By using the notations used in the note about the work of Yamabe, they are
For simplicity, we set
Schoen proved that:
Theorem. In any case, there holds
and the equality occurs if and only if is conformally equivalent to the sphere with standard metric.
In order to prove the above result, it suffices to consider the case either or is locally conformally flat at some point. The main ingredient of his proof is the positive mass theorem. This well-known result says that if the metric of is conformally flat in a neighborhood of , then the Green fucntion of the conformal Laplacian
say , has an expansion in suitable coordinates near as follows
with . Moreover, if and only if is isometric to the standard flat . In the case is locally conformally flat at some point, the positive mass theorem says that .
Instead of using the test functions
he used a small multiple of outside a neigborhood of , say , verifies
Precisely, he set
for some suitable , in such a way that is continuous across the boundary . Then after several long calculation but worth-doing, he realized that
where denotes the volume of . When is a conformally flat manifold, . Hence, by choosing small and much smaller than , we obtain
showing that as we expected. A careful study shows that in the case , the Green function still has the same expansion as showed above for the locally conformally flat case. Therefore, the above argument works in the case as well.
For the case , Schoen used a modified metric which agrees with outside a small ball centered in and Euclidean in that small ball. When is small, the corresponding Green function for has the following expansion
Schoen showed that one can conclude the theorem provided
This turns out to be the key point since we don’t have any positive mass theorem in hand. Nevertheless, Schoen proved that this is the case provided the original is not conformally isometric to .
Thus, in the case , we start with a modified metric instead of the original one and look for a metric conformally to the modified one to get the positive constant scalar curvature. Once we can conclude
in most of cases mentioned above, we can use the method of variation to complete the proof. More precise, having this estimate, we can get an uniform lower bound on the norm of the sequence of solutions of subcritical problems. Thus, this procedure generates a non-trivial solution of the critical problem.
Just a very quick remark, the sequence of solutions of subcritical problems always converges to a solution of the critical problem. Unfortunately, there is no reason to guarantee that this limiting solution is non-trivial. I have touched this part of argument in my recent paper published in Advances in Mathematics. In my case, we can have an uniform lower bound on the sequence of solutions of subcritical problems, thanks to the term with an negative exponent. In the Yamabe problem, the key argument to make sure that the limiting solution is non-trivial is to prove that
- 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