This is a great question. It is important to first understand what a patient means by under eye lines. One would think that it should be obvious enough but it often is not. For example starting with the lower eyelid lashes and looking below them we can in most of us see some lines and wrinkles. Those are indeed under eye lines but may not be what you mean. That skin I just described is very thin, gets thinner with age and wrinkles easily. The orbicularis oculi muscle is under the skin and our fat pads below that. There is for example no “floor” there and so no fillers can be put there. Putting fillers there would just make for swelling and migration. That skin that I just described, which is immediately below the eyelashes, is often tightened with RF such as Thermage or ThermiSmooth or various lasers both ablative and non-ablative. And the fat pads which bulge forward can be reduced from the inside of the eyelid in an office procedure called a transconjunctival blepharoplasty which requires on stiches. Sometimes we also remove a little pinch of skin starting from below the eyelashes and this is called a pinch blepharoplasty.
Or, do you mean that bit of skin that appears darker, a little depressed and is “U” shaped like a hammock? People often point that area out and say they look tired and they have dark circles under their eyes. If so, we are now talking about the ORL a.k.a. the orbital retaining ligament. The darkness due to the shadowing of the ORL is usually improved by reducing the fat pads, as described above, which are bulging the skin that is above the orbital rim outward. It can also be helpful to support the ORL by placing some filler just below it deep in the naso-jugal groove area. Doing that can lift it a bit and help to smooth out that valley above.
If there is volume loss as part of aging or being an athlete, etc. than a consideration can be made to do some form of filling and tightening in other areas of the face. It is helpful to see what is happening to the entire face rather than simply focusing on the “eye” area. A physician very familiar with facial aging can help you figure out what the best treatment plan would be. Often the plan is best when it is multifactorial. Best of luck!
Your vertical line down the forehead is something we see very often and is normal. It appears to be a vein. They are easier to see in slim people or those who have lost a lot of weight. Stress, pregnancy, high blood pressure and other things may make the vein(s) more prominent. Treatment options include long pulsed lasers such as the 1064 nm laser (e.g. GYag), sclerotherapy using a sclerosant such as Asclera (polydocinol), ambulatory microphlebectomy (removing the vein through a few tiny nicks in the skin). Removing the vein with microphlebectomy is definitive in that it is then gone. Laser and sclerotherapy treatment options are likely to take multiple sessions. The goal of the sessions is to irritate the inside of the vessel so that it closes and then the body needs to resorb that vein that is no longer working. After treatments things look worse before the begin to look better. A potential issue is that where the vein used to be can be left indented after the unwanted vein is absorbed or removed. This can be filled in with your own living fat (autologous fat transfer - AFT) after the vein procedure is fully healed. Good luck!