On curvature estimates for four-dimensional gradient Ricci solitons

Abstract: In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for $4$-dimensional gradient steady Ricci solitons, including the sharp curvature estimate $|Rm|\le C R$ for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 1.1).