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In this note, we obtain a sharp volume estimate for complete gradient Riccisolitons with scalar curvature bounded below by a positive constant. UsingChen-Yokota's argument we obtain a local lower bound estimate of the scalarcurvature for the Ricci flow on complete manifolds. Consequently, one has asharp estimate of the scalar curvature for expanding Ricci solitons; we alsoprovide a direct (elliptic) proof of this sharp estimate. Moreover, if thescalar curvature attains its minimum value at some point, then the manifold isEinstein.

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