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tokens: then actually expect less active nuclear star formation and/or inflows in NGC 1097, which will lead to the deficiency of a geometrically thick torus in this LLAGN as compared to luminous AGNs such as NGC 1068 due to the deficiency of the energy source(s): this is an expected trend by the evolutionary torus models (Section 1 ). Another remarkable thing is that the envelope of the CND traced by the CO(3–2) emission and that traced by the NIR H 2 emission (Davies et al. 2009 ; Hicks et al. 2009 ) are almost identical in NGC 1097 at our available resolutions (Figure 3 (d)). This indicates that the outer radial extent of the molecular CND is unrelated to gas temperature, although the vertical height of the CND could differ (see below), which is consistent with hydrodynamic simulations of CND-scale gas distribution (e.g., Wada et al. 2009 ). However, we also found that their internal distributions differ significantly: while the two knots stand out in the CO(3–2) map, the NIR H 2 emission is clearly concentrated toward the nucleus, likely reflecting complex excitation conditions including fluorescence around the AGN. Thus, we should be careful to simply assume a linear conversion from NIR H 2 line luminosity to CO-based (Müller-Sánchez et al. 2006 ) when using that luminosity as a tracer of molecular mass. Gas dynamics is also key for studying the nature of the torus. Figures 4 (a) and (b) show the observed line-of-sight velocity and velocity dispersion maps of the CO(3–2) line in NGC 1097, respectively. It is evident that rotation (kinematic P.A. ≃ 118°) dominates the global gas dynamics, while streaming motions as well as high-velocity dispersions along the gas spirals/arc are detected. Figure 4. Intensity-weighted (a) mean velocity and (b) velocity dispersion maps of the CO(3–2) emission in the central 400 pc of NGC 1097. The stars mark the AGN location. The systemic velocity is 1254 km s −1 (thick line). Thin contours indicate 1080–1400 km s −1 in steps of 20 km s −1 for (a), and 10–40 km s −1 in steps of 10 km s −1 for (b), respectively. (c) and (d) Position–velocity diagrams (PVDs) of the CO(3–2) line along the adopted major axis (P.A. = 118°) and minor axis (P.A. = 208°) shown in the blue contours. The overlaid red contours indicate the PVDs produced by our dynamical model (MODEL-1). Both contours are plotted at 5, 10, ..., 40 σ , where 1 σ = 0.84 mJy beam −1 . Due to the inclined geometry of the concentric rings, no model component exists at offset ≳11 along the minor axis. To extract basic beam-deconvolved dynamical information, particularly the rotation velocity () and the velocity dispersion ( σ ), we fitted tilted rings to these observed maps with the 3D Barolo code (Di Teodoro Fraternali 2015 ). The main parameters are dynamical center, systemic velocity (), , σ , galactic inclination ( i ), and P.A., all of which can be varied for each ring. Our initial fits indicated a dynamical center as the AGN position, as 1254 km s −1 . Thus , σ ,