Volume growth estimates for Ricci solitons and quasi-Einstein manifolds

In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. We prove a sharp volume growth estimate for complete noncompact gradient shrinking Ricci soliton. Moreover, we provide upper bound volume growth estimates for complete noncompact quasi-Einstein manifolds with $\lambda=0.$ In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with $\lambda<0$ and $\mu\leq 0$ have at most exponential volume growth.