Abstract
Let \((M^n, g, f)\), \(n\ge 5\), be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature \(Rc\ge 0\). In this paper, we show that if the asymptotic scalar curvature ratio of \((M^n, g, f)\) is finite (i.e., \( \limsup _{r\rightarrow \infty } R r^2< \infty \)), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, \(\limsup _{r\rightarrow \infty } |Rm| \ \! r^{\alpha }< \infty \) for any \(0<\alpha <2\).
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Notes
See also Corollary 4.11 in the very recent work of Chan et al. [21].
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Acknowledgements
We would like to thank Ovidiu Munteanu and Jiaping Wang for their interests in this work and their helpful comments and suggestions. We are also grateful to the referee for the careful reading of our paper and for providing valuable suggestions which led to a simpler version of Lemma 3.1 and a more streamlined proof of Lemmas 3.2 and 3.3 than in the previous version.
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Communicated by A. Mondino.
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Cao, HD., Liu, T. & Xie, J. Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio. Calc. Var. 62, 48 (2023). https://doi.org/10.1007/s00526-022-02387-1
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DOI: https://doi.org/10.1007/s00526-022-02387-1