On the classification of gradient Ricci solitons

Abstract

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor.

14 pages. case. v.4--Added formula for the f-laplacian of the curvature operator and applications

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• Ricci solitons on singly warped product manifolds and applications
• Classification of expanding and steady Ricci solitons with integral curvature decay
• A remark on Einstein warped products
• The Weyl Tensor of Gradient Ricci Solitons
• Gradient Ricci almost soliton warped product
• Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
• Some geometric analysis on generic Ricci solitons
• $\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons
• Isometry theorem of gradient Shrinking Ricci solitons
• Locally conformally flat ancient Ricci flows
• Half conformally flat gradient Ricci almost solitons
• Some triviality results for quasi-Einstein manifolds and Einstein warped products
• Bach-flat isotropic gradient Ricci solitons
• Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
• Kählerity of shrinking gradient Ricci solitons asymptotic to Kähler cones
• Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature
• On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization
• Cohomogeneity two Ricci solitons with sub-Euclidean volume