Let $(M,g)$ be a compact Riemannian manifold, then by the resolved Yamabe-problem, there exists a metric $\tilde{g}$ of constant scalar curvature in the conformal class $[g]$ of $g$. By normalizing volume, we have $s_{\tilde{g}}=Y(g)$ where $Y(g)=Y([g])$ is the Yamabe functional.
This might not be the only constant scalar curvature metric in $[g]$. So my question is: is there anything known about the set of constants $c$ where $c\equiv scal_{\tilde{g}}$ for a metric $\tilde{g}\in [g]$ of unit volume?
Best Answer
Plenty is known! For instance,
(Schoen 1989) For any $N$, there exists a product of round spheres whose conformal class' set of such constants has size at least $N$.
(Brendle-Marques 2009, based on earlier work of Brendle) In any dimension $n\geq 25$, there exist conformal classes on $S^n$ for which the set of such constants is infinite. Specifically, there exists a sequence of them tending upwards to the Yamabe constant of the round sphere.
(Khuri-Marques-Schoen 2009) For any spin manifold $M$ of dimension $\leq 24$, for any conformal class and any real $c$, the set of metrics with constant scalar curvature $c$ is compact in the $\mathcal{C}^2$ topology.
For a recent survey see this article of Brendle-Marques.