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

A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria