Fat Grafting Behind Eyeballs to Lessen Hollowing?

A hollow appearance can result when the eyes are deep set. This results from a relatlively large orbital socket vs. the volume inside it.
In traumatic cases, the volume can be enhanced with placement of tissue inside the orbital socket, to allow the globe to sit more forward. In thyroid patients when the eye is overly prominent, the opposite can be done.
These procedures all have very significant risk of injury to the optic nerve, the only nerve that goes to the eye. Disruption through bleeding will cause blindness.
Needless to say, this is not a common cosmetic procedure for this reason. Unless the deformity is severe (i.e. from an accident), and the surgeon performing the work is a craniofacial surgeon, it is not recommended.

Thanks for the question.
Eyes that are sunken in the the orbits can be described as enopthalmos.The anterior projection of the eye is usually measured relative to the outer edge of the orbit, the orbital rim, but can also be assessed relative to the maxillary prominences, or the contralateral eye.
There are congenital causes of enopthalmos but the most common reason is from trauma (orbital blow out fractures).
Fat grafting is not a technique that is typically used. For patients that have significant orbital volume the orbit itself can be reshaped by cutting the bones that comprise the orbit or layering additional bone or titanium plates to decrease the volume. These are serious operations with the rare risks including blindness or impairment of the movement of occular muscles. Again, usually, patients needing these treatments have experienced facial trauma or have congenital facial bone defects.
You should see a board certified plastic surgeon for a complete evaluation.
I hope this helps.

Injecting fat into the eye socket to correct a retruded eye or enophthalmos is not a good idea. As noted, this can be due to many conditions and may be your natural appearance. A better approach is to either correct the underlying abnormality or prop up the eye with an orbital implant. For purely cosmetic reasons, these surgeries should be avoided as they can have serious side effects including vision loss.