UNIQUENESS RESULTS FOR SPECIAL LAGRANGIANS AND LAGRANGIAN MEAN CURVATURE FLOW EXPANDERS IN Cm

We prove two main results. (1) Suppose that L is a closed, embedded, exact special Lagrangian m-fold in C-m asymptotic at infinity to the union Pi(1) boolean OR Pi(2) of two transverse special Lagrangian planes Pi(1,) Pi(2) in C-m for m >= 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 Cm asymptotic at infinity to the union Pi(1) boolean OR Pi(2) of two transverse Lagrangian planes Pi(1), Pi(2) in C-m for m >= 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), 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.