Deforming symplectomorphisms of complex projective spaces by the mean curvature flow

We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension n, there exists a constant $\Lambda$, explicitly computable, such that any $\Lambda$-pinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.