A carrier head for a chemical mechanical polisher includes a base, a substrate mounting surface, an annular inner ring and an outer ring. The inner ring has a lower surface configured to contact an upper surface of a substrate positioned on the substrate mounting surface, an outer surface, and an inwardly facing surface extending downwardly from the lower surface and is configured to circumferentially surround the edge of the substrate, the inner ring vertically movable relative to the substrate mounting surface. The outer ring has an inner surface circumferentially surrounding the inner ring, an outer surface, and a lower surface to contact the polishing pad, and the outer ring is vertically movable relative to and independently of the substrate mounting surface and the inner ring.