Following the previous topic where we was able to point out the serious mistake in the Yamabe paper found by Trudinger. Today, we discuss about the way Trudinger did correct that mistake. Trudinger published the result in a paper entitlde “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds” in Ann. Scuola Norm. Sup. Pisa in 1968. The paper can be downloaded from this link.
In the paper, he proved the following result
Theorem 2. There exists a positive constant (depending on , ) such that if , there exists a positive, solution of the equation
Let us discuss the proof of the above result. Again, the sub-critical approach was used in his argument and we refer the reader to the previous topic.
He said that we expect a subsequence of the converges in a certain sense to a smooth solution of the critical equation. However, the convergence is not strong enough to imply the non-triviality of the resulting solution. Fortunately, if is small enough, the convergence is sufficiently nice to guarantee a positive, smooth solution of the critical equation.
Recall that the function verifies the sub-critical equation in the weak sense, that is,
for all test functions .
We now choose the test function for some . The result is
Writing , the above inequality becomes
Let us suppose . Using the Sobolev and Holder inequalities, we thus obtain
since is uniformly bounded independent of . Keep in mind that the constant depends on and if is sufficiently small. This is because of the second Sobolev constant in the Sobolev inequality in .
Hence, if , for large , we get that
Clearly, this estimate continues to hold if . We now choose such that . With this, one can check that
which implies that is bounded. This and the previous estimate tell us that is also bounded, that is,
which proves the equiboundedness of . Note that this is reasonable by standard elliptic theory since
and (this is the most important part)
The proof completes.
Trudinger also proved a useful regularity result which we have discussed before, check this entry. As can be seen, for large , the solvablity of the Yamabe was still left open. In the next entry, we talk about the work of Aubin.