  We consider complete noncompact Riemannian manifolds with quadratically
decaying lower Ricci curvature bounds and minimal volume growth. We first prove
a rigidity result showing that ends with strongly minimal volume growth are
isometric to warped product manifolds. Next we consider the almost rigid case
in which manifolds with nonnegative and quadratically decaying lower Ricci
curvature bounds have minimal volume growth. Compact regions in such manifolds
are shown to be asymptotically close to warped products in the Gromov-Hausdorff
topology. Manifolds with nonnegative Ricci curvature and linear volume growth
are shown to have regions which are asymptotically close to being isometric
products. The proofs involve a careful analysis of the Busemann functions on
these manifolds using the recently developed Cheeger-Colding Almost Rigidity
Theory. In addition, we show that the diameters of the level sets of Busemann
functions in such manifolds grow sublinearly.
