  We study theoretically the transport through a finite-size sawtooth lattice
coupled to two fermionic reservoirs kept in the superfluid state. We focus on
the DC Josephson effect and find that the flat band states of the sawtooth
lattice can support larger critical current and at higher temperature than the
dispersive band states. However, for this to occur the boundary states of the
finite-size lattice need to be tuned at resonance with the bulk flat band
states by means of additional boundary potentials. We show that transport in a
two-terminal configuration can reveal the salient features of the geometric
contribution of flat band superconductivity, namely the linear dependence of
key quantities, such as the critical current and critical temperature, on the
interaction. Our results are based on parameters of a realistic experimental
lattice potential, and we discuss the conditions one needs to reach to observe
the predicted effects experimentally.
