# Ngô Quốc Anh

## June 28, 2011

### The Yamabe problem: A Story

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:47

I want to write a short survey about the Yamabe problem. Long time ago, I introduced the problem in this blog [here] but it turns out that the note was not rich enough to perform the importance of the problem.

Hidehiko Yamabe, in his famous paper entitled On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), pp. 21-37,  wanted to solve the Poincaré conjecture

Conjecture. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

For this he thought, as a first step, to exhibit a metric with constant scalar curvature. We refer the reader to this note for details. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement:

Theorem (Yamabe). On a compact Riemannian manifold $(M, g)$ of dimension $\geqslant 3$, there exists a metric $g'$ conformal to $g$, such that the corresponding scalar curvature $\text{Scal}_{g'}$ is constant.

As can be seen, the Yamabe problem is a special case of the prescribing scalar curvature problem that can be completely solved. For the prescribing scalar curvature, we also solve it completely when the invariant is non-positive.

1. Conformal metrics.

Definition (conformal). Two pseudo-Riemannian metrics $g$ and $\widetilde g$ on a manifold $M$ are said to be

• (pointwise) conformal if there exists a $C^\infty$ function $f$ on $M$ such that

$\displaystyle \widetilde g=e^{2f}g$;

• conformally equivalent if there exists a diffeomorphism $\alpha$ of $M$ such that $\alpha^* \widetilde g$ and $g$ are pointwise conformal.

Note that, if $g$ and $\widetilde g$ are conformally equivalent, then $\alpha$ is an isometry from $e^{2f}g$ onto $\widetilde g$. So we will only study below the case $\widetilde g = e^{2f}g$.