GRADIENT SHRINKING SOLITONS WITH VANISHING WEYL TENSOR

By Zhuhong Zhang. 2008

124 citations recorded by the source corpus.

We will give a local version of the Hamilton-Ivey-type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification of gradient shrinking solitons with vanishing Weyl tensor.

View Paper: https://www.semanticscholar.org/paper/f80e78383a3ec784fdc8c6b340132ba31227face

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Citations in the corpus

• On Bach-flat gradient shrinking Ricci solitons (2011, unknown): “…finite quotients of either the 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…”
• On Four-Dimensional Anti-self-dual Gradient Ricci Solitons (2011, background): “2, we quote here the main theorem in [33].”
• On locally conformally flat gradient steady Ricci solitons (2009, background): “In fact, the same arguments in [23] imply that a complete ancient solution gij(t) to the Ricci flow with vanishing Weyl tensor for each time t is necessarily of nonnegative curvature operator.”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “A proof in the compact case can be found in the paper of Eminenti-La Nave-Mantegazza [28] and, in the noncompact case, in the combined work of Ni-Wallach [46] and Zhang [56] (see also the papers of Petersen-Wylie [49] and Cao-Wang-Zhang [18]).”
• Recent Progress on Ricci Solitons (2009, background): “ient shrinking Ricci soliton with potential function f. Assume the Weyl tensor W = 0 and Z M |Rc|2e−fdvol < ∞, then (M n,g) is a finite quotient of Rn, Sn or S −1 ×R. On the other hand, Z.-H. Zhang [102] removed all curvature bound assumptions in Proposition 5.15 and Proposition 5.16, thus giving ”
• Remarks on non-compact gradient Ricci solitons (2009, background): “ich can take the form of classification (metric rigidity), or alternatively, triviality of the soliton structure (soliton rigidity). For instances of the former, see e.g. the recent far-reaching paper [23] and references therein. As forthe latter, it hasbeen known forsome time that compact, expanding”
• Ricci almost solitons (2010, background): “Zhu where a complete classification in the three-dimensional case is given, [33] by Z.”
• On Gradient Ricci Solitons (2009, background): “Besides being interesting on their own, such estimates would have as a consequence an alternate, simpler proof of the classification proved in [32] and should be useful in proving more general results.”
• 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.”