On the mean curvature flow of graphs of symplectomorphisms of Kahler-Einstein manifolds; Application to complex projective spaces

The presented work is a study of the mean curvature flow of graphs of symplectomorphisms of Kahler-Einstein manifolds in general, and of complex projective spaces in particular. We establish properties of singular values of symplectic linear maps. Using these observations, we derive, in a general Kahler-Einstein setting, the evolution equation of the Jacobian of the projection from the graph of a symplectomorphism onto the domain manifold under the flow. Finally, we apply this result to the case when the domain and the image manifolds are complex projective spaces with the Fubini-Study metric: we formulate a pinching condition for the singular values of the initial symplectomorphism, sufficient for the flow to exist and converge to the graph of a biholomorphic isometry.