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We use stellar dynamical bulge/disk/halo simulations to study whetherbarlenses (lens-like structures embedded in the narrow bar component) are justthe face-on counterparts of Boxy/Peanut/X-shapes (B/P/X) seen in edge-on bars,or if some additional physical parameter affects that morphology. A range ofbulge-to-disk mass and size ratios are explored: our nominal parameters($B/D=0.08$, $r_{\rm eff}/h_r=0.07$, disk comprising 2/3 of total force at$2.2h_r$) correspond to typical MW mass galaxies. In all models a bar withpronounced B/P/X forms in a few Gyrs, visible in edge-on view. However, thepure barlens morphology forms only in models with sufficiently steep innerrotation curves, $dV_{cir}/dr\gtrsim5V_{max}/h_r$, achieved when including asmall classical bulge with $B/D\gtrsim0.02$ and $r_{\rm eff}/h_r\lesssim0.1$.For shallower slopes the central structure still resembles a barlens, but showsa clear X-signature even in low inclinations. Similar result holds forbulgeless simulations, where the central slope is modified by changing the haloconcentration. The predicted sensitivity on inner rotation curve is consistentwith the slopes estimated from gravitational potentials calculated from the3.6$\mu$m images, for the observed barlens and X-shape galaxies in the SpitzerSurvey of Stellar Structure in Galaxies (S$^4$G). For inclinations $<60^\circ$the galaxies with barlenses have on average twice steeper inner rotation curvesthan galaxies with X-shapes: the limiting slope is $\sim250$km/s/kpc. Amongbarred galaxies, those with barlenses have both the strongest bars and thelargest relative excess of inner surface density, both in barlens region($\lesssim0.5h_r$) and near the center ($\lesssim0.1h_r$); this providesevidence for bar-driven secular evolution in galaxies.

Boxy/Peanut/X-shape bulges: steep inner rotation curve leads to barlens face-on morphology