复射影空间中高余维平均曲率流
Mean Curvature Flow of High Codimension in Complex Projective Space
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中文总结 AI 辅助
本文研究复射影空间中高余维紧致子流形的平均曲率流，在二次夹紧条件下建立余维估计，证明奇异时刻重缩放收敛到光滑余维一极限流，并推广了Pipoli和Sinestrari的工作。
中文摘要 AI 辅助
我们研究了黎曼流形 $\mathbb{C}P^n$ 中具有二次夹紧条件的 $m$ 维光滑紧致子流形的平均曲率流。我们的主要关注点是高余维情况，即 $k\geq 2$。我们建立了一个余维估计，表明在高曲率区域，子流形在可量化的意义上近似于余维一。该估计使我们能够在流的奇异时刻证明存在一个重缩放，该重缩放收敛到欧几里得空间中的光滑余维一极限流。在柱状夹紧条件下，我们证明该极限流是弱凸的并且通过平移运动。这些估计使我们能够分析流在奇点附近的行为，并建立极限流的存在性。最后，我们证明了一个衰减估计，表明重缩放随时间无限增长而光滑收敛到一个全测地极限。这种行为仅在子流形维数为偶数时可能发生。我们的方法依赖于二次夹紧条件沿流的保持以及一个在高曲率区域控制平均曲率的梯度估计。该结果推广了Pipoli和Sinestrari关于复射影空间中子流形平均曲率流的工作。
英文摘要
We study the mean curvature flow of smooth $m$-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold $\mathbb{C}P^n$. Our main focus is on the case of high codimension, $k\geq 2$. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and moves by translation. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Lastly, we prove a decay estimate that shows that the rescaling converges smoothly to a totally geodesic limit in infinite time. This behaviour is only possible if the dimension of the submanifold is even. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature. This result generalises the work of Pipoli and Sinestrari on the mean curvature flow of submanifolds of the complex projective space.
