  We present two-dimensional gas-kinematic maps of the central region in
Centaurus A. The adaptive optics (AO) assisted SINFONI data from the VLT have a
resolution of 0.12" in K-band. The ionized gas species (Br_gamma, [FeII],
[SiVI]) show a rotational pattern that is increasingly overlaid by
non-rotational motion for higher excitation lines in direction of Cen A's radio
jet. The emission lines of molecular hydrogen (H_2) show regular rotation and
no distortion due to the jet. The molecular gas seems to be well settled in the
gravitational potential of the stars and the central supermassive black hole
and we thus use it as a tracer to model the mass in the central +/-1.5". These
are the first AO integral-field observations on the nucleus of Cen A, enabling
us to study the regularity of the rotation around the black hole, well inside
the radius of influence, and to determine the inclination angle of the gas disk
in a robust way. The gas kinematics are best modeled through a tilted-ring
model that describes the warped gas disk; its mean inclination angle is ~34deg
and the mean position angle of the major axis is ~155deg. The best-fit black
hole mass is M_BH~4.5x10^7 Msolar, based on a "kinematically hot" disk model
where the velocity dispersion is included through the Jeans equation. This
black hole mass estimate is somewhat lower than, but consistent with the mass
values previously derived from ionized gas kinematics. It is also consistent
with the stellar dynamical measurement from the same AO observations, which we
present in a separate paper. It brings Cen A in agreement with the M_BH-sigma
relation.
