Lower Limits on $\mu \to e \gamma$ from new Measurements on $U_{e3}$

New data on the lepton mixing angle $\theta_{13}$ imply that the $e\mu$ element of the matrix $m_\nu m_\nu^\dagger$, where $m_\nu$ is the neutrino Majorana mass matrix, cannot vanish. This implies a lower limit on lepton flavor violating processes in the $e\mu$ sector in a variety of frameworks, including Higgs triplet models or the concept of minimal flavor violation in the lepton sector. We illustrate this for the branching ratio of $\mu \to e \gamma$ in the type II seesaw mechanism, in which a Higgs triplet is responsible for neutrino mass and also mediates lepton flavor violation. We also discuss processes like $\mu\to e\bar{e}e$ and $\mu\to e$ conversion in nuclei. Since these processes have sensitivity on the individual entries of $m_\nu$, their rates can still be vanishingly small.