  We present a detailed study of the lattice dynamics and electron-phonon
coupling for a (3,3) carbon nanotube which belongs to the class of small
diameter based nanotubes which have recently been claimed to be
superconducting. We treat the electronic and phononic degrees of freedom
completely by modern ab-initio methods without involving approximations beyond
the local density approximation. Using density functional perturbation theory
we find a mean-field Peierls transition temperature of approx 40K which is an
order of magnitude larger than the calculated superconducting transition
temperature. Thus in (3,3) tubes the Peierls transition might compete with
superconductivity. The Peierls instability is related to the special 2k_F
nesting feature of the Fermi surface. Due to the special topology of the (n,n)
tubes also a q=0 coupling between the two bands crossing the Fermi energy at
k_F is possible which leads to a phonon softening at the Gamma point.
