Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in Cm

Abstract

We prove two main results. (1) Suppose that $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb{C}}^m$ asymptotic at infinity to the union ${\Pi }_1\cup {\Pi }_2$ of two transverse special Lagrangian planes ${\Pi }_1,{\Pi }_2$ in ${\mathbb{C}}^m$ for $m\ge 3$. Then $L$ is one of the explicit Lawlor neck family of examples found by Lawlor. (2) Suppose that $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb{C}}^m$ asymptotic at infinity to the union ${\Pi }_1\cup {\Pi }_2$ of two transverse Lagrangian planes ${\Pi }_1,{\Pi }_2$ in ${\mathbb{C}}^m$ for $m\ge 3$. Then $L$ is one of the explicit family of examples in recent work by Joyce, Lee, and Tsui. If instead $L$ is immersed rather than embedded, the only extra possibility in (1), (2) 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.