  We present deep Halpha images of three nearby late-type spiral galaxies
(NGC628, NGC1058 and NGC6946), which reveal the presence of HII regions out to,
and beyond, two optical radii (defined by the 25th B-band isophote). The
outermost HII regions appear small, faint and isolated, compared to their inner
disk counterparts, and are distributed in organized spiral arm structures,
likely associated with underlying HI arms and faint stellar arms. The
relationship between the azimuthally--averaged Halpha surface brightness
(proportional to star formation rate per unit area) and the total gas surface
density is observed to steepen considerably at low gas surface densities. We
find that this effect is largely driven by a sharp decrease in the covering
factor of star formation at large radii, and not by changes in the rate at
which stars form locally. An azimuthally--averaged analysis of the
gravitational stability of the disk of NGC6946 reveals that while the existence
of star formation in the extreme outer disk is consistent with the Toomre-Q
instability model, the low rates observed are only compatible with the model
when a constant gaseous velocity dispersion is assumed. We suggest that
observed behaviour could also be explained by a model in which the star
formation rate has an intrinsic dependence on the azimuthally-averaged gas
volume density, which decreases rapidly in the outer disk due to the vertical
flaring of the gas layer.
