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The stellar surface density within the inner 1 kpc ($\Sigma_{1}$) has becomea popular tool for understanding the growth of galaxies and its connection withthe quenching of star formation. The emerging picture suggests that building acentral dense core is a necessary condition for quenching. However, it is notclear whether changes in $\Sigma_{1}$ trace changes in stellar kinematics andthe growth of dispersion-dominated bulges. In this paper, we combine imagingfrom the Sloan Digital Sky Survey with stellar kinematics from the Sydney-AAOMulti-object Integral-field unit (SAMI) and Mapping Nearby Galaxies at ApachePoint Observatory (MaNGA) surveys to quantify the correlation between$\Sigma_{1}$ and the proxy for stellar spin parameter within one effectiveradius ($\lambda_{re}$) for 1599 nearby galaxies. We show that, on thestar-forming main sequence and at fixed stellar mass, changes in $\Sigma_{1}$are mirrored by changes in $\lambda_{re}$. While forming stars, main sequencegalaxies remain rotationally-dominated systems, with their $\Sigma_{1}$increasing but their stellar spin staying either constant or slightlyincreasing. The picture changes below the main sequence, where $\Sigma_{1}$ and$\lambda_{re}$ are no longer correlated. Passive systems show a narrower rangeof $\Sigma_{1}$, but a wider range of $\lambda_{re}$ compared to star-forminggalaxies. Our results indicate that, from a structural point of view, passivegalaxies are a more heterogeneous population than star-forming systems, and mayhave followed a variety of evolutionary paths. This also suggests that, ifdispersion-dominated bulges still grow significantly at $z\sim$0, thisgenerally takes place during, or after, the quenching phase.

The physical connection between central stellar surface density and stellar spin in SAMI and MaNGA nearby galaxies