The volume growth of complete gradient shrinking Ricci solitons

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature. A complete Riemannian manifold M of dimension n is called gradient shrinking Ricci soliton if there exists f ∈ C (M) and a constant ρ > 0 such that Rij +∇i∇jf = ρgij , where Rij denotes the Ricci curvature tensor and ∇i∇jf denotes the Hessian of f . We can scale the metric on M such that ρ = 1 2 , which will always be assumed in this paper i.e. Rij +∇i∇jf = 1 2 gij. (1) Gradient Ricci solitons have been intensively studied in the context of the Ricci flow ( [H1], [H2]) and are natural generalizations of Einstein metrics. Since often the limit of dilations of singularities in the Ricci flow is a Ricci soliton, it is useful to have a good knowledge of their geometry. In a recent paper, H.-D. Cao and D. Zhou have studied the asymptotic behavior of the potential function f and the volume growth rate of complete noncompact gradient shrinking solitons [C-Z]. They proved, assuming the normalization (1), that the potential function satisfies 1 4 (r (x)− c) ≤ f (x) ≤ 1 4 (r (x) + c) ,