Honestly, I've been putting the 0.1% directly under my eyes (and even on the tops of my eyelids), with no problem, and I've only been using tretinoin for three weeks now. However, I am an outlier as my skin has always been very healthy. Also, I will admit that I had initial irritation, but once your face goes through the retinization period, you can go to town with tretinoin.
But for those getting into it, you are never supposed to put it directly under the eyes initially (or right under your nose, or at the sides of your lips). It will spread there on its own as you sleep. A lot of people just put it on the thicker parts of their skin initially, then use moisturizer to dilute the tretinol and spread it all over the place.
If you really wanna get rid of the dark circles (or other hyper pigmentation), you can also use a strong vitamin C serum. Products from ordinary are cult favourites among redditors because of their quality ingredients are barebones pricing:
https://theordinary.deciem.com/product/rdn-vitamin-c-suspension-23pct-ha-spheres-2pct-30ml
You should also use a solid facial moisturizer. The cult fav among redditors is Cerave PM lotion:
https://www.cerave.ca/our-products/moisturizers/pm-facial-moisturizing-lotion
Cerave is also a cult fav brand for the same reason as the ordinary. I use Cerave SA cream for my face (commonly referred to as "Cerave in the tub"), but that can be too intense for the face as it is intended for the body. My skin can handle it, but many break out.
https://www.cerave.ca/our-products/moisturizers/sa-cream-for-rough-and-bumpy-skin
While you're at it, you should also grab a gentle Cerave cleanser. If Cerave products do not agree with your skin, try its counterpart Cetaphil.
I've had dark circles under my eyes all my life, and minoxidil made them 200% worse. My regimen of 0.1% Tretinoin, vitamin C, and the Cerave products have made the skin under my under eye lighter then they have ever been in my entire life....by a lot. And that is s big deal, especially considering that all the skin on my face went about 3 shades lighter. (I am South Asian btw).
Cheers.
Edit: I will warn any who want to actually take the meaningful steps above to get rid of their dark eye circles, that you should treat this journey the same way you treat your hair loss journey. Research the crap out of everything. If you think people vary a lot based on their results to hairloss treatments, you'll be blown away at the variability of people's facial skin to different products. Expect lots of breaking out, burning, peeling, pimples, etc. The products I listed should be safe though, and pretty much all you need for 90% of the max results you can attain.
Like Min/finasteride, Tretinoin takes a minimum of three months to see some results, and a full year to see full results. Also, from week 1 to week 12 (longer or shorter for some people), much like shedding with finasteride/min, Tretinoin will turn your face red, make your skin peel, cause breakouts and make some look awful. Also like min/finasteride, you have to take Tretinoin everyday for the rest of your life.
You will also HAVEEE to start using sunscreen on your face everyday...even in the winter. So start to researching sunscreens because what you get in Canada/U.S. wont cut it for proper UVA and UVB filters. Get used to looking at European, or better yet, Japanese sunscreens. Pro tip: Rohoto Skin Aqua Super Moisture Milk SPF50 PA++++ are among the cult faves.
*No sunscreen while using Tretinoin = easier sunburn, hyperpigmentation, extreme dark eye circles, etc.
Get comfortable browsing:
https://www.reddit.com/r/SkincareAddiction/