J-holomorphic Disks and Lagrangian Squeezing

We define an invariant $l(M,W,\omega )$ for Lagrangian submanifold and prove that if the Lagrangian submanifold is embedded in the ball of radius $r_0$, then $l(M,W,\mathrm{\Omega })$ must be smaller than $4\pi t_0^2$. This improves Gromov’s Lagrangian embedding theorem.