Wait until the concrete countertop completely dries before beginning repairs, if curling has occurred on a newly-poured countertop. Fans placed on low can assist in providing gentle air flow over the concrete counter top. As drying occurs, moisture content will become consistent throughout the countertop, which may significantly reduce the amount of curling prior to the need for additional repair.
Pour water over the surface of the concrete countertop. Holding the disc of the grinder flat against the concrete surface, maintain steady pressure while slowly grinding the curled concrete. Continue applying water; the grinder must not dry out. Grind the curled concrete level with the rest of the countertop.
Rinse the countertop of any grinding residue; repeat as necessary, cleaning with a rag, to make sure that the concrete "grit" is completely removed. Allow the countertop to dry.
Fill any low spots with a concrete filler -- such as an epoxy filler that's easy to work with and adheres to clean concrete with little problems. Follow the manufacturer's directions and level off the filler to the height of the countertop with the putty knife. Remove any excess and allow the filler to dry as recommended.
Clean the counter thoroughly, once the filler has dried completely. Using the paintbrush, cover the countertop with a concrete sealer to protect it from stains, water marks and other types of damage. Allow the sealer to dry per the manufacturer's recommendations; if necessary, repeat with a second coat.