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

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

2009-09-03

arxiv.org/abs/0909.0716v2

Acta Mathematica Sinica, Einglish series. vol. 27, 2011, 871-882

Mathematics

Differential Geometry

15 pages. Added some references and an appendix to give a more simple computation of the volume estimate

Scientific paper

10.1007/s10114-011-9527-7

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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