论文标题
在封闭的Riemann表面上的线束上的平均场类型方程
Mean field type equations on line bundle over a closed Riemann surface
论文作者
论文摘要
令$（\ Mathcal {l}，\ Mathfrak {g}）$为封闭的Riemann Surface $（σ，G）$，$γ（\ Mathcal {l}）$是所有平滑段的集合，以及$ \ Mathcal {d}：d}：γ（\ Mathcal as） γ（\ mathcal {l}）$是独立于捆绑公制$ \ mathfrak {g} $的连接，其中$ t^\astς$是cotangengent bundle。假设存在$γ（{\ Mathcal {l}}）$上的全局单位框架$ζ$。完全适用于任何$σ\inγ（\ Mathcal {l}）$，存在一个唯一的平滑函数$ u：σ\ rightarrow \ mathbb {r} $，使得$σ=uζ$带有$ | embem | embem equiv 1 $ on $σ$。对于任何实际数字$ρ$，我们定义一个功能$ \ MATHCAL {J}_ρ：W^{1,2}（σ，\ Mathcal {l}）\ rightArrow \ rightarrow \ rightarrow \ Mathbb {r {r} $ by $$ \ MATHCAL {J}_ρ（σ）= \ frac {1} {2} \int_σ| \ Mathcal {d}σ|^2dv_g+\fracρ{|σ|} e^{\langleσ，ζ\ rangle} dv_g，其中$ w^{1,2}}（σ，\ mathcal {l}）$是$γ（\ Mathcal {l}）$的完成，在通常$ h：σ\ rightarrow \ mathbb {r} $是一个严格的正平滑功能，$ \ langle \ cdot，\ cdot \ rangle $是$ \ mathfrak {g} $引起的内部产品。 $ \ Mathcal {J}_ρ$的Euler-Lagrange方程称为平均字段类型方程。写$ \ Mathcal {h} _0 = \ {σ\ in W^{1,2}（σ，=，\ Mathcal {l}）：\ Mathcal {d} = 0 = 0 \} $ and $$ \ MATHCAL \ MATHCAL \ MATHCAL {H} w^{1,2}（σ，\ Mathcal {l}）：\int_σ\langleσ，τ\ rangle dv_g = 0，\ \，\，\，\ forallτ\ in \ nathcal {h}任何$ρ<8π$的空间上的临界点$ \ mathcal {h} _1 $;基于爆破分析，我们计算$ \ inf_ {σ\ in \ Mathcal {h} _1} \ Mathcal {J} _ {8π}（σ）$的确切值，前提
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 $σ\in\mathcal{H}_1$;
