Regularity of the Ricci Flow and Rigidity of Ricci Solitons

In this dissertation we consider problems related to the singularity analysis of the Ricci flow. We obtain curvature conditions which prevent the Ricci flow on a closed manifold from developing singularities in finite time. These can be viewed as regularity results for the Ricci flow as a weakly parabolic PDE system. We also prove some rigidity phenomena for Ricci-flat manifolds and gradient steady Ricci solitons. In particular, we improve an -rigidity theorem for Ricci-flat manifolds by removing all the volume growth assumptions in its previously known versions. Then we generalize this -rigidity result to gradient steady Ricci solitons with non-constant potential functions.