What would be a permanent solution for under eye hollowness and Dennie-Morgan folds?

You definitely have a few lower eyelid lines. You also have an deep under eye circle. The whole allergic shiner/ Dennie-Morgan line story seems to be a story sold to concerned parents to feel like their was a reason their babies had these aesthetic issue. The evidence that allergic shiners are really related to allergies in most is thin. I am posting to highly caution you about seeking a permanent solution for this. Surgeons will offer you nanofat, micro fat, grafted fat, mid face lift, orbital rim implants. It is essential to understand that most of these "solutions" will harm the glide function of the lower eyelid/midface junction. That damage is very hard to undo or fix. The state of the art treatment for this is actually hyaluronic acid filler treatment for the circles. This is an office service that lasts about 9 to 12 months for most individuals. The filler you get makes a very big difference, you need a high G prime hyaluronic acid filler. I only use Restylane for this application. 

A permanent treatment for under eye tear trough hollowness is infraorbital rim silicone implant. Need personal evaluation to determine if good candidate.