  We study the energetic stability and the local electronic structure of the
general twisted trilayer graphene (TTG) with the top and bottom layers rotated
with respect to the middle layer respectively by $\theta$ and $\theta'$.
Approximate supercells of the moir\'{e}-of-moir\'{e} superlattices with
$\theta$ and $\theta^{\prime}$ within $1^{\circ}\sim 2^{\circ}$ are established
to describe the structural and electronic properties of relaxed TTG with the
periodic boundary condition. Full relaxation demonstrates that the commensurate
TTG with $\theta=\theta^{\prime}$ has the local minimum total energy
($E_{tol}$) at a fixed $\theta$, while $E_{tol}$ first reaches a local maximum
and begins to drop with decreasing $\theta^{\prime}$ for $\theta^{\prime} <
\theta$. Some regions exhibit enhanced in-plane relaxation in the top and
bottom layers but suppressed relaxation in the middle layer and form a
hexagonal network with the moir\'{e}-of-moir\'{e} length scale. The stacking
configurations with the atoms in the three layers vertically aligned at the
origin of the relaxed TTG supercells at $\theta$ around $1.6^{\circ}$ and
$\theta^{\prime}$ around $1.4^{\circ}$ have a high density of states (DOS) near
the Fermi level ($E_F$), which can reach that of the mirror symmetric TTG with
equal twist angles of about $1.7^{\circ}$. In contrast, some other stackings
can have rather low DOS around $E_F$. The significant stacking dependence of
DOS for some TTG supercells demonstrates that the local electronic structure of
TTG can exhibit strong spatial inhomogeneity when the twist angles are slightly
away from those of the small supercells with large variations of DOS among
different stackings. Moreover, the structural relaxation of TTG plays a crucial
role in the high DOS and its strong stacking dependence.
