4 weeks ago I had a bulging blue vein under my eye cauterized. I have this pits in my skin. Is this normal?

Thank you for your question. The pitting may just be the result of the vein collapsing, and this will fill in over 3-6 months. We see this on noses where veins are removed, and a temporary indentation is gradually filled in. I hope this helps!

I am sure the oculoplastic surgeon is well experienced to treat this problem and I encourage to see him/her for a follow up soon, especially if you have concerns about healing/scarring and so on. 

Cautery works by heating the vein and the skin overlying the vein.  The heat evaporates the vein and causes a burn reaction in the skin which will scab and then heal.  This process could take several months.  The veins and skin look worse before they look better.  A topical ointment such as vitamin E, aloe or even hydrocortisone may help.  Follow up with your treating physician and give it more time to heal.