Non-Monotonic Relations of Galaxy Star Formation, Radius, and Structure at Fixed Stellar Mass

We investigate the relation between galaxy structure and star formation rate (SFR) in a sample of $\sim2.9\times10^{4}$ central galaxies with $z<0.0674$ and axial ratios $b/a>0.5$. The star-forming main sequence (SFMS) shows a bend around the stellar mass of $M_\ast\leq{}M_c=2\times10^{10}{}M_{\odot}$. At $M_\ast\leq{}M_c$ the SFMS follows a power-law $\text{SFR}\propto{}M_\ast^{0.85}$, while at higher masses it flattens. $M_c$ corresponds to a dark matter halo mass of $M_\text{vir}\sim{}10^{11.8}M_{\odot}$ where virial shocks occurs. Some galaxy structure (e.g., half-light radius, $R_e$) exhibits a non-monotonic dependence across the SFMS at a fixed $M_\ast$. We find $\text{SFR}\propto{R_e^{-0.28}}$ at fixed $M_\ast$, consistent with the global Kennicutt-Schmidt (KS) law. This finding suggests that galaxy sizes contribute to the scatter of the SFMS. However, at $M_\ast>M_c$ the relationship between SFR and $R_e$ diminishes. Low-mass galaxies above the mean of the SFMS have smaller radii, exhibit compact and centrally concentrated profiles resembling green valley (GV) and quiescent galaxies at the same mass, and have higher $M_{\text{H}_2}/M_\text{HI}$. Conversely, those below the SFMS exhibit larger radii, lower densities, have no GV or quiescent counterparts at their mass and have lower $M_{\text{H}_2}/M_\text{HI}$. The above data suggest two pathways for quenching low-mass galaxies, $M_\ast\leq{}M_c$: a fast one that changes the morphology on the SFMS and a slow one that does not. Above $M_c$, galaxies below the SFMS resemble GV and quiescent galaxies structurally, implying that they undergo a structural transformation already within the SFMS. For these massive galaxies, CG are strongly bimodal, with SFMS galaxies exhibiting negative color gradients, suggesting most star formation occurs in their outskirts, maintaining them within the SFMS.