  A critical result in superconductivity is that flat bands, though
dispersionless, can still host nonzero superfluid weight due to quantum
geometry. We show that the derivation of the mean field superfluid weight in
previous literature is incomplete, which can lead to severe quantitative and
even qualitative errors. We derive the complete equations and demonstrate that
the minimal quantum metric -- the metric with minimum trace -- is related to
the superfluid weight in isolated flat bands. We complement this result with an
exact calculation of the Cooper pair mass in attractive Hubbard models with the
uniform pairing condition. When the orbitals are located at high symmetry
positions, the Cooper pair mass is exactly given by the quantum metric, which
is guaranteed to be minimal. Moreover, we study the effect of closing the band
gap between the flat and dispersive bands, and develop a mean-field theory of
pairing for different band-touching points via the $S$-matrix construction. In
mean field, we show that a non-isolated flat band can actually be beneficial
for superconductivity. This is a promising result in the search for high
temperature superconductivity as the material does not need to have flat bands
that are isolated from other bands by the thermal energy. Our work resolves a
fundamental caveat in understanding the relation of multiband superconductivity
to quantum geometry, and the results on band touchings widen the class of
systems advantageous for the search of high temperature flat band
superconductivity.
