Singularity models in the three-dimensional Ricci flow
Singularity models in the three-dimensional Ricci flow
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发表时间：
2022-01
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通讯作者：
S. Brendle
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作者：
S. Brendle
。里奇流是给定流形上黎曼度量的自然演化方程。主要目标是理解奇点的形成。佩雷尔曼在2002年取得了惊人的突破，他对三维奇点的形成有了定性的理解。更准确地说，佩雷尔曼表明，里奇流在3维的每一个有限时间奇点都是在一个古老的κ解上建模的。此外，Perelman还证明了3维古代κ -解的结构定理。在这个调查中，我们讨论了最近的发展，导致了所有奇点模型的完整分类在维度3。此外，我们给出了三维非坍缩稳定梯度Ricci孤子分类的另一种证明（最初由作者在2012年证明）。
. The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3. More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension 3 is modeled on an ancient κ -solution. Moreover, Perelman proved a structure theorem for ancient κ -solutions in dimension 3. In this survey, we discuss recent developments which have led to a complete classification of all the singularity models in dimension 3. Moreover, we give an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3 (originally proved by the author in 2012).