Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces

Abstract

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of ℝℙ ⁿ, 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 ℝℙ ^m. 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 ℝℙ ^m 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 → ∞. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

Keywords

Mean curvature flow / Submanifolds of arbitrary codimension / Complex projective space / Convergence theorem / Differentiable sphere theorem

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