On Locally Conformally Flat Gradient Shrinking Ricci Solitons

By Xiaodong Cao, Biao Wang, Zhou Zhang. 2008

87 citations recorded by the source corpus.

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

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

View Paper: https://www.semanticscholar.org/paper/15099bdafbcb5a09f4867912aeec6b6423ba1e82

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Ricci solitons: the equation point of view (2008, methodology): “During the publication of this paper, there appeared several preprints extending our results (or presenting different proofs) to the complete noncompact case [8,13, 22–24,26].”
On the geometry of gradient einstein-type manifolds (2014, unknown): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”
Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “In the shrinking case, classification results for gradient Ricci solitons have been obtained by many authors under various curvature conditions on the Weyl tensor, e.g., [7; 9; 10; 17; 20; 21; 24; 28; 29; 30; 31; 33; 35; 36], etc.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “Also, from [16], [26], [30], [38], [43], [46], and [53], it follows that the only complete nonflat shrinking solitons of vanishing Weyl tensor (even harmonic Weyl tensor) in dimensions n ≥ 4 are finite quotients of the standard metrics on S or Sn−1 × R.”
On Bach-flat gradient shrinking Ricci solitons (2011, background): “…round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n − 1 × R (ﬁrst due to Z. H. Zhang [26] based on the work of Ni-Wallach [21], see also the works of Eminenti-La Nave-Mantegazza [15], Petersen-Wylie [24], X. Cao, B. Wang and Z. Zhang [10], and Munteanu-Sesum [20]).”