Bi-Maximal Mixing of Three Neutrinos

We show that if the solar and atmospheric data are both described by maximal vacuum oscillations at the relevant mass scales then there exists a unique mixing matrix for three neutrino flavors. The solution necessarily conserves CP and automatically implies that there is no disappearance of atmospheric ${\nu }_e$, consistent with indications from the Super-Kamiokande experiment. We also investigate the consequences for three-neutrino mixing if the solar and atmospheric oscillations exhibit mixing that is large but not maximal. For non-maximal mixing ${\nu }_e\leftrightarrow {\nu }_{\tau }$ and ${\nu }_e\leftrightarrow {\nu }_{\mu }$ oscillations are predicted that may be observable in future long-baseline experiments.