球面中平均曲率流的精确四次曲率夹挤
Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere
浏览论文内容
中文总结 AI 辅助
本研究针对球面中平均曲率流，运用爆破论证等方法证明了精确四次曲率夹挤定理，推广了Pu的相关收敛性成果，无需Stampacchia迭代或积分分析即得到重标度的光滑收敛结果。
中文摘要 AI 辅助
我们证明了$\mathbb{S}^{n+m}$、$m\ge2$中平均曲率流的精确四次曲率夹挤定理，该结果推广了Pu关于$\mathbb{S}^{n+m}$中子流形收敛至圆点的研究工作。通过爆破论证，我们证明了余维数估计与柱形估计：在高曲率区域，子流形定量地近似为余维一，且呈弱凸性，要么沿平移方向运动，要么为自收缩子。结合衰减估计，重标度流在无穷时间内光滑收敛至全测地极限，整个过程无需使用Stampacchia迭代或积分分析。
英文摘要
We prove a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, which generalises Pu's work on the convergence of submanifolds in $\mathbb{S}^{n+m}$ to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.
