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On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

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Abstract

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.

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  1. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, P. R. China

    Shi Jin Zhang

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  1. Shi Jin Zhang
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Correspondence to Shi Jin Zhang.

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Zhang, S.J. On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below. Acta. Math. Sin.-English Ser. 27, 871–882 (2011). https://doi.org/10.1007/s10114-011-9527-7

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  • DOI: https://doi.org/10.1007/s10114-011-9527-7

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