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
Note that, if and are conformally equivalent, then is an isometry from onto . So we will only study below the case .
Followed by this note, the trick use in that note also works for the following equation
In fact, by letting , then our PDE becomes
Let be a solution of the following PDE
where is the average of over . We let . Then it is easy to verify that solves the following
we get that
Unfortunately, it is hard to see the relation between and , the characteristic of .
Let us consider the following so-called Lichnerowicz equation in
Recently, Brezis [here] proved the following
Theorem. Any solution of the Lichnerowicz equation with satisfies in .
Let us study the trick used in his paper.
Then . Fix any point and consider the function
Since as , is achieved at some . We have
Since is increasing we deduce that
By sending we deduce that . In other words, .
As can be seen, he only uses the fact that is monotone increasing in his argument, therefore, this approach can be used for a wider class of nonlinearity.
Today, we shall prove the following identity
First, by the Jacobi formula, we know that for any matrix
where is a differential of . Since
we can rewrite the Jacobi formula as follows
We now make use this rule with replaced by metric and replaced by . Obviously,
where we have used the fact that and that
As an application of the principle of least action to the Einstein-Hilbert action, in this short note, we discuss a question: the variation with respect to metric of the scalar curvature.
To calculate the variation of the scalar curvature we calculate first the variation of the Riemann curvature tensor, and then the variation of the Ricci tensor.
The variation of the Riemann curvature tensor. So, the Riemann curvature tensor is defined as,
Since the Riemann curvature depends only on the Levi-Civita connection , the variation of the Riemann tensor can be calculated as,
Now, since is the difference of two connections, it is a tensor and we can thus calculate its covariant derivative,
We can now cleverly observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms,