Procedure for Drooping Eye Corners?

The procedure for correcting droopiness of the corners of the eyes is known as "canthopexy". Alternatively, a "canthoplasty" may be required to reconstruct the lower lid, the goal being to bring the tension and support, as well as the position, back to its usual, normal position. Both of these procedures can be performed through a very small incision at the corner of the eye. Canthopexy or canthoplasty are commonly performed at the same time as a rejuvenating eyelid surgery/blepharoplasty.

It is hard to see clearing from the picture that you provided. However, if your eyelid is drooping for pulling away from your eye, then there are procedures that can help. Have you had any previous procedures? Is this a new finding or has it been present your whole life? It is best that you see a plastic surgeon who specializes in eyes. Good luck.

I enlarged your photo and looks like you have ectropion. You did not mention any previous surgeries. Usually it happens after lower eyelid surgery. If this is the case then I am concerned about some medical problem like thyroid etc. You should see an Internist and get checked for that and an ophthalmologist. '
Once you have done the above, you can get it fixed by getting lower eyelid canthoplasty procedure.
Regards

Correcting a drooping outer eye is a challenging position. It often combines the lower eyelid, the cheek, the cannthus, the upper eyelid and the lateral brow. Unless the lower eyelid alone has a problem (looseness or prior surgery pulling it down), all five of these measures are usually necessary to achieve a haronious, balanced result.
It can make a dramatic difference in a patient's appearance, particularly in reducing the "sad" appearance many patients have.
Below two articles on the topic, including some extreme examples:

There are many potential reasons for "drooping eye corners"
You must look at the brow, upper eyelid, canthal support mechanisms, lower eyelid, and cheek.
Any and all of these anatomic structures can contribute to your problem. Therefore, I would highly suggest that you see a plastic surgeon to discuss your specific problem and recommended treatment options.

From the picture this looks like a normal variant, but it may still concern you.  The corners of the eyes can be picked up with a canthopexy or canthoplasty, and reshape the eyelid "aperture" into a more pleasing shape.

In the photo I noted that your corners of the eyelids are droopy and loose. This is your normal anatomy that you had when you were born. I also see that the whites of your eyes are visible the entire length of your eyelid, not just on the corners. A cantopexy or canthoplasty may help but to lift the entire lid, a lower eyelid spacer may need to be placed also. You may need to see an eyelid specialist for this one. 

A canthoplasty and/or canthopexy can be performed to bring up the outer corners of the lower lids. This is performed as an outpatient procedure where the lower lid is brought up and resutured to the orbital rim to make a more natural fitting contour. An oculoplastic surgeon with many years of experience is probably the best person for this type of problem.

Dear Gareth,
The photo you submitted accentuates the droop to the lower eyelid (when your face is pointing down and you are looking up, your lower lid will appear lower.) You should evaluate yourself with your face in a horizontal plane (parallel to a wall mounted mirror.)
Fine points aside, I'm sure you are correct that you have a lateral sweep to your lower lid. This is commonly due to lower lid laxity, and can be corrected with a canthopexy (minor weakness), or canthoplasty (for correction of major weakness). Visit with a surgeon who is versed in lower eyelid surgery and have an evaluation.
Best regards,
Dr.B

You do have some droop, but also, it has to do with head position and gaze.  Still, there is some droop.
 
You also appear to have under developed cheek bones.  You would do well with a lower lid elevation procedure and maybe cheek inplant.
 
sek