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Four-dimensional gradient shrinking solitons with pinched curvature


Author: Zhu-Hong Zhang
Journal: Proc. Amer. Math. Soc. 146 (2018), 3049-3056
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/proc/13859
Published electronically: March 14, 2018
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Abstract: We show that any four-dimensional gradient shrinking soliton with pinched Weyl curvature $ (*)$ and satisfying $ c_1 \le R \le c_2$ for some positive constant $ c_1$ and $ c_2$, will have nonnegative Ricci curvature. As a consequence, we prove that it must be a finite quotient of $ \mathbb{S}^4$, $ \mathbb{CP}^2$, or $ \mathbb{S}^3 \times \mathbb{R}$. In particular, a compact four-dimensional gradient shrinking soliton with pinched Weyl curvature $ (*)$ must be $ \mathbb{S}^4$, $ RP^4$ or $ \mathbb{CP}^2$.


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Additional Information

Zhu-Hong Zhang
Affiliation: School of Mathematical Sciences, South China Normal Univeristy, Guangzhou, People’s Republic of China 510275
Email: juhoncheung@sina.com

DOI: https://doi.org/10.1090/proc/13859
Keywords: Ricci flow, maximum principle, pinched Weyl tensor
Received by editor(s): December 5, 2015
Received by editor(s) in revised form: June 9, 2017
Published electronically: March 14, 2018
Additional Notes: The author was supported in part by NSFC 11301191 and NSFC 11371377.
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society

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