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We study the correlation between galaxy structure and the quenching of starformation using a sample of SDSS central galaxies with stellar masses 9.75< logM_*/M_sun<11.25 and redshifts z<0.075. GALEX UV data are used to cleanly dividethe sample into star-forming and quenched galaxies, and to identify galaxies intransition (the green valley). Despite a stark difference in visual appearancebetween blue and red galaxies, their average radial stellar mass densityprofiles are remarkably similar (especially in the outer regions) at fixedmass. The inner stellar mass surface density within a radius of 1 kpc,\Sigma_1, is used to quantify the growth of the bulge as galaxies evolve. Whengalaxies are divided into narrow mass bins, their distribution in thecolor-\Sigma_1 plane at fixed mass forms plausible evolutionary tracks.\Sigma_1 seems to grow as galaxies evolve through the blue cloud, and once itcrosses a threshold value, galaxies are seen to quench at fixed \Sigma_1. The\Sigma_1 threshold for quenching grows with stellar mass, \Sigma_1 ~M_*^{0.64}. However, the existence of some star-forming galaxies above thethreshold \Sigma_1 implies that a dense bulge is necessary but not sufficientto quench a galaxy fully. This would be consistent with a two-step quenchingprocess in which gas within a galaxy is removed or stabilized against starformation by bulge-driven processes (such as a starburst, AGN feedback, ormorphological quenching), whereas external gas accretion is suppressed byseparate halo-driven processes (such as halo gas shock heating). Quenching thusdepends on an interplay between the inner structure of a galaxy and itssurrounding dark matter halo, and lack of perfect synchrony between the twocould produce the observed scatter in color vs. \Sigma_1. (Abridged)

A Link Between Star Formation Quenching and Inner Stellar Mass Density in SDSS Central Galaxies