A cut? A wide gaping gash? What do you do with it?
From personal experience, unfortunatly there is not much you can do about it.
Unless your lip is hanging off your face because it is so split, then the chances are you will not get stitches, or anything for that matter. You will have to suck it up for a few days, plus some vaseline or polysporin on it and try not to smile and split it further.
Why is this you may ask? Any other part of your body with a gash in it would easily require stitched, why not your lip? Well because there are hundreds of little blood vessels that run through your lips and enough nerve ending that it will heal at remarkable speed. With the helpful power of your own saliva and the team of blood vessels working on your cut you should be good to go in no time. This healing that goes on in your mouth and around your lips is incredible and the doctors like to take advantage of this and let it naturally repair itself. You may not be quite so appriciative as your lip sits there sliced right open, but if you care to take notice, your body will do a fantastic job getting it healed in no time.