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Shrinkers with Curvature-Pinching Conditions are Compact

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Abstract

In this paper, we give various curvature-pinching conditions such that shrinkers are compact. On one hand, we prove that shrinkers with positive Ricci curvature are compact when they have bounded curvature and certain curvature-pinching conditions. On the other hand, we prove that shrinkers with certain asymptotically nonnegative sectional curvature are compact. As applications, some related classifications of shrinkers are provided.

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Acknowledgements

The authors thank Xiaolong Li for making them aware of the work of his joint work [14]. We would like to thank Jianyu Ou and Yuanyuan Qu for helpful discussion. The authors thank the referee for making valuable comments and suggestions which helped improve the presentation of this work. The first author is supported by Natural Science Foundation of Zhejiang Province (Grant No. LY23A010016).

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Authors and Affiliations

  1. School of Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Guoqiang Wu

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Jia-Yong Wu

  3. Newtouch Center for Mathematics of Shanghai University, Shanghai, 200444, China

    Jia-Yong Wu

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  1. Guoqiang Wu
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  2. Jia-Yong Wu
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Correspondence to Jia-Yong Wu.

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Wu, G., Wu, JY. Shrinkers with Curvature-Pinching Conditions are Compact. J Geom Anal 33, 285 (2023). https://doi.org/10.1007/s12220-023-01352-4

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