Loading...

The Ricci Flow with Applications to the Poincare and Calabi Conjectures

Safdari, Mohammad | 2009

273 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 39622 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Bahreini, Alireza
  7. Abstract:
  8. Poincare conjecture is one of the rst eorts for classifying 3-manifolds that was stated in the begining of 20th century. Eorts for proving this conjecture continued for about 100 years and nally Perelman has solved it in 2003. He has used Ricci ow which has been invented by Richard Hamilton in the early 80's. In this thesis rst we will introduce Ricci ow and will state the results before Perelman such as short time and long time existence of solution. Then we will mention some applications of Ricci ow and will state an explanation of the proof of uniformization and Calabi conjecture by Ricci ow. After that we start the main part of the thesis. In this part we will montion one of the important inventions of Perelman. i.e. L-Length. In fact we will explain one of the steps in the proof of Poincare conjecture. L-Length is a natural length for paths in the space-time and by using it we can dene a notion of volume called reduced volume which is decreasing under ow. This will help to prove the little loop conjecture and enables us to use Cheeger-Gromov type compactness theorems that results in understanding the structure of singularities of the ow. One of the important works of Perelman was removing singularities by surgery which we will explain brie y about it and other steps of the proof in the third chapter.
  9. Keywords:
  10. Ricci Flow ; L-Length ; Reduced Volume ; Poincare Conjecture ; Calabi Conjecture

 Digital Object List

 Bookmark