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

Abstract

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.Comment: 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 page