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,
and (see this note for a similar calculation for Laplacian)
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.
- 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