Gravitationally violated U(1) symmetry and neutrino anomalies

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.