As on arXiv
Abstract
We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial -flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao–Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
The rigity of spaces with cyclic parallel Ricci tensor
Topics Almost & η-Ricci solitons, Einstein metrics, Boundaries & non-compact · Same author Fengyuan Zhang · Shared terms “cyclic parallel”, “parallel Ricci”, “Ricci tensor”
-
Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow
Topics Almost & η-Ricci solitons, Einstein metrics, Stability & uniqueness · Shared terms “quasi Einstein”, “identity”, “gradient Ricci”
-
-Ricci Solitons on Kenmotsu 3-Manifolds
Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “cyclic parallel”, “parallel Ricci”, “Ricci tensor”
-
The extended quasi-Einstein manifolds with generalised Ricci solitons
Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”
-
Almost Gradient Ricci Solitons on Static Spacetime
Topics Almost & η-Ricci solitons, Einstein metrics, Boundaries & non-compact · Shared terms “vacuum static”, “gradient Ricci”
-
Back to almost Ricci solitons
Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “Ricci tensor”
-
Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds
Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “compact quasi”
-
Almost Ricci solitons on Finsler spaces
Topics Almost & η-Ricci solitons, Einstein metrics, Stability & uniqueness · Shared terms “gradient Ricci”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.