Aubin's conjecture for the Yamabe invariant

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$. For nonzero $\psi\in H_1(M)$, define the Yamabe functional

$$I_g(\psi)=\frac{\displaystyle\int_M \left(|\nabla\psi|^2+\frac{n-2}{4(n-1)}R_g\psi^2\right)\,\mathrm{d}v}{\|\psi\|_N^2},\qquad N=\frac{2n}{n-2},$$

and set

$$\mu(M,g)=\inf_{\psi\in H_1(M)-\{0\}}I_g(\psi).$$

Here $R_g$ is the scalar curvature of $g$, and $(S_n,g_{can})$ denotes the round sphere with its canonical metric. Aubin's conjecture. If $(M,g)$ is not conformal to $(S_n,g_{can})$, then

$$\mu(M,g)<\mu(S_n,g_{can}).$$

This strict inequality is the key estimate in the solution of the Yamabe problem: it ensures that the infimum defining the Yamabe invariant is attained outside the conformal class of the round sphere. The conjecture was subsequently proved, so its status is solved.