Deformable substrate chuck

A chemical mechanical polishing system, comprising:

• a platen with a polishing surface;
• a carrier head including
• a housing,
• a support assembly connected to the housing so as to be vertically movable relative to the housing, the support assembly including a support plate and a plurality of fluid-impermeable flexible pieces that are more flexible than the support plate and that are secured to and extend below the support plate to provide a plurality of individually pressurizable lower chambers below the support plate to apply pressure to a substrate, and
• at least one upper barrier that divides a volume between the support plate and the housing into a plurality of individually pressurizable upper chambers to apply pressure on a top surface of the support plate in a plurality of first zones, and wherein the support plate is sufficiently flexible to bend under a pressure differential between the upper chambers;
• one or more pressure sources connected to the upper chambers and the lower chambers;
• an in-situ monitoring system configured to measure a shape of the substrate, and
• a controller connected to the in-situ monitoring system and the one or more pressure sources, wherein the controller is configured to control pressure based on the shape of the substrate.