# Hamilton-Ivey estimates for gradient Ricci solitons

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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.