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

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