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The Ricci flow on the sphere with marked points
Journal of Differential Geometry ( IF 1.3 ) Pub Date : 2020-01-01 , DOI: 10.4310/jdg/1577502023
D. H. Phong 1 , Jian Song 2 , Jacob Sturm 3 , Xiaowei Wang 3
Affiliation  

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a unique conical shrinking gradient Ricci soliton in the unstable case.

中文翻译:

带有标记点的球体上的 Ricci 流

带有标记点的 2 球面上的 Ricci 流显示在所有三种稳定、半稳定和不稳定情况下都收敛。在稳定的情况下,已知流无需任何重新参数化即可收敛,并给出了这一事实的新证明。半稳定和不稳定的情况是新的,它表明流动在 Gromov-Hausdorff 拓扑中收敛到一个极限度量空间,该空间也是一个 2 球体,但具有不同的标记点,因此具有不同的复杂结构。极限度量空间在半稳定情况下具有唯一的锥形常曲率度量,在不稳定情况下具有唯一的锥形收缩梯度 Ricci 孤子。
更新日期:2020-01-01
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