  We perform a detailed analysis of the band structure, phonon dispersion, and
electron-phonon coupling of three types of small-radius carbon nanotubes
(CNTs): (5,0), (6,0), and (5,5) with diameters 3.9, 4.7, and 6.8 \AA
respectively. The large curvature of the (5,0) CNTs makes them metallic with a
large density of states at the Fermi energy. The density of states is also
strongly enhanced for the (6,0) CNTs compared to the results obtained from the
zone-folding method. For the (5,5) CNTs the electron-phonon interaction is
dominated by the in-plane optical phonons, while for the ultrasmall (5,0) and
(6,0) CNTs the main coupling is to the out-of-plane optical phonon modes. We
calculate electron-phonon interaction strengths for all three types of CNTs and
analyze possible instabilities toward superconducting and charge-density wave
phases. For the smallest (5,0) nanotube, in the mean-field approximation and
neglecting Coulomb interactions, we find that the charge-density wave
transition temperature greatly exceeds the superconducting one. When we include
a realistic model of the Coulomb interaction we find that the charge-density
wave is suppressed to very low temperatures, making superconductivity dominant
with the mean-field transition temperature around one K. For the (6,0) nanotube
the charge-density wave dominates even with the inclusion of Coulomb
interactions and we find the mean-field transition temperature to be around
five Kelvin. We find that the larger radius (5,5) nanotube is stable against
superconducting and charge-density wave orders at all realistic temperatures.
