Best Permanent Under Eye Filler for Hollowness?

Although fat is an excellent permanent facial filler, it can behave unpredictably in the tear trough area with a risk of lumpiness, which would require further surgery to correct.
As the tear trough & lid cheek junction ageing is dynamic and progressive, I do not recommend permanent fillers in this area.  

Fat grafting can help achieve permanent results. Fat is taken from a donor site on your body, such as the midsection, and injected into the undereye area. However, I strongly recommend you reconsider getting permanent fillers, particularly in this area. This is because if you don’t like the results, surgery would probably be necessary to correct them.

Silikon-1000 is a permanent filler that is amazing at filling lower eyelid hollows. You must find a board-certified dermatologist or facial plastic surgeon experienced in the micro-droplet technique of injection. You should see many favorable, attractive photos from your doctor before proceeding.
I've attached a link to my Silikon-1000 photos for your perusal.
I hope this helps, and best regards.

Structural fat grafting can be performed on upper eyelid hollows, to smooth hollows under the eyes, to fill sunken cheeks, for filling sunken hollows under eyes eyes and is often used for blepharoplasty correction when too much skin and or fat has been removed in an eyelid surgical procedure.

Fat Grafting requires specialized training and specialized surgical instruments, as well as patience and attention to detail on the part of the surgeon. When performed properly, permanent improvements in facial plastic surgery are possible.

The technique I use, structural fat grafting, in which small amounts (less than 0.1 cc at a time) of fat are carefully microinjected in a series of discrete layers to gradually 'build' new soft tissue structure. As there is space between each microinjection, new blood vessels are able to grow into the grafted fat, allowing it to persist. If this process of blood vessel ingrowth (neovascularization) does not occur, then the injected tissue cannot truly be considered a 'graft' and is instead just another 'soft tissue filler' of limited duration.

It is important select a plastic surgeon with specialized training who performs fat grafting often.