  A significant fraction of nearby elliptical galaxies are known to have high
density gas disks in their circumnuclear (CN) region (0.1 to a few kpc). Yet,
ellipticals, especially luminous ones, show little signs of recent star
formation (SF). To investigate the possible cause of the dearth of SF in these
systems, we study the gravitational stability of CN gas disks embedded within
the potentials of both the stellar bulge and the central massive black hole
(BH) in ellipticals. We find that CN disks in higher mass galaxies are
generally more stable than those in lower mass galaxies, because higher mass
galaxies tend to have more massive BHs and more centrally concentrated stellar
density profiles. We also consider the case in which the central stellar
density profile has a core, which is often observed for ellipticals whose total
stellar mass is higher than about 10^11 Msun. Such a cored stellar density
profile leads to more unstable CN disks than the power-law density profile
characteristic of less massive galaxies. However, the more massive BHs in
high-mass galaxies act to stabilize the CN disk. Our results demonstrate that
the gravitational potentials of both the central BH and the stellar component
should be taken into account when studying the properties of CN disks, as their
stability is sensitive to both the BH mass and the stellar density profile. Our
results could explain the observed trend that less luminous ellipticals have a
greater tendency to exhibit ongoing SF than giant ellipticals.
