Arbitrarily large symplectic balls in a cotangent cylinder
= Arbitrarily large symplectic balls in a cotangent cylinder
The symplectic manifold $T^2\times\mathbb R^2$ contains symplectic balls of arbitrarily large radius. A steep linear Lagrangian graph in $\mathbb R^4=T^*\mathbb R^2$ has a thin neighborhood that injects under the quotient $\mathbb R^2\to T^2$, while a symplectic base-fiber dilation fits a large round ball into that long thin neighborhood.