The Central Molecular Zone (CMZ; inner ~100 pc) hosts some of the most dense and massive molecular clouds of the Milky Way. These clouds might serve as local templates for dense clouds seen in nearby starburst galaxies or in the early universe. The clouds have a striking feature: they form stars at a very slow pace, considering their mass and high average density. Here we use interferometer data from ALMA and the SMA to show that this slow star formation is a consequence of the cloud density structure: CMZ clouds have a very flat density structure. They might, for example, exceed the average density of the Orion A molecular cloud by an order of magnitude on spatial scales ~5 pc, but CMZ “cores” of ~0.1 pc radius have masses and densities lower than what is found in the Orion KL region. This absence of highest–density gas probably explains the suppression of star formation. The clouds are relatively turbulent, and ALMA observations of H2CO and SiO indicate that the turbulence is induced by high–velocity shocks. We speculate that these shocks might prevent the formation of high–mass cores. It has been argued that the state of CMZ clouds depends on their position along the orbit around Sgr A*. Our incomplete data indicate no evolution in the density structure, and only a modest evolution in star formation activity per unit mass.