On the classification of four-dimensional gradient Ricci solitons

Abstract

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with div4Rm±=0, we show that it is either Einstein or a finite quotient of R4, S2×R2 or S3×R. The same result can be obtained under the condition of div4W±=0. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.