Squeezing Lagrangian tori in dimension 4

Abstract

Let C 2 be the standard symplectic vector space and L(a, b) ⊂ C 2 be the product Lagrangian torus, that is, a product of two circles of area a and b in C. We give a complete answer to the question of knowing the minimal ball into which these Lagrangians may be squeezed. The result is that there is full rigidity when b ≤ 2a, which disappears almost completely when b > 2a.