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A survey on Ricci solitons on Riemannian submanifolds

机译：黎曼子苗条的RICCI孤子调查

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摘要

A smooth vector field ￡ on a Riemannian manifold (M,g) is called a Ricci soliton if it satisfies the following Ricci soliton equation: {formula} where Lξg denotes the Lie-derivative of the metric tensor g with respect to ξ, Ric is the Ricci tensor and A is a constant. Compact Ricci solitons are the fixed points of the Ricci flow projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise as blow-up limits for the Ricci flow on compact manifolds. Further, Ricci solitons model the formation of singularities in the Ricci flow and they correspond to self-similar solutions. In this survey article we present recent results on Ricci solitons which occur naturally on certain Riemannian submanifolds. In addition, we also present recent criteria of trivial compact shrinking Ricci solitons via Poisson's equation.
机译：如果它满足以下Ricci Soliton方程，则riemannian歧管（m，g）上的平滑矢量字段£称为Ricci孤子：{公式}其中Lξg表示公制张量G相对于ξ，ric是的RICCI张量和A是常数。紧凑型RICCI孤子是从度量空间从度量的RICCI流程到其商量模子群和缩放的固定点，并且通常由于COMPOCH歧管上的RICCI流量而被爆破限制。此外，RICCI孤子模型在RICCI流中形成奇点，它们对应于自相似的解决方案。在本调查文章中，我们最近在某些黎曼子宫内发生的RICCI孤子的结果。此外，我们还通过泊松等式提出了最近的近距离紧凑次萎缩的Ricci孤子标准。

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