When the manifold is the standard sphere , the Yamabe is interesting. First, we cover the so-called Kazdan-Warner obstruction. The original statement is for the prescribing scalar curvature problem, i.e. the following PDE
where and are scalar curvatures of the metrics and respectively. We note that, instead of using the condition to avoid the triviality of the limiting solution, in this generalized equation, one has to use
Theorem (Kazdan-Warner obstruction). If is a solution of the preceding PDE on with , then
for any spherical harmonics of degree .
When , such an obstruction has been mentioned once here. We shall not provide any proof for this obstruction here, it is similar to that for the case and basically is integration by parts. We note that all spherical harmonics of degree satisfy
Interestingly, on the sphere having unit volume, constants appearing in the Sobolev inequality are optimal
Note that as we have already discussed in the previous note that the Sobolev inequality takes the following form
with . When the sectional curvature is constant, is allowed as the following holds
In fact, Aubin obtained on the following
if . Suppose has unit volume, using the constant function we easily obtain and this answers why the constants are optimal.
It is important to note that these findings due to Aubin are not optimal at the moment. By a remarkable joint work with Vaugon, Hebey was able to prove that for any compact Riemannian manifold of dimension
for some positive constant .
For the sphere , there holds
To see this, we can set where . Then on one hand, the equation
has no solution due to the Kazdan-Warner obstruction. However, on the other hand,
as long as and is small enough.
In the case , the equation
has infinitely many solutions. For example, with solves