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：

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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。