field type equations. Write ${\mathcal{H}}_0=\{\sigma \in W^{1,2}(\Sigma ,\mathcal{L}):\mathcal{D}\sigma =0\}$ and $${\mathcal{H}}_1=\left\{\sigma \in W^{1,2}(\Sigma ,\mathcal{L}):{\int }_{\Sigma }⟨\sigma ,\tau ⟩d{v}_g=0,\forall \tau \in {\mathcal{H}}_0\right\}.$$ Based on the variational method, we prove that ${\mathcal{J}}_{\rho }$ has a constraint critical point on the space ${\mathcal{H}}_1$ for any $\rho <8\pi$; Based on blow-up analysis, we calculate the exact value of ${\inf }_{\sigma \in {\mathcal{H}}_1}{\mathcal{J}}_{8\pi }(\sigma )$, provided that it is not achieved by any $\sigma \in {\mathcal{H}}_1$;

25 pages