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Trilayer graphene with a twisted middle layer has recently emerged as a newplatform exhibiting correlated phases and superconductivity near its magicangle. A detailed characterization of its electronic structure in the parameterspace of twist angle $\theta$, interlayer potential difference $\Delta$, andtop-bottom layer stacking $\tau$ reveals that flat bands with large Coulombenergy vs bandwidth $U/W > 1$ are expected within a range of $\pm 0.2^{\circ}$near $\theta \simeq1.5^{\circ}$ and $\theta \simeq1.2^{\circ}$ for $\tau_{\rmAA}$ top-bottom layer stacking, between a wider $1^{\circ} \sim 1.7^{\circ}$range for $\tau_{\rm AB}$ stacking, whose bands often have finite valley Chernnumbers thanks to the opening of primary and secondary band gaps in thepresence of a finite $\Delta$, and below $\theta \lesssim 0.6^{\circ}$ for all$\tau$ considered. The largest $U/W$ ratios are expected at the magic angle$\sim 1.5^{\circ}$ when $|\Delta| \sim 0$~meV for AA, and slightly below near$\sim 1.4^{\circ}$ for finite $|\Delta| \sim 25$~meV for AB stackings, and near$\theta \sim 0.4^{\circ}$ for both stackings. When ${\tau}$ is the saddle pointstacking vector between AB and BA we observe pronounced anisotropic localdensity of states (LDOS) strip patterns with broken triangular rotationalsymmetry. We present optical conductivity calculations that reflect the changesin the electronic structure introduced by the stacking and gate tunable systemparameters.

Stacking and gate tunable topological flat bands, gaps and anisotropic strip patterns in twisted trilayer graphene