A Convergence Theorem for Mean Curvature Flow of Submanifolds in Complex Projective Spaces

Abstract

Mean curvature flow (MCF) is an effective tool for investigating the geometry and topology of submanifolds under suitable curvature pinching conditions. While most existing convergence results in complex projective spaces rely on pointwise curvature assumptions, much less is known about the case of integral curvature constraints. In this paper, we consider the MCF of smooth closed submanifolds of small codimension immersed in CPn+k2. We establish our main theorem under an explicit integral curvature pinching condition: the Lp-norm of the second fundamental form of the initial submanifold is bounded above by a small constant depending only on the dimension and the exponent p. The proofs are based on evolution equations for geometric quantities, Sobolev inequalities, and Moser iteration, which together yield uniform curvature estimates and preserve the required integral pinching condition along the flow. These estimates allow us to reduce the problem to previously established convergence criteria under pointwise curvature pinching conditions. Consequently, the MCF either shrinks to a round point in finite time or converges smoothly to a totally geodesic submanifold as t→∞. As a consequence, we obtain a differentiable sphere theorem: any such submanifold satisfying the same integral curvature pinching condition is diffeomorphic to either the standard sphere Sn or the complex projective space CPn2.