dc.contributor.author Tran, Hung en_US dc.date.accessioned 2014-07-28T19:24:49Z dc.date.available 2019-05-26T06:00:28Z dc.date.issued 2014-05-25 en_US dc.identifier.other bibid: 8641148 dc.identifier.uri https://hdl.handle.net/1813/37064 dc.description.abstract 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, Wilking [98] 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 as in [81]. 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 [63] and F. He [59], and display a close connection to the time slice analysis as in [97]. Applications are obtained for a Ricci flow with nonnegative isotropic curvature assumption. Finally, we investigate the Weyl tensor within a gradient Ricci soliton struc¨ ture. First, we prove a Bochner-Weitzenbock 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. en_US dc.language.iso en_US en_US dc.subject Ricci flow en_US dc.subject Weyl tensor en_US dc.subject Harnack estimates en_US dc.title Aspects Of The Ricci Flow en_US dc.type dissertation or thesis en_US thesis.degree.discipline Mathematics thesis.degree.grantor Cornell University en_US thesis.degree.level Doctor of Philosophy thesis.degree.name Ph. D., Mathematics dc.contributor.chair Cao, Xiaodong en_US dc.contributor.committeeMember Saloff-Coste, Laurent Pascal en_US dc.contributor.committeeMember Gross, Leonard en_US
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