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 of dimension , there exists a metric conformal to , such that the corresponding scalar curvature 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 and on a manifold are said to be

- (pointwise) conformal if there exists a function on such that
;

- conformally equivalent if there exists a diffeomorphism of such that and are pointwise conformal.

Note that, if and are conformally equivalent, then is an isometry from onto . So we will only study below the case .

**2. Scalar curvature under conformal changes of Riemannian metrics.**

We have already shown in this entry that under the conformal change , the scalar curvatures verify the following equation

If we consider the conformal deformation in the form (with , ), that is,

we then have

In other words,

Thus

and (see this note for a similar calculation for Laplacian)

Therefore,

Hence,

**3. The Yamabe approach.**

From the previous section, the scalar curvature satisfies the equation

So, Yamabe problem is equivalent to solving the above equation with and the solution must be smooth and strictly positive.

**4. The mistake and…**

The Yamabe problem was born, since there is a gap in Yamabe’s proof.

See also: Conformal invariant operators: Laplacian operators, 2

- The Yamabe problem: A Story
- The Yamabe problem: The work by Hidehiko Yamabe
- The Yamabe problem: The work by Neil Sidney Trudinger
- The Yamabe problem: The work by Thierry Aubin
- The Yamabe problem: The work by Richard Melvin Schoen

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