Symplectic ball chart in complex projective space ($w\mapsto [w:\sqrt{1-|w{|}^2}]$)

Normalize the Fubini-Study form so that a complex projective line has symplectic area $\pi$. The displayed map takes the unit symplectic ball in ${\mathbb{C}}^n$ diffeomorphically onto the complement of the last-coordinate hyperplane. Its lift to the unit sphere has last coordinate positive real, so that coordinate contributes zero to the pulled-back standard symplectic form. The map is therefore a symplectomorphism.

Two disjoint symplectic balls of radii $r_1,r_2$ embedding into the unit four-ball must satisfy the displayed inequality. Extend their transported standard complex structures on smaller concentric balls to a compatible almost complex structure on the projective plane. A degree-one J-holomorphic curve through both centres has area $\pi$. The Monotonicity theorem for a J-holomorphic curve bounds its area in each smaller ball below by $\pi {\rho }_i^2$. Disjointness allows these bounds to be added; letting ${\rho }_i$ increase to $r_i$ proves the result.