No, after aerating you should wait a few days before mowing in order to give the lawn time to heal. Lawn aeration is the process of removing small plugs of soil from the ground in order to improve drainage, allow more air and water penetration, and help the grass grow thicker. If you mow too soon after aerating, you could damage the newly exposed grass roots.

Yes, you can mow right after aerating. In fact, it's a good idea to do so. Aerating your lawn will help to improve drainage and help the grass to grow better. Mowing right after aerating will help to cut the plugs that were created by the aerator and will help them to decompose more quickly.