Supervisors info:
Παναγιώτης Γιαννιώτης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Summary:
The main subject of this graduate thesis is the Yamabe problem, which is about conformal deformation of Riemannian metrics to ones of constant scalar curvature. In Chapter 1, we give an overview of the problem, as well as a solution for the 2-dimensional case using methods of Riemann surfaces. In Chapter 2, we give a review of the prerequisites, which are the classical theories of Riemannian manifolds and elliptic PDEs, up to the point that they are usually treated in graduate-level courses. Chapter 3, the main one in this thesis, systematically treats the Yamabe problem. Section 3.1 shifts the problem to the value of the Yamabe invariant, while Section 3.2 is concerned with the determination of this value, completing the solution to the problem. Finally, Chapter 4 is a survey on the spinorial Yamabe problem, which can be considered a first-order analogue of the Yamabe problem and shares a lot of similarities.
