Evolving pinched submanifolds of the sphere by mean curvature flow
Evolving pinched submanifolds of the sphere by mean curvature flow
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DOI：
10.1007/s00209-022-03179-1
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发表时间：
2020-04
影响因子：
0.8
通讯作者：
C. Baker;H. Nguyen
中科院分区：
文献类型：
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作者：
C. Baker;H. Nguyen
本文在第二基本形式的范数平方与平均曲率和背景曲率的范数平方之间存在一个拼挤条件的条件下，证明了球面上的高余维平均曲率流收敛于一个圆点或一个全测地球面.我们证明了这种箍缩对于维数是尖锐的，但对于维数不是尖锐的。对于维数和余维数2，我们考虑了一个包含法丛的法曲率的可供选择的拼挤条件。最后，我们锐化的陈道嘉莫-小林曲率条件的表面在四个领域-这个曲率条件是尖锐的极小曲面，我们猜想它是尖锐的曲率流在球体。
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 dimensionbut is not sharp for dimension. For dimensionand 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.