I realize this is an old thread, but there is a lot of bad information here, and it should be corrected.
Macbook airs DO have an LVDS cable connecting the motherboard to the built in monitor. Display port is used if one attaches an EXTERNAL monitor. And it is very common for the LVDS to get messed up, resulting in exactly what we see here in the pics. It's frequently the result of picking up the laptop by the monitor frame, such can cause stress on the cable. In other words, it's the cable, not the monitor.
Unfortunately fixing it almost always involves opening up the monitor frame, which requires a hot air gun, some spatulate tools, and great deal of patience. It's delicate work and easy to screw up. Check out Louis Rossmann's youtube videos on the procedure. It's a pain in the ass.
Some folks have had luck kludging a solution by applying light pressure to various parts of the frame, or by inserting a business card between the inside edge of the frame and the monitor, but it seems like a temporary fix.
If you can't open the frame yourself, or find a ship to do it for you, see if you can find a whole replacement lid assembly on eBay or something. Unfortunately, they aren't cheap typically, and neither would a repair shops fee. $150 or more, from what I recall.