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Gradient steady Ricci solitons are natural generalizations of Ricci-flatmanifolds. In this article, we prove a curvature gap theorem for gradientsteady Ricci solitons with nonconstant potential functions; and a curvature gaptheorem for Ricci-flat manifolds, removing the volume growth assumptions inknown results.

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