As mentioned by Willie Wong, I modified to the following version:


Let $M$ be a closed smooth $4$ manifold.


**Q**
Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int\_MScal^2\_gdv\_g=c$, where $Scal\_g$ denotes the scalar curvature of $g$? If not, for any small $\epsilon>0$, can we find a metric $g\_\epsilon$ such that $|\int\_MScal^2\_{g\_\epsilon}dv\_{g\_\epsilon}-c|<\epsilon$?


PS I do not know whether the question is trivial or not. Any reference is welcome.



Can we prescribe the $L^2$ norm of the scalar curvature on a four-manifold?
This is not always possible.


Let $M$ be a compact smooth manifold of dimension $n$. Consider the Einstein-Hilbert functional $\mathcal{E}$ given by


$$\mathcal{E}(g) = \dfrac{\displaystyle\int\_Ms\_g d\mu\_g}{\operatorname{Vol}(M, g)^{\frac{n-2}{n}}}.$$


If $\mathcal{C}$ is a conformal class, then by using the conformal Laplacian and Hölder's inequality, one can show that $\mathcal{E}|\_{\mathcal{C}}$ is bounded below. So we can define the *Yamabe constant of $\mathcal{C}$* to be the finite quantity $Y(M, \mathcal{C}) = \inf\_{g\in\mathcal{C}}\mathcal{E}(g)$. A result of Aubin shows that $Y(M, \mathcal{C}) \leq Y(S^n, [g\_{\text{round}}])$, so we can define the *Yamabe invariant of $M$* to be the finite quantity


$$Y(M) = \sup\_{\mathcal{C}}\mathcal{E}(g) = \sup\_{\mathcal{C}}\inf\_{g\in\mathcal{C}}\mathcal{E}(g).$$


The following result is Proposition 1 from [*Kodaira Dimension and the Yamabe Problem*](https://arxiv.org/pdf/dg-ga/9702012.pdf) by LeBrun.



> 
> Let $M$ be a smooth compact $n$-manifold, $n \geq 3$. Then $$\inf\_{g}\int\_M|s\_g|^{n/2} d\mu\_g= \begin{cases}0 & \text{if}\ Y(M) \geq 0\\ |Y(M)|^{n/2} & \text{if}\ Y(M) < 0.\end{cases}$$ Here the infimum on the left hand side is taken over all smooth Riemannian metrics $g$ on $M$.
> 
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In particular, if $n = 4$ and $Y(M) < 0$, then for all $c$ with $c < Y(M)^2$, there is no metric $g$ with $\displaystyle\int\_Ms\_g^2 d\mu\_g = c$.


The question now becomes: is there an example of a compact smooth four-dimensional manifold with $Y(M) < 0$? The answer is yes as is shown in Theorem 2 of the same paper:



> 
> Let $M$ be the underlying $4$-manifold of a complex surface with Kodaira dimension $2$. Then $Y(M) < 0$. Moreover, if $X$ is the minimal model of $M$, then $$Y(M) = Y(X) = -4\pi\sqrt{2c\_1(X)^2}.$$
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For complex surfaces, $c\_1(X)^2 = 2\chi(X) + 3\sigma(X)$, so we can actually compute the Yamabe invariant for the manifolds in the above theorem precisely.


**Example:** Let $X$ be the product of two complex curves of genus $2$. Then $X$ has Kodaira dimension $2$ and is minimal. Moreover, it has Euler characteristic $4$ and signature $0$, so $c\_1^2(X) = 8$. Therefore


$$Y(X) = -4\pi\sqrt{2c\_1(X)^2} = -4\pi\sqrt{16} = -16\pi.$$


The underlying smooth $4$-manifold is $\Sigma\_2\times \Sigma\_2$. So for any $c < (-16\pi)^2 = 256\pi^2$, we see there is no metric $g$ on $\Sigma\_2\times\Sigma\_2$ with $\displaystyle\int\_{\Sigma\_2\times\Sigma\_2}s\_g^2 d\mu\_g = c$.


