Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Curvature estimates for steady Ricci solitons


Author: Pak-Yeung Chan
Journal: Trans. Amer. Math. Soc. 372 (2019), 8985-9008
MSC (2010): Primary 53C21, 53C25, 53C44
DOI: https://doi.org/10.1090/tran/7954
Published electronically: September 25, 2019
MathSciNet review: 4029719
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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\vert Rm\vert<\frac {1}{5}$, $ \vert Rm\vert\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 $ \vert Rm\vert\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, $ \vert Rm\vert$ decays exponentially provided that $ \overline {\lim }_{r\to \infty } rS$ is sufficiently small and $ f$ is bounded above by a constant.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C21, 53C25, 53C44

Retrieve articles in all journals with MSC (2010): 53C21, 53C25, 53C44


Additional Information

Pak-Yeung Chan
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: chanx305@umn.edu

DOI: https://doi.org/10.1090/tran/7954
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.
Article copyright: © Copyright 2019 American Mathematical Society