  The merger of binary neutron stars is likely to lead to differentially
rotating remnants. In this paper, we survey several cold nuclear equations of
state (EOSs) and numerically construct models of differentially rotating
neutron stars in general relativity. For each EOS we tabulate maximum allowed
masses as a function of the degree of differential rotation. We also determine
effective polytropic indices, and compare the maximum allowed masses with those
for the corresponding polytropes. We consistently find larger mass increases
for the polytropes, but even for the nuclear EOSs we typically find maximum
masses 50% higher than the corresponding values for nonrotating (TOV) stars. We
evaluate our findings for the six observed binary neutron star (pulsar)
systems, including the recently discovered binary pulsar J0737-3039. For each
EOS we determine whether their merger will automatically lead to prompt
collapse to a black hole, or whether the remnant can be supported against
collapse by uniform rotation (possibly as a supramassive star) or differential
rotation (possibly as a hypermassive star). For hypermassive stars, delayed
collapse to a black hole is likely. For the most recent EOSs we survey the
merger remnants can all be supported by rotation against prompt collapse, but
their actual fate will depend on the nonequilibrium dynamics of the coalescence
event. Gravitational wave observations of coalescing binary neutron stars may
be able to distinguish these outcomes -- no, delayed or prompt collapse -- and
thereby constrain possible EOSs.
