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We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. Upper bound for the scalar curvature of expander with nonpositive Ricci curvature will also be given. Furthermore, we provide a sufficient condition for the scalar curvature of expanding soliton being non-negative. Curvature estimates of expanding solitons in dimensions three and four will also be established. As an application, we prove a gap theorem on 3D gradient expander.

Curvature Estimates and Gap Theorems for Expanding Ricci Solitons