  We study the steady-state low-temperature dynamics of an elastic line in a
disordered medium below the depinning threshold. Analogously to the equilibrium
dynamics, in the limit T->0, the steady state is dominated by a single
configuration which is occupied with probability one. We develop an exact
algorithm to target this dominant configuration and to analyze its geometrical
properties as a function of the driving force. The roughness exponent of the
line at large scales is identical to the one at depinning. No length scale
diverges in the steady state regime as the depinning threshold is approached
from below. We do find, a divergent length, but it is associated only with the
transient relaxation between metastable states.
