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

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. Introduction The Ricci flow ∂ ∂t g(x, t) = −2Ric(x, t), was introduced by Hamilton in [6]. We say that a quadruple (M, g, f, ε), where (M, g) is a Riemannian manifold, f is a smooth function on M and ε ∈ R, is a gradient Ricci soliton if Rij +∇i∇jf + ε 2 gij = 0. (0.1) We call f the potential function. We say that g is shrinking, steady, or expanding if ε < 0, ε = 0, or ε > 0, respectively. The following volume growth estimate for complete shrinking gradient Ricci solitons was proved by O. Munteanu [8], with an important special case was proved by H.-D. Cao and D.-T. Zhou [5]. Let (M, g, f,−1) be a complete shrinking gradient Ricci soliton. Given o ∈ M, there exists a constant C < ∞ such that Shijin Zhang is currently a visiting PhD student at the Department of Mathematics UCSD. He is partially supported by China Scholarship Council.