Does a cut that doesn't heal properly leave a scar?

There can be a variety of reason for a cut to not heal properly. When there is an open wound that does not heal is is referred to as an ulcer. Whatever the cause may be, the best thing is to go see a doctor right away. If there is any kind of bacterial infection it can be very serious.

you wrote: " have a scar under my eye from a glass cut " (but I think you meant to say you have a CUT not scar.) The non scabbing won't determine whether a scar will be left or not. The severity to the tissue and the kind of tissue it is on will determine that. I think it will leave a scar because the skin under the eye is not really elastic to begin with. Plus, you are saying it is a rather 'deep' cut. You can apply something to the cut to promote healing and once it is healed over, then you can put something on that to restore some of the skin's resilience...and lessen the scar's appearance. But the reason skin scars is to heal it and make it stronger. Scar tissue is not elastic.(won't stretch=is not flexible). If your cut is not healing, you should keep it clean, put Polysporin on it and cover it with a clean dressing...then go to see the doctor.