# Ngô Quốc Anh

## July 21, 2012

### The Yamabe problem: The work by Thierry Aubin

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:05

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

$\displaystyle -\Delta \varphi + R\varphi = C_0 \varphi^\frac{n+2}{n-2}.$

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 $n\geqslant 6$.

By using the notations used in the note about the work of Yamabe, they are

$\displaystyle {F_q}(u) = \frac{{\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla u{|^2} + R{u^2}} \right)d{v_g}} }}{{{{\left( {\displaystyle\int_M {|u{|^q}d{v_g}} } \right)}^{\frac{2}{q}}}}}$

where $q \leqslant \frac{2n}{n-2}$ and

$\displaystyle {\mu _q} = \mathop {\inf }\limits_{u \in {H^1}(M)} {F_q}(u),$

Aubin proved that

Theorem. If $\mu_\frac{2n}{n-2}$ satisfies

$\displaystyle\mu_\frac{2n}{n-2}

then the Yamabe problem is solvable where $\omega_n$ is the volume of the unit sphere in $\mathbb R^n$.

In fact, he proved a stronger result saying that in any case, there holds

$\displaystyle\mu_\frac{2n}{n-2} \leqslant n(n-1)\omega_n^\frac{2}{n},$

and the equality occurs if and only if $M$ 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 $\psi$ such that $F_\frac{2n}{n-2}(\psi).

To conclude the paper, Aubin proved the following

Theorem. If $(M, g)$ ($n \geqslant 6$) is a compact nonlocally conformally flat Riemannian manifold, then

$\displaystyle\mu_\frac{2n}{n-2}

Therefore, the cases that $n=3, 4, 5$ and that $M$ is locally conformally flat are still open.