  While bright, blue, compact galaxies are common at $\rm z \sim 1$, they are
relatively rare in the local universe, and their evolutionary paths are
uncertain. We have obtained resolved H I observations of nine $\rm z \sim 0$
luminous compact blue galaxies (LCBGs) using the Giant Metrewave Radio
Telescope and Very Large Array in order to measure their kinematic and
dynamical properties and better constrain their evolutionary possibilities. We
find that the LCBGs in our sample are rotating galaxies that tend to have
nearby companions, relatively high central velocity dispersions, and can have
disturbed velocity fields. We calculate rotation velocities for each galaxy by
measuring half of the velocity gradient along their major axes and correcting
for inclination using axis ratios derived from SDSS images of each galaxy. We
compare our measurements to those previously made with single dishes and find
that single dish measurements tend to overestimate LCBGs' rotation velocities
and H I masses. We also compare the ratio of LCBGs' rotation velocities and
velocity dispersions to those of other types of galaxies and find that LCBGs
are strongly rotationally supported at large radii, similar to other disk
galaxies, though within their half-light radii the $\rm V_{rot}/ \sigma$ values
of their H I are comparable to stellar $\rm V_{rot}/ \sigma$ values of dwarf
elliptical galaxies. We find that LCBGs' disks on average are gravitationally
stable, though conditions may be conducive to local gravitational instabilities
at the largest radii. Such instabilities could lead to the formation of
star-forming gas clumps in the disk, resulting eventually in a small central
bulge or bar.
