  We derive the effective low-energy theory for interacting electrons in
metallic single-wall carbon nanotubes taking into account acoustic phonon
exchange within a continuum elastic description. In many cases, the nanotube
can be described as a standard Luttinger liquid with possibly attractive
interactions. We predict surprisingly strong attractive interactions for thin
nanotubes. Once the tube radius reaches a critical value $R_0 \approx 3.6\pm
1.4$ \AA, the Wentzel-Bardeen singularity is approached, accompanied by strong
superconducting fluctuations. The surprisingly large $R_0$ indicates that this
singularity could be reached experimentally. We also discuss the conditions for
a Peierls transition due to acoustic phonons.
