TITLE: Some questions about scalar curvature
QUESTION [33 upvotes]: Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor.  I will now summarize everything that I know about scalar curvature in three sentences:

The scalar curvature at a point relates the volume of an infinitesimal ball centered at that point to the volume of the ball with the same radius in Euclidean space.
There are no topological obstructions to negative scalar curvature.
On a compact spin manifold of positive scalar curvature, the index of the Dirac operator vanishes (equivalently, the $\hat{A}$ genus vanishes).

The third item is of course part of a larger story - one can use higher index theory to produce more subtle positive scalar curvature obstructions (e.g. on non-compact manifolds) - but all of these variations on the compact case.
I am also aware that the scalar curvature is an important invariant in general relativity, but that is not what I want to ask about here.  This is what I would like to know:

Are there any interesting theorems about metrics with constant scalar curvature?  For example, are there topological obstructions to the existence of constant scalar curvature metrics, or are there interesting geometric consequences of constant scalar curvature?
Can anything be said about manifolds with scalar curvature bounds (other than the result I quoted above about spin manifolds with positive scalar curvature), analogous to the plentiful theorems about manifolds with sectional curvature bounds?  (Thus additional hypotheses like simple connectedness are allowed)
Is anything particular known about positive scalar curvature for non-spin manifolds?

REPLY [6 votes]: There is an important relation between variation of scalar curvature Kahler metric and Lichnerowicz operator. Lets explain it here
Let $(X,J,\omega)$ be Fano Kahler manifold with $\omega_0\in [\omega_0]=\kappa$, then by $\partial\bar\partial$-lemma any other Kahler 2-form can be written as $\omega_0+\sqrt{-1}\partial\bar\partial \varphi$. Now the set of Kahler metrics in $[\omega_0]$ can be identified with $$\mathcal H=\{\varphi \in C^\infty(X,\mathbb R)|\omega_0+\sqrt{-1}\partial\bar\partial \varphi>0\}/\mathbb R$$. Note that $T_\varphi\mathcal H=C^\infty(X,\mathbb R)/\mathbb R$
The Scalar curvature can de define as 
$$Sc:\mathcal H\to C^{\infty}(X,\mathbb R)$$
$$Sc(\varphi)=\Lambda_\varphi\sqrt{-1}\partial\bar\partial\log \omega^n$$
and for $\psi\in T_{\varphi}\mathcal H$ we have 
$$d \text{Sc}(\varphi).\psi=\Delta_{\omega_\varphi}^2\psi-Sc(\varphi)\Delta_{\omega_{\varphi}}\psi$$ and moreover by using following known relation of Lichnerowicz  
$$\mathcal D_\varphi^*\mathcal D_\varphi \psi=\Delta_{\omega_\varphi}^2\psi-Sc(\varphi)\Delta_{\omega_\varphi}\psi+\frac{1}{2}g(dSc(\varphi),d\psi)$$ so we can rewrite the derivative of scalar curvature metric as follows
$$d \text{Sc}(\varphi).\psi=\mathcal D_\varphi^*\mathcal D_\varphi \psi-\frac{1}{2}g(dSc(\varphi),d\psi)$$
See also Oliver Biquard expository paper