  We study the structural and environmental dependence of the star formation on
the plane of stellar mass versus central core density ($\Sigma_{\rm 1\ kpc}$)
in the nearby universe. We study the central galaxies in the sparse environment
and find a characteristic population-averaged $\rm \Sigma_{1\ kpc} \sim
10^9-10^{9.2}\ M_{\odot}\ kpc^{-2}$, above which quenching is operating. This
$\rm \Sigma^{crit}_{1\ kpc}$ only weakly depends on the stellar mass,
suggesting that the mass-quenching of the central galaxies is more closely
related to the processes that operate in the central regions than over the
entire galaxies. For satellites, at a given stellar mass, environment-quenching
appears to operate in a similar fashion as mass-quenching in centrals, also
starting from galaxies with high $\rm \Sigma_{1\ kpc}$ to low $\rm \Sigma_{1\
kpc}$, and $\rm \Sigma^{crit}_{1\ kpc}$ becomes strongly mass-dependent, in
particular in dense regions. This is because (1) more low-mass satellites are
quenched by the environmental effects in denser regions and (2) at fixed
stellar mass and environment, the environment-quenched satellites have, on
average, larger $\Sigma_{\rm 1\ kpc}$, $\rm M_{1\ kpc}/M_{\star}$ and Sersic
index $n$, and as well as smaller size. These results imply that either some
dynamical processes change the structure of the satellites during quenching or
the satellites with higher $\Sigma_{\rm 1\ kpc}$ are more susceptible to the
environmental effects.
