Department of Mathematics - Seminar on Pure Mathematics - Hamilton-Ivey estimates for gradient Ricci solitons

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.