  We present exact results that give insight into how interactions lead to
transport and superconductivity in a flat band where the electrons have no
kinetic energy. We obtain bounds for the optical spectral weight for flat band
superconductors, that lead to upper bounds for the superfluid stiffness and the
2D $T_c$. We focus on on-site attraction on the Lieb lattice with trivial flat
bands and on the $\pi$-flux model with topological flat bands. For trivial flat
bands, the low-energy optical spectral weight $\widetilde{D}_\text{low} \leq n
|U| \Omega/2$, where $n$ is the density and $\Omega$ the Marzari-Vanderbilt
spread of the Wannier functions (WFs). We also obtain a lower bound involving
the quantum metric. For topological flat bands, with an obstruction to
localized WFs respecting all symmetries, we nevertheless obtain an upper bound
for $D_{\rm low}$ linear in $|U|$ and $n$. We discuss the insights obtained
from our bounds by comparing them with mean-field and quantum Monte-Carlo
results.
