Bach-flat gradient steady Ricci solitons

In this paper we prove that any n -dimensional ( n  ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric...

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Published inCalculus of variations and partial differential equations Vol. 49; no. 1-2; pp. 125 - 138
Main Authors Cao, Huai-Dong, Catino, Giovanni, Chen, Qiang, Mantegazza, Carlo, Mazzieri, Lorenzo
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2014
Springer Nature B.V
Subjects
Analysis
Calculus
Calculus of variations
Calculus of Variations and Optimal Control; Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
53C44
53C25
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Summary:In this paper we prove that any n -dimensional ( n  ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012 ) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011 ).
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-012-0575-3