  We derive a precise estimate on the volume growth of the level set of a
potential function on a complete noncompact Riemannian manifold. As
applications, we obtain the volume growth rate of a complete noncompact
self-shrinker and a gradient shrinking Ricci soliton. We also prove the
equivalence of weighted volume finiteness, polynomial volume growth and
properness of an immersed self-shrinker in Euclidean space.
