  Temperature fluctuations in the normal direction of planar crystals such as
graphene are quite violent and may be expected to influence strongly their
melting properties. In particular, they will modify the Lindemann melting
criterium. We calculate this modification in a self-consistent Born
approximation. The result is applied to graphene and its wrapped version
represented by single-walled carbon nanotubes (SWNTs). It is found that the
out-of-plane fluctuations dominate over the in-plane fluctuations. This makes
strong restrictions to possible Lindemann parameters. Astonishing we find that
these large out-of-plane fluctuations have only a small influence upon the
melting temperature.
