Heuristics for the Yamabe Problem I am trying to understand the Yamabe problem, and I was naturally lead to a question: given a manifold with a Riemannian metric on it, why is it interesting to find a conformally related metric that has constant scalar curvature? Do you know any situation in which this is applied? Could you give me any reference in which some heuristics are explained? I think a good understanding of the 2-dimensional case would be enough...
 A: Let me outline my understanding of the big picture.
In two dimensions the scalar curvature has a topological obstruction due to the Gauss-Bonnet theorem, this means that not every smooth function on 2-dimensional closed manifold can be a scalar curvature of some metric.
On the other hand, the uniformization theorem implies that every Riemann surface has a metric conformal to a metric of constant Gaussian curvature.
Locally, any two metrics on a two-dimensional manifold are conformally equivalent, so one may say that there is no local conformal structure in this case.
Rephrasing the uniformization theorem, every Riemann surface has the best metric, namely the one with constant curvature.
As there is a lot of possible metrics on a manifold $M$, it would be good if similar to two dimensional case we could have a notion of the best possible metric for any dimension.
What is the "best" is not quite clear. Asking just for a metric with a constant scalar curvature is not too impressive since according to Aubin's theorem every compact manifold can be given a metric with constant negative curvature. See e.g. J.L. Kazdan "Prescribing the curvature of a Riemannian manifold" (AMS, 1985), p.10.
More on scalar curvature please see in this discussion on MathOverflow.
A wiser approach would be define the "best" as a critical point of some functional. As we are concerned with the scalar curvature, such a functional would be
$$
\mathrm{EH}(g) = \int \mathrm{S}^g \, \mathrm{dvol}_g
$$
which is known as the Einstein-Hilbert action. Here $ \mathrm{S}^g$ is the scalar curvature of metric $g$, and $\mathrm{dvol}_g$ is the corresponding volume element.
Since we can scale the metrics with constant factors ("homotheties"), it makes sense to consider critical points of this functional over the space of all metrics with fixed volume. The critical points in this setting are known to be Einstein metrics. In dimension $n \ge 3$ the scalar curvature of an Einstein metric $g$ is constant. Unfortunately, not all compact manifolds admit an Einstein metric.
So, Yamabe asked for something weaker: given a metric, can we deforming it in the simplest possible way, namely by multiplying it by a positive smooth function, obtain a new metric which has constant scalar curvature? Such metrics are again critical points of $\mathrm{EH}(g)$ but now over all metrics within a conformal class. More on this story can be recovered from the Wikipedia's article on the Yamabe invariant.
I also recommend this short overview of the Yamabe problem.
I used the following sources when preparing this answer.
1. H. Yamabe, On a deformation of Riemannian structures on compact manifolds.
2. J.M. Lee, T.H. Parker, The Yamabe problem
3. J.L. Kazdan, Applications of Partial Differential Equations to Problems in Geometry
4. J.M. Lee, Riemannian Manifolds, An Introduction to Curvature, pp 124-127.
(I am also deeply indebted to Heather Macbeth for an enlightening talk on the topic given in the University of Auckland on 19.12.2012)
