A Variational Approach to the Yamabe Problem: Conformal Transformations and Scalar Curvature on Compact Riemannian Manifolds

We start by taking the analytical approach to discuss how the minimizer of Yamabe functional provides constant scalar curvature and its relationship with the Sobolev Space $W^{1,2}.$ Then, after demonstrating the importance of the sphere $S^n$, with stereographic projection and dilation, we show that the minimizer of Yamabe functional on standard sphere is obtained from a standard round metric $\bar{g}$ by a conformal diffeomorphism, thus giving us the constraint $\lambda (M) < \lambda(S^n),$ which leads us to the final theorem that the Yamabe problem is solvable when $\lambda(M) < \lambda(S^n).$ For the proof of this theorem, we adopt the approach of Concentration-Compactness.