Volume Growth Estimates for Ricci Solitons and Quasi-Einstein Manifolds

Abstract

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 $$\mu \le 0$$ have at most exponential volume growth.