The Yamabe problem

Abstract

Differential geometry is a field of mathematics which deals with the study of smooth manifolds, which generalize the notion of smooth surfaces to n dimensions. The importance of differential geometry, and its subfield of Riemannian geometry, have only increased over time as they have become the language of one of the foundational theories of modern physics, general relativity [27]. A problem of great prominence in the field of Riemannian geometry is called the Yamabe problem, which was formulated as both as an extension to the uniformisation theorem, proved independently by H. Poincaré and P. Koebe in 1907. as well as a step towards solving the Poincaré conjecture. The statement is as follows [29]: The Yamabe Problem. Given a compact Riemannian Fmanifold (M, g) of dimension n > 3, find a metric conformal to g with constant scalar curvature. A solution was presented by Hidehiko Yamabe in 1960 [45], but an error was discovered in the proof and the problem remained as a conjecture until it was proved for all cases in 1984 by R. Schoen, following the work of Yamabe, N. Trudinger and T. Aubin. In this dissertation, we start by presenting a solution to the uniformisation theorem with an emphasis on its historical context, as well as any necessary analytic preliminaries. Next, we develop the language of differential geometry and more specifically geometric analysis, which rose to prominence in no small part due to the Yamabe problem and the Poincaré conjecture. We then present the solution to the Yamabe problem as it developed over time, highlighting the various geometric, analytical and physical concepts which were required for its total solution. Finally, we present some of the mathematical developments which followed the solution of the Yamabe problem, while also exploring some of the connections between the Yamabe problem and the Poincaré conjecture.