Curvature estimates for steady Ricci solitons
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Abstract:
We show that for an $n$ dimensional complete non-Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline {\lim }_{r\to \infty } r|Rm|<\frac {1}{5}$, $|Rm|\leq Ce^{-r}$ for some constant $C>0$, improving on a result of Munteanu, Sung, and Wang’s. For any four dimensional complete non-Ricci flat gradient steady Ricci soliton with scalar curvature $S\to 0$ as $r\to \infty$, we prove that $|Rm|\leq cS$ for some constant $c>0$, improving on an estimate by Cao and Cui. As an application, we show that for a four dimensional complete non-Ricci flat gradient steady Ricci soliton, $|Rm|$ decays exponentially provided that $\overline {\lim }_{r\to \infty } rS$ is sufficiently small and $f$ is bounded above by a constant.References
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Additional Information
- Pak-Yeung Chan
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 1168570
- Email: chanx305@umn.edu
- Received by editor(s): April 29, 2019
- Received by editor(s) in revised form: July 13, 2019
- Published electronically: September 25, 2019
- Additional Notes: The author was partially supported by NSF grant DMS-1606820.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8985-9008
- MSC (2010): Primary 53C21, 53C25, 53C44
- DOI: https://doi.org/10.1090/tran/7954
- MathSciNet review: 4029719