One special feature for the Ricci flow in dimension 3 is the
Hamilton-Ivey estimate. The curvature pinching estimate provides a lot of
information about the ancient solution and plays a crucial role in the
singularity formation of the flow in dimension 3. We study the pinching
estimate on 3 dimensional expanding and 4 dimensional steady gradient Ricci
solitons. A sufficient condition for a 3-dimensional expanding soliton to
have positive curvature is established. This condition is satisfied by a
large class of conical expanders. As an application, we show that any
3-dimensional gradient Ricci expander C^2 asymptotic to certain cones is
rotationally symmetric. We also prove that the norm of the curvature tensor
is bounded by the scalar curvature on 4 dimensional non Ricci flat steady
soliton singularity model and derive a quantitative lower bound of the
curvature operator for 4-dimensional steady solitons with linear scalar
curvature decay and proper potential function. This talk is based on a joint
work with Zilu Ma and Yongjia Zhang.