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Curvature estimates for 4-dimensional complete gradient expanding Ricci solitons

  • Huai-Dong Cao ORCID logo EMAIL logo and Tianbo Liu

Abstract

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm and its covariant derivative Rm can be bounded by the scalar curvature R by | Rm | C a R a and | Rm | C a R a (on M K ), for any 0 a < 1 and some constant C a > 0 . Moreover, if the scalar curvature has at most polynomial decay at infinity, then | Rm | C R (on M K ). As an application, it follows that if a 4-dimensional complete gradient expanding Ricci soliton ( M 4 , g , f ) has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, hence admits C 1 , α asymptotic cones at infinity ( 0 < α < 1 ) according to Chen and Deruelle (2015).[21].

Funding source: Simons Foundation

Award Identifier / Grant number: 586694 HC

Funding statement: Research of Huai-Dong Cao was partially supported by a Simons Foundation Collaboration Grant (No. #586694 HC).

Acknowledgements

We would like to thank Pak-Yeung Chan, Chih-Wei Chen, Alix Deruelle, Ovidiu Munteanu, Jiaping Wang, Junming Xie and Detang Zhou for their interests and their helpful comments on an earlier version of the paper. We also thank the referee for carefully reading our paper and providing helpful suggestions.

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Received: 2021-09-24
Revised: 2022-05-12
Published Online: 2022-07-28
Published in Print: 2022-09-01

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