There several treatments that can improve the fine lines and wrinkles under the eyes.  Starting from the least invasive:
Medical micro needling.  This can be a great way to improve fine lines as well as sun damage.
Medical chemical peels.  The chemical peels that are done in a spa are usually not deep enough to help very much but the peels that a medical skin care specialist working under the guidance of a physician like a plastic surgeon can get great results with minimal downtime.
Plasma Pen or Fibroblasting.  This is a great treatment to improve wrinkles around the eyes.  There is down time of about a week to 19 days.
Croton Oil Peel.  This is very effective for fine lines and sun damage around the eye.
Surgery or a lower eyelid tuck/lift.