Lepton Flavor Violation and the Tau Neutrino Mass

We point out that, in the left-right symmetric model of weak interaction, if ${\nu }_{\tau }$ mass is in the keV to MeV range, there is a strong correlation between rare decays such as $\tau \to 3\mu ,\tau \to 3e$ and the ${\nu }_{\tau }$ mass. In particular, we point out that a large range of ${\nu }_{\tau }$ masses are forbidden by the cosmological constraints on $m_{{\nu }_{\tau }}$ in combination with the present upper limits on these processes.