论文标题
通过平均曲率流动的球体的捏夹子延伸
Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow
论文作者
论文摘要
在本文中，我们证明了球体中高的编质量平均曲率流与圆点或完全测量的球体的收敛，假设在第二个基本形式的标准平方与平均值曲率的标准平方和球体的背景曲率之间存在捏合条件。我们表明，对于尺寸$ n \ geq 4 $，这种夹紧是锋利的，但对于尺寸$ n = 2,3 $并不锋利。对于尺寸，$ n = 2 $和CODIMENSION $ 2 $，我们考虑了一种替代条件，其中包括正常捆绑包的正常曲率。最后，我们锐化了Chern-Do Carmo-Kobayashi曲率条件的四个球体表面 - 对于最小表面，这种曲率条件很清晰，我们猜测它是在球体中的曲率流动。
In this paper, we prove convergence of the high codimension mean curvature flow in the sphere to either a round point or a totally geodesic sphere assuming a pinching condition between the norm squared of the second fundamental form and the norm squared of the mean curvature and the background curvature of the sphere. We show that this pinching is sharp for dimension $n\geq 4$ but is not sharp for dimension $n=2,3$. For dimension $n=2$ and codimension $2$, we consider an alternative pinching condition which includes the normal curvature of the normal bundle. Finally, we sharpen the Chern-do Carmo-Kobayashi curvature condition for surfaces in the four sphere - this curvature condition is sharp for minimal surfaces and we conjecture it to be sharp for curvature flows in the sphere.
