  Driven elastic manifolds in random media exhibit a depinning transition to a
state with non-vanishing velocity at a critical driving force. We study the
depinning of stiff directed lines, which are governed by a bending rigidity
rather than line tension. Their equation of motion is the (quenched)
Herring-Mullins equation, which also describes surface growth governed by
surface diffusion. Stiff directed lines are particularly interesting as there
is a localization transition in the static problem at a finite temperature and
the commonly exploited time ordering of states by means of Middleton's theorems
(A. Middleton, Phys. Rev. Lett. 68, 670 (1992)) is not applicable. We employ
analytical arguments and numerical simulations to determine the critical
exponents and compare our findings with previous works and functional
renormalization group results, which we extend to the different line
elasticity. We see evidence for two distinct correlation length exponents.
