Broken iPad? Curve in screen?

As described, there is a potential issue of swelling/swollen battery - this effectively forcing the screen away from the unibody chassis.

If the battery is swelling, this is potentially serious safety issue. If this is the case, the battery could rupture and cause a fire. Li-ion batteries are generally very safe, but if the battery should be damaged, they can internally short-circuit and spontaneously combust.

On the basis of your description alone, until the iPad is inspected by an Apple Approved technician, I would recommend that you immediately stop using the device. Do not connect an external charger. The iPad should be carefully removed from anywhere containing combustible materials and, ideally, moved outside of your home.

You should contact Apple Support to seek explicit advice. The link to contact support is at top-right of this page.

Li-ion batteries don’t like heat. If your device is becoming uncomfortably hot during use, this may be indicative of a fault.

Continued use a damaged or defective device, prior to examination by a qualified technician, is unwise - and entirely at your own risk.

If you need to be convinced of the severity of a Li-ion battery fire - due to battery failure - just “google”.

Okay - so your reported bend is actually tiny - and you’re not bothered (despite your earlier comments) about overheating or overcharging. I clearly misinterpreted your call for help.

No problem then, nothing to fix, you’re happy. If only everyone’s problem were so trivial...