We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy \(|{{{\,\textrm{Rm}\,}}}|\le cR\) for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvature decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Math. Ann. 364(3–4), 777–792 (2014) in dimension three and show that any 3-dimensional gradient Ricci expander \(C^2\) asymptotic to \(\left( C(\mathbb S^2), dt^2+\alpha t^2 g_{\mathbb {S}^2}\right) \) is rotationally symmetric, where \(\alpha \in (0,1]\) is a constant and \(g_{\mathbb {S}^2}\) is the standard metric on \(\mathbb {S}^2\) with constant curvature 1.
