On locally conformally flat gradient steady Ricci solitons

By Huaihuo Cao, Qiang Chen. 2009

198 citations recorded by the source corpus.

In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/07635e39eefc51aaa2b913836234fd60cb1ff656

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In Applied Mathematics

1. A. Barros
2. E. Ribeiro
3. G. Catino
4. S. Brendle
5. Qiang Chen
6. L. Mazzieri
7. M. Rigoli
8. Huaihuo Cao

In Geometry and Topology

1. A. Barros
2. E. Ribeiro
3. S. Brendle
4. G. Catino
5. Qiang Chen
6. L. Mazzieri
7. Huaihuo Cao
8. M. Rigoli

Citations in the corpus

• On Four-Dimensional Anti-self-dual Gradient Ricci Solitons (2011, background): “2 both originate from a wonderful idea in Cao–Chen [8].”
• Bach-flat gradient steady Ricci solitons (2011, methodology): “which was first introduced in [8] and played the key role in classifying locally conformally flat gradient steady Ricci solitons [8] and Bach-flat gradient shrinking Ricci solitons [7].”
• On Bach-flat gradient shrinking Ricci solitons (2011, methodology): “1(i) and the 4-dimension classification theorem for locally conformally flat gradient steady Ricci solitons (see [8], [11]), we have the following.”
• 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).”
• Generalized quasi-Einstein manifolds with harmonic Weyl tensor (2010, unknown): “In particular, we can prove a local characterization for locally conformally flat Ricci almost solitons (which have been introduced in [11]), similar to the one for Ricci solitons [2,4].”
• Critical point equation on four‐dimensional compact manifolds (2014, methodology): “3 of [4] follow that H is constant on , so is μ, this also appeared recently in Corollary 3.”
• On the geometry of gradient einstein-type manifolds (2014, background): “It turns out that, as in the case of gradient Ricci solitons (see [7], [8] and [6]), the leading actor is a three tensor, D, that plays a fundamental role in relating the Einstein-type structure to the geometry of the underlying manifold.”
• Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary (2014, background): “[7] and [8]) to study the level sets of the potential function of Miao-Tam critical metrics.”
• Rotational symmetry of self-similar solutions to the Ricci flow (2012, background): “We refer to [4], [5], [6], and [8] for details.”