Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Μελάς Αντώνιος Καθηγ.(Επιβλέπων), Λάππας Διονύσιος Αναπλ. Καθηγ., Τσαρπαλιάς Αθανάσιος Καθηγ.
Original Title:
Η ροή Ricci και εφαρμογές
Summary:
The goal of the current thesis is a detailed presentation of the
results and techniques used by Richard Hamilton in his groundbreaking work
“Three-manifolds with positive Ricci curvature”, in order to prove the
following theorem:
Theorem 0.0.1. Let X be a compact 3-manifold which admits a Riemannian
metric with strictly positive Ricci curvature. Then X also admits a metric of
constant positive curvature.
In particular, we try to describe the main new tool he created, the Ricci flow.
The first chapter is a comprehensive introduction to the theory of Riemannian
manifolds with emphasis on the aspects of the theory that will be used heavily
in what follows. Chapter 2 consists of an overview of the history of the
Poincare and Thurston conjectures. The main goal here is to highlight the Ricci
flow as the tool through which Hamilton carried a topological-geometric problem
into the area of differential equations, but also to describe the steps made by
him towards the solution of Thurston’s conjecture and also of those made by
Grigori Perelman who managed to overcome the difficulties that stopped Hamilton
thus completing the proof.
The final three chapters contain a step by step presentation of the proof of
theorem (0.0.1). In the 3rd chapter we introduce the concept of a flow of
manifolds and a few examples are given with that of the Ricci flow being the
most important. The next chapter is about the maximum principle which is the
main tool used by Hamilton to prove many of the results that follow. We mention
several special cases of the maximum principle, but the emphasis is given on
those concerning maps between vector bundles. Finally, in the last chapter we
describe the proof of all the intermediate results leading to the completion of
the proof of theorem (0.0.1).
Keywords:
Ricci flow, Poincare conjecture, Geometrization conjecture, Thurston, Perelman
