设$(X, ω)$是紧致辛流形，$L$是拉格朗日子流形.假设$(X, L)$有一个哈密顿$S^1$动作，带有矩映射$μ$。对于一个不变的$ω$相容的几乎复结构，我们考虑元组$(C, P, A, φ)$，其中$C$是一个固定拓扑类型的光滑有边Riemann曲面，$P\to C$是一个$S^1$ -主丛，$A$是$P$上的一个联络，$φ$是$P\times_{S^1} X$的一个截面，满足$\ov\partial_A φ=0,\ ι_νF_A+ μ(φ)=c$和边界条件$φ(\partial C) \subset P \times_{S^1} L$ .这里$F_A$是$A$的曲率，$ν$是$C$上的体积形式，$c\in i{\mb R}$是常数。我们用能量$\leq K$紧化了这类物体的同构类的模空间，其中能量定义为Yang-Mills-Higgs泛函$\| F_A\|_{L^2}^2+ \| d_Aφ\|_{L^2}^2+ \| μ(φ)-c \|_{L^2}^2.$.这推广了Mundet-Tian \cite{Mundet_Tian_2009}在闭Riemann曲面情况下的紧性定理.
英文摘要：
Let $(X, ω)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $μ$. Take an invariant $ω$-compatible almost complex structure, we consider tuples $(C, P, A, φ)$ where $C$ is a smooth bordered Riemann surface of fixed topological type, $P\to C$ is an $S^1$-principal bundle, $A$ is a connection on $P$ and $φ$ is a section of $P\times_{S^1} X$ satisfying $\ov\partial_A φ=0,\ ι_νF_A+ μ(φ)=c$ with boundary condition $φ(\partial C) \subset P \times_{S^1} L$. Here $F_A$ is the curvature of $A$ and $ν$ is a volume form on $C$ and $c\in i{\mb R}$ is a constant.
We compactify the moduli space of isomorphism classes of such objects with energy $\leq K$, where the energy is defined to be the Yang-Mills-Higgs functional $\| F_A\|_{L^2}^2+ \| d_Aφ\|_{L^2}^2+ \| μ(φ)-c \|_{L^2}^2.$ This generalizes the compactness theorem of Mundet-Tian \cite{Mundet_Tian_2009} in the case of closed Riemann surfaces.
A reverse isoperimetric inequality for J-holomorphic curves
证明了具有拉格朗日边界条件的$J$ -全纯曲线的边界长度由一个常数乘以其面积决定。该常数取决于辛形式、殆复结构、拉格朗日边界条件和亏格。对于实$J$ -全纯曲线的实部的长度也有类似的结果。适当正规化的常数在$J$上的下确界给出了拉格朗日子流形的不变量。我们计算了拉格朗日子流形$\mathbb R P^n \subset \mathbb C P^n.$的这个不变量为$2π$。我们应用我们的结果证明了$J$ -全纯映射到渐近精确的非紧目标空间的模的紧性。在不同的方向上，我们的结果意味着adic收敛的超势。
