We calculate the coupled hydrostatic and ionization structures of spherically symmetric , pressure-supported gas clouds that are confined by gravitationally dominant dark-matter ( DM ) mini-halos and by an external bounding pressure provided by a hot medium .
We focus on clouds that are photoionized and heated by the present-day background metagalactic field and determine the conditions for the formation of warm ( WNM ) , and multi-phased ( CNM/WNM ) neutral atomic hydrogen ( HI ) cores in the DM-dominated clouds .

We consider \Lambda CDM dark-matter halos with cuspy ( NFW ) and constant density ( Burkert ) cores .
We compute models for a wide range of halo masses , total cloud gas masses , and external bounding pressures .

We present models for the pressure-supported HI structures observed in the Local Group dwarf irregular galaxies Leo A and Sag DIG .
We find that the hydrogen gas becomes neutral for projected HI column densities exceeding 10 ^ { 19 } cm ^ { -2 } .
We identify the HI cloud boundaries observed in Leo A and Sag DIG with the ionization fronts , and we derive an upper limit of P _ { HIM } / k \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 100 cm ^ { -3 } K for the ambient pressure of the intergalactic medium in the Local Group .
The observed HI gas scale heights in Leo A and Sag DIG imply characteristic DM densities of 1.2 amu cm ^ { -3 } ( or 0.03 M _ { \odot } pc ^ { -3 } ) , consistent with the DM densities previously inferred via HI rotation curve studies of dwarf and low-surface brightness galaxies .
Leo A and Sag DIG obey the scaling correlations that are expected for typical ( median ) DM halos in a \Lambda CDM cosmology , provided the halos contain constant density cores , as suggested by Burkert .

We construct explicit “ mini-halo ” models for the multi-phased ( and low-metallicity ) compact high-velocity HI clouds ( CHVCs ) .
If the CHVC halos are drawn from the same family of halos that successfully reproduce the dwarf galaxy observations , then the CHVCs must be “ circumgalactic ” objects , with characteristic distances of 150 kpc .
For such systems we find that multi-phased behavior occurs for peak WNM HI column densities between 2 \times 10 ^ { 19 } and 1 \times 10 ^ { 20 } cm ^ { -2 } , consistent with observations .
In contrast , if the CHVCs are “ extragalactic ” objects with distances \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 750 kpc , then their associated halos must be very “ underconcentrated ” , with characteristic DM densities \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 0.08 cm ^ { -3 } , much smaller than expected for their mass , and significantly smaller than observed in the dwarf galaxies .
Furthermore , multi-phased cores then require higher shielding columns .
We favor the circumgalactic hypothesis .
If the large population of CHVCs represent “ missing low-mass DM satellites ” of the Galaxy , then these HI clouds must be pressure-confined to keep the gas neutral .
For an implied CHVC mini-halo scale velocity of v _ { s } = 12 km s ^ { -1 } , the confining pressure must exceed \sim 50 cm ^ { -3 } K. A hot ( \sim 2 \times 10 ^ { 6 } K ) Galactic corona could provide the required pressure at 150 kpc .

Our static mini-halo models are able to account for many properties of the CHVCs , including their observed peak HI columns , core sizes , and multi-phased behavior .
However , important difficulties remain , including the presence in some objects of extended low column density HI wings , and H \alpha emission line fluxes in several CHVCs that are significantly larger than expected .