Squeezing Lagrangian tori in dimension 4

Abstract

Let ${\mathbb{C}}^2$ be the standard symplectic vector space and $L(a,b)\subset {\mathbb{C}}^2$ be the product Lagrangian torus, that is, a product of two circles of areas $a$ and $b$ in $\mathbb{C}$. We give a complete answer to the question of finding the minimal ball into which these Lagrangians may be squeezed by a Hamiltonian flow. The result is that there is full rigidity when $a\le b\le 2a$, which disappears almost completely when $b>2a$.