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arXiv · 0909.0716

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

Shijin Zhang
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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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Shijin Zhang. 2009-09-03. On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below. https://doi.org/10.1007/s10114-011-9527-7

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