  We present new results on the dynamics and gravitational-wave emission from
the collapse of differentially rotating neutron stars. We have considered a
number of polytropic stellar models having different values of the
dimensionless angular momentum J/M^2, where J and M are the asymptotic angular
momentum and mass of the star, respectively. For neutron stars with J/M^2<1,
i.e., "sub-Kerr" models, we were able to find models that are dynamically
unstable and that collapse promptly to a rotating black hole. Both the dynamics
of the collapse and the consequent emission of gravitational waves resemble the
one seen for uniformly rotating stars, although with an overall decrease in the
efficiency of gravitational-wave emission. For stellar models with J/M^2>1,
i.e., "supra-Kerr" models, on the other hand, we were not able to find models
that are dynamically unstable and all of the computed supra-Kerr models were
found to be far from the stability threshold. For these models a gravitational
collapse is possible only after a very severe and artificial reduction of the
pressure, which then leads to a torus developing nonaxisymmetric instabilities
and eventually contracting to a stable axisymmetric stellar configuration.
While this does not exclude the possibility that a naked singularity can be
produced by the collapse of a differentially rotating star, it also suggests
that cosmic censorship is not violated and that generic conditions for a
supra-Kerr progenitor do not lead to a naked singularity.
