This can be treated with a variety of lasers and RF devices, go to a vein specialist if you are going to have sclerotherapy especially around the eyes. Best, Dr. Emer.
There are several treatment options depending upon the exact size and location of the blood vessel. Laser vein therapy might be effective and possibly sclerotherapy. Other options may include simple excision of the vein. I would see a specialist in your area to discuss your concerns.
Treatment under eye veins
These veins can be treated in multiple ways including sclerotherapy, fine needle RF, VeinWave, VeinGogh, laser and removal through micro incisions. I would use a combination of sclerotherapy and fine needle RF.
Under eye veins can be treated with RF, thermolysis, sclerotherapy and long pulsed lasers .. Buffalo Niagara NY
Under eye veins can be treated with RF, thermolysis, sclerotherapy and long pulsed lasers.
Much larger veins can be removed with ultracision microphlebectomy.
Given the size of the vein under your eye, I recommend scerotherapy or a long pulsed ND YAG laser treatment.