Underneath my eyes are swollen, no pain, could it be sinus cavities? What to do to to reduce swelling?

Possible. Under each eye is the maxillary sinus. The floor of the sinus rests on the upper posterior teeth. An infection of the sinus or conjestion in the sinus with cause the area to swell. The use of a cold compress will help reduce the swelling however if infection is present then antibiotics are indicated.
Need more info. How long have you had the swelling. It's unlikely to be sinus cavities, and is most likely sub dermal fat that become more prominent with time. Other that plastic surgery there usually isn't much that you can do.