Some rigidity results for noncompact gradient steady Ricci solitons and Ricci-flat manifolds

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
The result concerning ACyl manifolds was removed from the previous version since it can be generalized and does not depend on either Ricci soliton or Ricci-flat condition. 18 pages