  A new monotone quantity in graphical mean curvature flows of higher
codimensions is identified in this work. The submanifold deformed by the mean
curvature flow is the graph of a map between Riemannian manifolds, and the
quantity is monotone increasing under the area-decreasing condition of the map.
The flow provides a natural homotopy of the corresponding map and leads to
sharp criteria regarding the homotopic class of maps between complex projective
spaces, and maps from spheres to complex projective spaces, among others.
