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Weak scalar curvature lower bounds along Ricci flow

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Abstract

In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon (2002) that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W1,p for some n < p ⩽ ∞. As an application, we use this result to study the relation between the Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense, then the manifold is isometric to a Ricci flat manifold.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 12125105 and 12071425) and the Fundamental Research Funds for the Central Universities. The second author was supported by National Natural Science Foundation of China (Grant Nos. 11971424 and 12031017). The third author was supported by National Natural Science Foundation of China (Grant No. 11971424). The authors thank Professor Dan Lee and Professor Christina Sormani for many helpful suggestions.

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  1. Huaiyu Zhang

    Present address: Current address: School of Mathematics and Statistics, Nanjing University of Science & Technology, Nanjing, 210094, China

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310058, China

    Wenshuai Jiang, Weimin Sheng & Huaiyu Zhang

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  1. Wenshuai Jiang
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  2. Weimin Sheng
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  3. Huaiyu Zhang
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Correspondence to Weimin Sheng.

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Jiang, W., Sheng, W. & Zhang, H. Weak scalar curvature lower bounds along Ricci flow. Sci. China Math. 66, 1141–1160 (2023). https://doi.org/10.1007/s11425-021-2037-7

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  • DOI: https://doi.org/10.1007/s11425-021-2037-7

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