On Bach-flat gradient shrinking Ricci solitons

By Huaihuo Cao, Qiang Chen. 2011

207 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6ad0ece7e9026e8fd290697281eb1c444458aeb3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. A. Barros
2. E. Ribeiro
3. Brett Kotschwar
4. Lu Wang
5. G. Catino
6. Qiang Chen
7. Ovidiu Munteanu
8. M. Rigoli

In Astronomy and Astrophysics

1. A. Barros
2. E. Ribeiro
3. Lu Wang
4. Qiang Chen
5. G. Catino
6. M. Rigoli
7. D. Monticelli
8. P. Mastrolia

In Geometry and Topology

1. A. Barros
2. E. Ribeiro
3. Lu Wang
4. Qiang Chen
5. Ovidiu Munteanu
6. G. Catino
7. M. Rigoli
8. D. Monticelli

Citations in the corpus

• On the geometry of gradient einstein-type manifolds (2014, methodology): “In particular, we extend a recent result of Cao and Chen [8].”
• On Bach flat warped product Einstein manifolds (2011, methodology): “On a (λ, n+m)-Einstein manifold (M, g, f ) for any X, Y, Z ∈ TM we define the D-tensor, which is identical to the one in [Cao and Chen 2011] for Ricci solitons, as follows: (2-5) D(X, Y, Z)= 1 (n− 1)(n− 2) ( Ric(X,∇ f )g(Y, Z)−Ric(Y,∇ f )g(X, Z) )”
• Geometry of shrinking Ricci solitons (2014, background): “[4] H. D. Cao and Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math.”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “Hamilton [33] proved that the only complete nonflat twodimensional shrinking solitons are the standard round metrics on S2 and RP 2, and, with a combination of results from his later paper [34] and the work of Ivey [35], Perelman [47], Ni-Wallach [46], and Cao-ChenZhu [16], it follows that the only ”
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “His method of proof was motivated by the work of Cao and Chen [6; 7], and also based on Derdzi´nski’s study on Codazzi tensors [19] (the harmonicity of the Weyl tensor is equivalent to the Schouten tensor being Codazzi).”
• Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary (2014, methodology): “12) It is important to highlight that Tijk was defined similarly to Dijk in [7].”