Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of Double-struck capital RDOUBLE-STRUCK CAPITAL Pn, Comm. Anal. Geom., 25, 2017, 799-846] initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space Double-struck capital RDOUBLE-STRUCK CAPITAL Pm. The purpose of this paper is to develop the work due to Pipoli and Sinestrari, and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space. Namely, the authors prove that if the initial submanifold in Double-struck capital RDOUBLE-STRUCK CAPITAL Pm satisfies a suitable pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t -> infinity. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space.