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.

73 pages. To appear in Duke Mathematical Journal

References in corpus (3)

• Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow
• Recent Progress on Singularities of Lagrangian Mean Curvature Flow
• Floer homology of Lagrangian submanifolds

Cited by in corpus (4)

• On short time existence of Lagrangian mean curvature flow
• Constructing soliton solutions of geometric flows by separation of variables
• On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow
• Quantitative Thomas-Yau uniqueness