  In a sample of local active galactic nuclei studied at a spatial resolution
on the order of 10 pc we show that the interstellar medium traced by the
molecular hydrogen v=1-0 S(1) 2.1um line forms a geometrically thick, clumpy
disk. The kinematics of the molecular gas reveals general rotation, although an
additional significant component of random bulk motion is required by the high
local velocity dispersion. The size scale of the typical gas disk is found to
have a radius of ~30 pc with a comparable vertical height. Within this radius
the average gas mass is estimated to be ~10^7 Msun based on a typical gas mass
fraction of 10%, which suggests column densities of Nh ~ 5x10^23 cm^-2.
Extinction of the stellar continuum within this same region suggest lower
column densities of Nh ~ 2x10^22 cm^-2, indicating that the gas distribution on
these scales is dominated by dense clumps. In half of the observed Seyfert
galaxies this lower column density is still great enough to obscure the AGN at
optical/infrared wavelengths. We conclude, based on the spatial distribution,
kinematics, and column densities that the molecular gas observed is spatially
mixed with the nuclear stellar population and is likely to be associated with
the outer extent of any smaller scale nuclear obscuring structure. Furthermore,
we find that the velocity dispersion of the molecular gas is correlated with
the star formation rate per unit area, suggesting a link between the two
phenomena, and that the gas surface density follows known "Schmidt-Kennicutt"
relations. The molecular/dusty structure on these scales may be dynamic since
it is possible that the velocity dispersion of the gas, and hence the vertical
disk height, is maintained by a short, massive inflow of material into the
nuclear region and/or by intense, short-lived nuclear star formation.
