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 $λ=0.$ In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with $λ<0$ and $μ\leq 0$ have at most exponential volume growth.

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