Some Uniqueness and Rigidity Results on Gradient Ricci Solitons

This thesis contains my work during Ph.D. studies under the guidance of my advisor Huai-Dong Cao.We initiated our research on Perelman\u27s Conjecture stating that the three-dimensional steady gradient Ricci soliton is the Bryant soliton up to scaling, and we managed to prove this with the assumption that the metric is locally conformally flat.Later, exploring the Bach tensor, we managed to show that a four-dimensional Bach flat shrinking Ricci soliton is either Einstein, the quotient of a Gaussian soliton R^4 or the product of S^3 with R. For dimension n\u3e4, a Bach flat Ricci soliton is either Einstein, the quotient of Gaussian soliton R^4 or the product of an Einstein manifold with a line. A similar argument can be carried over to steady Ricci solitons with some additional assumptions.In the proof we constructed a covariant 3-tensor called the $D$-tensor which is verified to be a key link for the geometry of Ricci solitons and the well-known Weyl curvature, Cotton tensor and Bach tensor. As an extended study, joint with Meng Zhu, we establish the rigidity result for Kahler-Ricci solitons with harmonic Bochner tensor. Joint with Chenxu He, we also applied the Bach-flat argument to quasi-Einstein manifolds and prove the classification theorem