  We present the first results of an analysis of the properties of the
molecular gas in the nuclear regions (r < 300 pc) of a sample of six nearby
galaxies, based on new high spatial resolution observations obtained in the
K-band with the near-infrared integral field spectrograph SINFONI at the Very
Large Telescope. We derive two-dimensional distributions of the warm molecular
and ionized gas from the H2, Br_gamma and HeI emission lines present in the
spectra of the galaxies. We find a range of morphologies, including bar- and
ring-like distributions and either centrally peaked or off-centre emission. The
morphologies of the molecular and the ionized gas are not necessarily
coincident. The observed emission-line ratios point towards thermal processes
as the principal mechanism responsible for the H2 excitation in the nuclear and
circumnuclear regions of the galaxies, independently of the presence of an
active nucleus. We find that a rescaling of the H2 2.12 microns emission-line
luminosity by a factor beta~1200 gives a good estimate (within a factor of 2)
of the total (cold) molecular gas mass. The galaxies of the sample contain
large quantities of molecular gas in their centres, with total masses in the ~
105 - 108 Msol range. Never the less, these masses correspond to less than 3
per cent of the stellar masses derived for the galaxies in these regions,
indicating that the presence of gas should not affect black hole mass estimates
based on the dynamical modelling of the stars. The high-spatial resolution
provided by the SINFONI data allowed us to resolve a circumnuclear ring (with a
radius of ~270 pc) in the galaxy NGC 4536. The measured values of the Br_gamma
equivalent width and the HeI/Br_gamma emission-line ratio suggests that bursts
of star formation occurred throughout this ring as recently as 6.5 Myr ago.
