Constant scalar curvature metrics in a conformal class

Question

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?

Answer 1

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.

Answer 2

Yes, as the OP mentions, in many cases the solution to the Yamabe problem may not be unique (we obviously ignore the effect of rescaling a metric), in the sense that there may be constant scalar metrics different from the minimizer of the total scalar curvature. Recall that the solution to the Yamabe problem was proved by finding a minimizer of the Hilbert-Einstein functional (total scalar curvature) on unit volume metrics in the conformal class $[g]$, however any critical point of this functional is a constant scalar curvature metric.

Perhaps a quick review of some uniqueness/non-uniqueness results could be of interest. For example, in the conformal class of an Einstein metric (except the round sphere), the solution is unique. Moreover,

• Anderson proved that on generic conformal classes, the solution is unique.

On the other hand, there are many non-uniqueness results:

• Ambrosetti and Malchiodi (JFA, 1999) and Berti and Malchiodi (JFA, 2001) proved non-uniqueness results for conformal classes of deformations of the round metric on spheres;

• Pollack (CAG, 1993) proved existence of arbitrarily small $C^0$ perturbations of any given metric (on any closed manifold!), with arbitrarily large number of solutions in its conformal class;

• Schoen (1991) proved that taking the product $S^1\times S^n$ with the product of round metrics, and making the radius $r$ of $S^1$ go to $+\infty$, the number of solutions also goes to $+\infty$ as $r\to+\infty$;

• Recently, de Lima, Piccione and Zedda obtained a sort of generalization, proving existence of infinitely many bifurcation points for many $1$-parameter family of product metrics, obtained by scaling one of the factors. In particular, this yields existence of a countable set of metrics in this family in which conformal class there exist at least three distinct solutions to the Yamabe problem;

• More recently, in a joint paper with Piccione, we obtained a similar result (infinitely many bifurcations) for families of homogeneous metrics on spheres. Recall that the homogeneous metrics on spheres are given by scaling the round metric in the direction of the Hopf fibration, e.g., take $S^3\to S^{4n+3}\to HP^n$ and shrink the round metric on $S^{4n+3}$ in the vertical directions by a factor of $t^2$, obtaining a metric $g_t$. Then, as $t\to 0$ (i.e., as the fibers collapse), our result implies that there are infinitely many values of $t$ for which $[g_t]$ has at least 3 constant scalar curvature metrics. In a subsequent paper, we extend this result to any homogeneous fibration $K/H\to G/H\to G/K$ where $K/H$ has positive scalar curvature and either $H$ is normal in $K$ or $K$ is normal in $G$.

In most cases above, where explicit non-uniqueness of solutions to the Yamabe problem is proven, it is easy to give a qualitative description of the values of the (constant) scalar curvatures of the various solutions. For general results, however, the best available are the type of compactness results of Brendle, Khuri, Marques, Schoen etc mentioned in macbeth's answer.