Rigidity of four-dimensional gradient shrinking Ricci solitons

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-05-22 DOI:10.1515/crelle-2023-0042

Xu Cheng, Detang Zhou

引用次数: 5

Abstract

Abstract Let ( M , g , f ) {{(M,g,f)}} be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric + ∇ 2 ⁡ f = λ ⁢ g {{\mathrm{Ric}+\nabla^{2}f=\lambda g}} , where λ {{\lambda}} is a positive real number. We prove that if M {{M}} has constant scalar curvature S = 2 ⁢ λ {{S=2\lambda}} , it must be a quotient of 𝕊 2 × ℝ 2 {{\mathbb{S}^{2}\times\mathbb{R}^{2}}} . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ 4 {{\mathbb{R}^{4}}} , 𝕊 2 × ℝ 2 {{\mathbb{S}^{2}\times\mathbb{R}^{2}}} or 𝕊 3 × ℝ {{\mathbb{S}^{3}\times\mathbb{R}}} .

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四维梯度收缩里奇孤子的刚性

设(M,g,f) {{(M,g,f)}}是一个四维完全非紧梯度收缩Ricci孤子，方程为Ric +∇2 λ f= λ¹g {{\mathrm{Ric} + \nabla ^{2f}= \lambda g，}}其中λ {{\lambda}}为正实数。我们证明了如果M {{M}}具有恒定的标量曲率S=2²λ {{S=2\lambda}}，它一定是 2 ×∈2 {{\mathbb{S} ^{2}\times\mathbb{R} ^{2}}}的商。结合已知的结果，这意味着一个四维完全梯度收缩里奇孤子具有恒定的标量曲率当且仅当它是刚性的，也就是说，它要么是爱因斯坦孤子，要么是高斯收缩孤子的有限商(∈4 {{\mathbb{R} ^{4}}}， 2 ×∈2 {{\mathbb{S} ^{2}\times\mathbb{R} ^{2}}}或 3 ×∈{{\mathbb{S} ^{3}\times\mathbb{R}}})。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.