The Yamabe Problem

Abstract

In this thesis, we study the proof of the so-called Yamabe Problem. This problem was proposed by Yamabe in an attempt to solve the Poincaré conjecture eventually. The problem was to prove whether, given any compact Riemannian manifold M_n(n ≥ 3), a conformal change of metric exists such that the manifold has a constant scalar curvature. This geometric problem reduces to proving the existence of smooth, positive solutions to a semilinear elliptic PDE of the form ∆u + h(x)u = λf (x)u^(2^∗−1) where h, f are smooth and f is strictly positive. In this thesis, we study the solution to Yamabe’s problem. This includes studying many prerequisites such as Sobolev spaces, Regularity theory for uniformly elliptic equations, and a little Calculus of Variations. In the end, we study Lee-Parker’s paper for a solution to Yamabe’s problem.