Year: 2026
Type: preprint
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 $Q$-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.
Source: arXiv (Cornell University)
Author: Zhang, Fengyuan
Institution: Academy of Mathematics and Systems Science
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221512051