  The current searches for neutrino oscillations seem to suggest an approximate
$L_e-L_\m-L_{\tau}$ flavor symmetry. This symmetry implies a pair of degenerate
neutrinos with mass $m_0$ and large leptonic mixing. We explore the possibility
that gravitational interactions break this global symmetry. The Planck scale
suppressed breaking of the $L_e-L_\m-L_{\tau}$ symmetry is shown to lead to the
right amount of splitting among the degenerate neutrinos needed in order to
solve the solar neutrino problem. The common mass $m_0$ of the pair can be
identified with the atmospheric neutrino scale. A concrete model is proposed in
which smallness of $m_0$ and hierarchy in the solar and atmospheric neutrino
scales get linked to hierarchies in the weak, grand unification and the Planck
scales.
