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 $\tau \rightarrow 3 \mu, \tau \rightarrow 3 e$ 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.