  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 $|U|$ 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 $\widetilde{D}_\text{low}
\leq \widetilde{n} |U| \Omega/2$ with $\widetilde{n} = \min\left(n,2-n\right)$,
where $n$ is the flat band 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 again obtain an upper bound for $D_{\rm low}$
linear in $|U|$. We discuss the insights obtained from our bounds by comparing
them with mean-field and quantum Monte-Carlo results.
