I’ve found that laser treatments are the most effective way to diminish the appearance of blue veins under the eyes. Lasers allow your doctor to accurately target the desired treatment area without damaging any surrounding tissue. The CoolTouch Varia, and Pulsed Dye Lasers are safe and effective in removing unwanted facial veins. The exact laser treatment used varies from patient to patient and should be determined by a board-certified dermatologist who has extensive expertise in laser therapy.
Your timing in answering this question is great as Dr Raffy Karamanoukian of the Santa Monica Vein Center explained all of this on The Doctors TV Show in early May 2014 and this video can be seen on my website or on The Doctors TV Show website by placing the kewy words varicose veins under the eyes. Dr Raffy is Board certified in Plastic Surgery and from the American Board of Venous and Lymphatic Medicine and he practices in Santa Monica, CA.
The treatment of blue veins under the eyes has several options including sclerotherapy, removal through small incisions (microphlebectomy), laser or sometimes fine needle radiofrequency and sometimes a combination of these treatments. The cosmetic results are excellent. You should see someone experienced in treating veins in this area.