Aspects Of The Ricci Flow

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.

Name thesis:degree_name

Ph. D., Mathematics

Level thesis:degree_level

Doctor of Philosophy

Discipline thesis:degree_discipline

Mathematics

Grantor

Cornell University

Year dc:date.issued

2014

Author dc:creator

• Tran, Hung

Committee members dc:contributor.committeemember

• Saloff-Coste, Laurent Pascal
• Gross, Leonard

• Ricci flow
• Weyl tensor
• Harnack estimates

Language dc:language.iso

en_US

Handle dc:identifier.uri

https://hdl.handle.net/1813/37064

OAI identifier oai:identifier

oai:ecommons.cornell.edu:1813/37064