| dc.description.abstract | In this thesis, we study the analytical properties of harmonic Ricci flows and Ricci flows in presence of a fi nite time singularity. After recalling some well-known results from the theories of these flows, we start our analysis considering Type I harmonic Ricci flows. We can characterise the pointwise singular behaviour of the flow by means of several natural curvatures associated to the flow, generalising results by Enders, Muller and Topping. Next, we move our attention to Ricci flows, and prove some extension results for Ricci flows under suitable space-time integral curvature assumptions, extending results by Wang. For this purpose, a new and delicate parabolic regularity theory argument is used. Then, a re ned local singularity analysis is developed, introduced jointly with R. Buzano in order to get rid of the Type I condition assumed in the work of Enders, Muller and Topping, obtaining thus a theory for general Ricci flows. This relies on some new concepts of regularity scales and their continuity properties. We give some applications of this new theory to the case of Ricci flows with scalar curvature bounded uniformly along the flow. Finally, we outline a plan for future research on the topic, shortly presenting some research in progress. | en_US |
