Following the previous note about the work of Trudinger, today we talk about the work of Aubin regarding to the Yamabe problem, that is the following simple PDE
In his elegant paper entitlde “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire” published in J. Math. Pures Appl. in 1976, Aubin proved the existence for almost all manifolds for .
By using the notations used in the note about the work of Yamabe, they are
Aubin proved that
Theorem. If satisfies
then the Yamabe problem is solvable where is the volume of the unit sphere in .
In fact, he proved a stronger result saying that in any case, there holds
and the equality occurs if and only if is conformally equivalent to the sphere with standard metric. Having this result, to solve the Yamabe problem, we have only to exhibit a test function such that .
To conclude the paper, Aubin proved the following
Theorem. If () is a compact nonlocally conformally flat Riemannian manifold, then
Therefore, the cases that and that is locally conformally flat are still open.
- 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