Four-dimensional gradient shrinking solitons with pinched curvature
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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 $\mathbb {CP}^2$.References
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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
- MR Author ID: 868125
- Email: juhoncheung@sina.com
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3049-3056
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/proc/13859
- MathSciNet review: 3787365