On Bach-flat gradient shrinking Ricci solitons

Abstract

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cylinder $S^3\times R$. More generally, for n>4, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $R^n$ or the product $N^{n-1}\times R$, where $N^{n-1}$ is Einstein.

Revised version, to appear in Duke Math Journal

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• Gradient shrinking Ricci solitons of half harmonic Weyl curvature
• Half conformally flat gradient Ricci almost solitons
• Rigidity of asymptotically conical shrinking gradient Ricci solitons
• A note on gradient Einstein-type manifolds
• On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor
• Four-dimensional complete gradient shrinking Ricci solitons
• Conical structure for shrinking Ricci solitons
• On the geometry of electrovacuum spaces in higher dimensions
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• Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
• Vacuum Static Spaces with Harmonic Curvature
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• V-static spaces with positive isotropic curvature
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature
• Critical metrics on $4$-manifolds with harmonic anti-self dual Weyl tensor
• Rigidity theorem for a static triple with half harmonic Weyl curvature
• Some rigidity characterizations of Einstein metrics as critical points for quadratic curvature functionals
• Gap theorems on critical point equation of the total scalar curvature with divergence-free Bach tensor
• 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor
• Sharp upper diameter bounds for compact shrinking Ricci solitons
• Mean-stable surfaces in Static Einstein-Maxwell theory