On curvature, volume growth and uniqueness of steady Ricci solitons

This thesis contains my work during my Ph.D. studies at Lehigh University under the guidance of my advisor Huai-Dong Cao. The work is related to objects called Ricci solitons which serve as singularity models of Ricci ow. We are going to study Ricci solitons in this thesis from the following aspects: 1. Curvature properties. 2. Volume growth properties. 3. Uniqueness under constraints of the asymptotic geometry.;We first explore the curvature estimate for four dimensional steady Ricci solitons. The main result is about control of the full curvature tensor Rm by scalar curvature R..;We are then going to study curvature and volume growth properties of complete steady Kahler Ricci solitons with positive Ricci curvature. The main result is that volume growth is at least half dimensional and scalar curvature behaves like 1/r in average where r is the geodesic distance to some point.;In the third part, we are going to study the uniqueness of the steady Kahler Ricci soliton constructed by Huai-Dong Cao under constraints of the asymptotic geometry. The main result says that it is unique if we ask that the metric tensor be C1 close in some sense to the model.