  We present new high angular resolution observations at near-IR wavelengths of
the core of the Luminous Blue Variable Eta Carinae, using NAOS-CONICA at the
VLT and VINCI at the VLT Interferometer (VLTI). The latter observations provide
spatial information on a scale of 5 milli-arcsec or ~11 AU at the distance of
Eta Carinae. The present-day stellar wind of Eta Carinae is resolved on a scale
of several stellar radii. Assuming spherical symmetry, we find a mass loss rate
of 1.6x10^{-3} M_sun/yr and a wind clumping factor of 0.26. The VLTI data taken
at a baseline of 24 meter show that the object is elongated with a de-projected
axis ratio of approximately 1.5; the major axis is aligned with that of the
large bi-polar nebula that was ejected in the 19th century. The most likely
explanation for this observation is a counter-intuitive model in which stellar
rotation near the critical velocity causes enhanced mass loss along the
rotation axis. This results from the large temperature difference between pole
and equator in rapidly rotating stars. Eta Carinae must rotate in excess of 90
per cent of its critical velocity to account for the observed shape. The large
outburst may have been shaped in a similar way. Our observations provide strong
support for the existence of a theoretically predicted rotational instability,
known as the Omega limit.
