  We prove theoretically the possibility of electric-field controlled polaron
formation involving flexural (bending) modes in suspended carbon nanotubes.
Upon increasing the field, the ground state of the system with a single extra
electron undergoes a first order phase transition between an extended state and
a localized polaron state. For a common experimental setup, the threshold
electric field is only of order $\simeq 10^{-2}$ V/$\mu$m.
