What should I do to improve the eye area?

I don't think you have droopy eyelids, however you do have under eye hollowness which is best treated with filler injection. Other options are also available. See link and video below.

To treat the hollowness under your eyes, I would recommend a filler like Restylane.  It does not look like you have droopy eyelids nor do you appear to have a low brow that would require a lift.  I would try the filler injection alone and see how you like it. 

Fillers, lasers, and microneedling/PRP can improve under eye hollows. My patients try to get non surgical treatments to avoid surgery. We use PRP and stem cells there often. Botox can be injected above the eyebrow for a lift. I suggest consulting an expert. Best, Dr. Emer.

Botox and Restylane to your tear troughs would be a great
option to help your eye area look more awake and youthful. Starting with a
conservative treatment could help you achieve the results you desire. A
certified experienced injector can help provide a consultation, plan, and
safe/effective treatment.

You're on the right track. Filler to the lower eyelid hollows will work well. Use something soft like belotero. The uppers may benefit from a Botox brow lift. You can also do radiofrequebcy skin tightening for the eyelids and brows. In addition, you have a relative hollow of the central upper eyelid, which makes the kids slant down. Filler for the upper eyelids centrally could add a nice touch. 

It's best to do very minimal interventions in your case at this point.Botox and Restylane, if subtly done, could improve the shape of your eye area slightly.Preventative maintenance is also important to minimize the effects of aging!However, don't jump into surgery until there is a clear indication, or you will join the ranks of those who look 'odd'.

Thank you for sharing your question. You are correct that there are noninvasive procedures that can help refresh your eyes. Botox can elevate your brows and pull up the extra skin. Restylane can be used to fill in the hollows beneath your eyes. See an experienced Oculoplatic Surgeon for detailed evaluation and discussion of possible treatments.Good luck,

Botox used in a specific pattern could give your lateral brow a lift that may give you a refreshed look! Also the thinnest formulation of Juvederm or Restylane, CAREFULLY injected into the hollows, can make a world of difference in helping you look more alert. IMPORTANT NOTE: When getting injections in this area of the eye in particular, make sure to schedule a time when your doctor is not rushed and you are not in a hurry. Because the skin is thinner in this area of the face/body than in other areas, care must be taken to go slowly and to deposit only little bits of Juvederm or Restylane at a time. This takes time and patience!! Hasty injections can cause bad bruising and even worse, permanent blindness.

Yes, Botox is a great option for a more "open eye" look.  Be sure to go to a reputable provider. In my opinion Restylane is, as of right now, the best filler for under eye hollowness.  This will give you a more "refreshed look"... if done correctly.  You can also consider Ultherapy to help raise your brows nonsurgically.   Good luck.  Hope this helps.  

Botox and other neuromodulators are great ways to elevate the lateral brow. Botox works by effectively weakening the muscles around the eye that pull the brows downward, allowing forehead muscles to lift the brow. With Botox around the "crow's feet" area you should be able to get a slight upward lift of the tail regions of your brow. Looking at your undereye photos, it does seem that you have some hollowing. I prefer belotero for the undereye region and it would serve to fill in the tear troughs that you have and smooth out that whole area. It is always advised to visit a board certified facial plastic surgeon to get a complete consultation for your facial rejuvenation. Good luck!