C^m中特殊拉格朗日量和拉格朗日中曲率流展开式的唯一性结果
Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in C^m
摘要：
我们证明了两个主要结果：（a）设$L$是${\mathbb C}^m$中的一个封闭的嵌入的精确特殊拉格朗日$m$ -折叠，其中$m\ge 3$在无穷远处渐近于${\mathbb C}^m$中两个横截特殊拉格朗日平面$Π_1,Π_2$的并$Π_1\cupΠ_2$.那么$L$是Lawlor发现的明确的“Lawlor颈”家族的例子之一（Invent. 95，1989）。(b)假设$L$是${\mathbb C}^m$中封闭的嵌入式精确拉格朗日中曲率流动展开器，对于$m\ge 3$，在无穷远处渐近于${\mathbb C}^m$中两个横向拉格朗日平面$Π_1,Π_2$的并集$Π_1\cupΠ_2$。那么$L$就是Joyce、Lee和Tsui发现的显式例子家族之一，arXiv：0801.3721。如果$L$被浸入而不是嵌入，则（a）、（b）中唯一额外的可能性是$L=Π_1\cupΠ_2$。我们的方法，这是新的，可能会被用来证明其他类似的唯一性定理，涉及$J$ -全纯曲线，拉格朗日弗洛尔上同调，和辛拓扑的类。当$m=2$，（a）是很容易证明使用hyperkahler几何，和（b）是证明了Lotay和Neves，arXiv：1208.2729。
英文摘要：
We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=Π_1\cupΠ_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.
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