Gravitationally violated U(1) symmetry and neutrino anomalies

The current searches for neutrino oscillations seem to suggest an approximate $L_e-L_{\mu }-L_{\tau }$ flavor symmetry. This symmetry implies a pair of degenerate neutrinos with a 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 $L_e-L_{\mu }-L_{\tau }$ symmetry is shown to lead to the right amount of splitting between 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 the smallness of $m_0$ and the hierarchy in the solar and atmospheric neutrino scales get linked to hierarchies in the weak, grand unification, and Planck scales.