  If neutrinos are to play a relevant cosmological role, they must be
essentially degenerate. We study whether radiative corrections can or cannot be
responsible for the small mass splittings, in agreement with all the available
experimental data. We perform an exhaustive exploration of the bimaximal mixing
scenario, finding that (i) the vacuum oscillations solution to the solar
neutrino problem is always excluded; (ii) if the mass matrix is produced by a
see-saw mechanism, there are large regions of the parameter space consistent
with the large angle MSW solution, providing a natural origin for the $\Delta
m^2_{sol} << \Delta m^2_{atm}$ hierarchy; (iii) the bimaximal structure becomes
then stable under radiative corrections. We also provide analytical expressions
for the mass splittings and mixing angles and present a particularly simple
see-saw ansatz consistent with all observations.
