Some properties of non-compact complete Riemannian manifolds

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold $X$. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on $X$, for any $q\in (0,\infty)$, every non-negative $L^q$ subharmonic function is constant under a natural decay condition on the Ricci curvature.