  The stability of the linear chain structure of three $\alpha$ clusters for
$^{12}$C against the bending and fission is investigated in the cranking
covariant density functional theory, in which the equation of motion is solved
on a 3D lattice with the inverse Hamiltonian and the Fourier spectral methods.
Starting from a twisted three $\alpha$ initial configuration, it is found that
the linear chain structure is stable when the rotational frequency is within
the range of $\sim$2.0 MeV to $\sim$2.5 MeV. Beyond this range, the final
states are not stable against fission. By examining the density distributions
and the occupation of single-particle levels, however, these fissions are found
to arise from the occupation of unphysical continuum with large angular
momenta. To properly remove these unphysical continuum, a damping function for
the cranking term is introduced. Eventually, the stable linear chain structure
could survive up to the rotational frequency $\sim$3.5 MeV, but the fission
still occurs when the rotational frequency approaches to $\sim$4.0 MeV.
