Hung Tran

This thesis contains several projects investigating aspects of the Ricci flow (RF), from preserved curvature conditions, Harnack estimates, long-time existence results, to gradient Ricci solitons. Recently, B. Wilking proved a theorem giving a simple criterion to check if a curvature condition is preserved along the RF. Using his approach, we show another criterion with slightly different flavor (interpolations of cone conditions). The abstract formulation also recovers a known preserved condition. Another project was initially concerned with the Ricci flow on a manifold with a warped product structure. Interestingly, that led to a dual problem of studying more abstract flows. Using the monotone framework, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's differential Harnack inequality. The behavior of the curvature towards the first finite singular time is also a topic of great interest. Here we provide a systematic approach to the mean value inequality method, suggested by N. Le and F. He, and display a close connection to the time slice analysis developed by B. Wang. Applications are obtained for a Ricci flow with nonnegative isotropic curvature assumption. Finally, we investigate the Weyl tensor within a gradient Ricci soliton structure. First, we prove a Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor and discuss its applications. We are also concerned with the interplay of curvature components and the potential function.