If the mangoes are not fully ripe, they will contain more stringy fibers, reducing the yield. Mangoes that are somewhat unripe are easier to work with if you peel them before cutting the flesh from the stones.
Slice each mango in half by making a cut on either side of the large flat stone. Use a spoon to scrape the flesh away from the skin and the stones. Place the flesh in a food processor. Add the lime juice and puree the fruit until smooth. Strain through a chinois and discard the solids.
Stir in the simple syrup and enough of the water to bring the mixture to between 16° and 20° Baumé.
Process in an ice cream freezer according to the manufacturer’s directions. Transfer the sorbet to a chilled container, cover, and store in the freezer.