The Ricci flow on the sphere with marked points
The Ricci flow on the sphere with marked points
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DOI：
10.4310/jdg/1577502023
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发表时间：
2014-07
影响因子：
2.5
通讯作者：
D. Phong;Jian Song;J. Sturm;Xiaowei Wang
中科院分区：
文献类型：
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作者：
D. Phong;Jian Song;J. Sturm;Xiaowei Wang
证明了在稳定、半稳定和不稳定三种情况下，2-球面上的Ricci流都是收敛的。在稳定的情况下，流动是已知的收敛没有任何重新参数化，并给出了这一事实的一个新的证明。半稳定和不稳定的情况下是新的，它表明，流收敛在Gromov-Hausdorff拓扑的限制度量空间，这也是一个2-球，但不同的标记点，因此不同的复杂结构。在半稳定情形下，极限度量空间具有唯一的圆锥常曲率度量;在不稳定情形下，极限度量空间具有唯一的圆锥收缩梯度Ricci孤子。
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.