High Codimension Mean Curvature Flow in the Complex Projective Space

Abstract We study the mean curvature flow of smooth, compact submanifolds of high codimension with quadratic pinching in $$\mathbb {C}P^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> </mml:math> . We establish a codimension estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, in a quantifiable way. Under a cylindrical type pinching, this limiting flow is weakly convex and moves by translation or is a self shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.