Following the previous post, we are interested in solving the following equation
where (with , ) is a conformal metric conformally to . In this entry, we introduce the Hidehiko Yamabe approach. His approach is variational. To keep his notation used, we rewrite the PDE as the following
Yamabe tried to minimize the following
over the Sobolev space where . Let us say
In the first stage, he showed that
Theorem B. For any , there exists a positive function satisfying
In the second stage, he sent to to obtain a solution. He proved
Theorem C. As tends to , a uniform limit of such exists, is positive and satisfies
Once the preceding PDE is solved, one can replace by any constant, thus, finishing the prescribing scalar curvature in the constant case.
While the proof of Theorem B is correct, a mistake was found in the proof of Theorem C by Trudinger. As can be expected, in order to send to , one needs to derive some uniform bound for the sequence of solutions of subcritical equations. In order to do that, Yamabe used the Green function for the operator
It is important to note that if and we define
- for any satisfying
- Therefore where is an absolute constant if is larger than a fixed constant.
Having this result, Yamabe used an iteration to derive an uniform boundedness. Take a positive fixed . Starting at
he then defined
It is immediately to check that
Clearly, using the Green function, there holds
Therefore, due to the presence of the high order term, if , we would have
Using this, we get that
Here we have used the following fact
to get the estimate. Clearly,
In other words, we would have
If we repeat this procedure, we would have
Unfortunately, this estimate was not calculated correctly in the Yamabe paper. To be precise, he claimed that
This points out a serious mistake found by Trudinger. In other words, in view of the iteration procedure, to compare norms between and , we first compare norms between and and then compare norms between and . It turns out that Yamabe made a mistake in the latter part where he missed to raise to power .
Now let us see the rest of Yamabe’s argument if this mistake were correct. Indeed, by denoting
we get by the Holder inequality, since , that
However, from the choice of , we get
Hence, for small and since , there holds
Combining all, we arrive at
which immediately implies that is bounded since . Notice that if we presume that Yamabe did correct, then we can go though the proof of uniformly boundedness of the sequence of solutions since the bound for is independent of so that the sequence of solutions is uniformly bounded.
If we modify the above argument using the correct estimate, we would have
But then we get
which gives us nothing since the exponent is too big, in fact, converging to as and we have no way to control from above, even for the case when no iteration happens.
We close this note by quoting the following in a paper by Trudinger: “More intuitively speaking, his proof of Theorem C cannot be expected to work since it does not distinguish at all two facts related to the problem
- the presence of the term in the equation and
- the compactness of “.
- 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