Although it is somewhat difficult to see from your picture, it appears that some of your problem is hereditary pigmentation which might improve with various topical or laser treatments. The hollowness can be improved as suggested, quickly with various fillers most of which can be injected in a single session and will last for about one year.
I really like using the product Redensity 2 for filling under eye hollows or tear troughs as they are called. This procedure is relatively painless and takes minutes to do. With the use of micro cannulas there may be minimal bruising. Results can last over 12 months.
It is hard to tell in this picture what you look like since you have covered up the lower eyelids. Fat does not usually go away quickly. It will initially look overcorrected, and then some of the fat will get resorbed. Whatever remains usually lasts many years. You can always try an HA filler there first to make sure you like the result you would obtain with filling the lower eyelids. The HA will gradually go away .