  We investigate the nature of the star formation law at low gas surface
densities using a sample of 19 low surface brightness (LSB) galaxies with
existing HI maps in the literature, UV imaging from the Galaxy Evolution
Explorer satellite, and optical images from the Sloan Digital Sky Survey. All
of the LSB galaxies have (NUV-r) colors similar to those for higher surface
brightness star-forming galaxies of similar luminosity indicating that their
average star formation histories are not very different. Based upon four LSB
galaxies with both UV and FIR data, we find FIR/UV ratios significantly less
than one, implying low amounts of internal UV extinction in LSB galaxies. We
use the UV images and HI maps to measure the star formation rate and hydrogen
gas surface densities within the same region for all of the galaxies. The LSB
galaxy star formation rate surface densities lie below the extrapolation of the
power law fit to the star formation rate surface density as a function of the
total gas density for higher surface brightness galaxies. Although there is
more scatter, the LSB galaxies also lie below a second version of the star
formation law in which the star formation rate surface density is correlated
with the gas density divided by the orbital time in the disk. The downturn seen
in both star formation laws is consistent with theoretical models that predict
lower star formation efficiencies in LSB galaxies due to the declining
molecular fraction with decreasing density.
