2020/04/29 by Xu Cheng, Cheng, Xu, Ernani Ribeiro Jr +3 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.2004.14426
arxiv created 2020/04/29 · arxiv updated 2020/05/01
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.