  We perform analysis of the band structure, phonon dispersion, and
electron-phonon interactions in three types of small-radius carbon nanotubes.
We find that the (5,5) can be described well by the zone-folding method and the
electron-phonon interaction is too small to support either a charge-density
wave or superconductivity at realistic temperatures. For ultra-small (5,0) and
(6,0) nanotubes we find that the large curvature makes these tubes metallic
with a large density of states at the Fermi energy and leads to unusual
electron-phonon interactions, with the dominant coupling coming from the
out-of-plane phonon modes. By combining the frozen-phonon approximation with
the RPA analysis of the giant Kohn anomaly in 1d we find parameters of the
effective Fr\"{o}lich Hamiltonian for the conduction electrons. Neglecting
Coulomb interactions, we find that the (5,5) CNT remains stable to
instabilities of the Fermi surface down to very low temperatures while for the
(5,0) and (6,0) CNTs a CDW instability will occur. When we include a realistic
model of Coulomb interaction we find that the charge-density wave remains
dominant in the (6,0) CNT with $T_{\rm CDW}$ around 5 K while the
charge-density wave instability is suppressed to very low temperatures in the
(5,0) CNT, making superconductivity dominant with transition temperature around
one Kelvin.
