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 ≤ 0 have at most exponential volume growth.