Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in C^m

Abstract

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 $\Pi_1\cup\Pi_2$ of two transverse special Lagrangian planes $\Pi_1,\Pi_2$ 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 $\Pi_1\cup\Pi_2$ of two transverse Lagrangian planes $\Pi_1,\Pi_2$ 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 $L=\Pi_1\cup\Pi_2$. 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.