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

TL;DR

This paper investigates the Yamabe problem on compact Riemannian manifolds, demonstrating that minimizers of the Yamabe functional yield constant scalar curvature and establishing conditions for solvability using conformal transformations and concentration-compactness.

Contribution

It provides a variational framework for solving the Yamabe problem, showing the existence of solutions under specific scalar curvature constraints on compact manifolds.

Findings

Minimizers of the Yamabe functional produce constant scalar curvature.

The Yamabe problem is solvable when the Yamabe invariant of the manifold is less than that of the sphere.

Concentration-compactness method is effective in proving the main theorem.

Abstract

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.