  We reconsider the problem of the static thermal roughening of an elastic
manifold at the critical dimension $d=2$ in a periodic potential, using a
perturbative Functional Renormalization Group approach. Our aim is to describe
the effective potential seen by the manifold below the roughening temperature
on large length scales. We obtain analytically a flow equation for the
potential and surface tension of the manifold, valid at all temperatures. On a
length scale $L$, the renormalized potential is made up of a succession of
quasi parabolic wells, matching onto one another in a singular region of width
$\sim L^{-6/5}$ for large $L$. We also obtain numerically the step energy as a
function of temperature, and relate our results to the existing experimental
data on $^4$He. Finally, we sketch the scenario expected for an arbitrary
dimension $d<2$ and examine the case of a non local elasticity which is
realized physically for the contact line.
