Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

Abstract

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient K\"{a}hler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question in case of K\"{a}hler-Ricci solitons proposed by Chow, Lu and Ni in a book.