The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:


(A) Every (smooth) function is a scalar curvature.


(B) The manifold is strongly scalar flat. 


(C) The manifold only admits scalar curvatures which are negative somewhere.


Of course class (A) is nonempty in all dimensions because it contains $S^n$. Gromov and Lawson showed that (B) contains all tori $T^n$. However, it's not clear to me that (C) is nonempty in all dimensions. Kazdan and Warner (Prescribing Curvatures, *Proc. Symp. Pure Math.* **27**) showed:


Let $M$ be a spin manifold with $\hat A(M)\ne 0$ and $b\_1(M)=\dim M$. Then $M$ does not admit a metric of zero scalar curvature.


Consequently, any such manifold must be type (C). They only give the example $T^4\#K3$. Are there examples in dimensions $3$ and $\ge 5$ of type (C) manifolds? Presumably one could use the Kazdan-Warner result above and then apply some knowledge of manifolds with nonzero A-roof genus. They mention Hitchin told them one can strengthen the hypothesis to $b\_1(M)\ne 0$. 



Examples of manifolds that do not admit scalar flat metrics
Christos Mantoulidis showed me how to construct examples in (C) in all dimensions. Namely, if $\Sigma\_g^2$ denotes a genus $g$ surface with $g\ge 2$, then $\Sigma\_g^2\times T^{n-2}$ does is in class (C). 


It does not carry a PSC metric because it is enlargeable (because it carries a metric of nonpositive sectional curvature or because it is a product of enlargeable manifolds). It is not in class (B) because it does not carry a Ricci flat metric. This follows from the general fact that $\mathrm{Ric}\ge 0\implies b\_1\le n$. But by the Kuenneth formula,
$$b\_1(\Sigma\_g^2\times T^{n-2})=b\_1(\Sigma\_g^2)+b\_1(T^{n-2})=2g+n-2,$$
which is $>n$ when $g\ge 2$. Therefore these manifolds lie in class (C). 


