Bluets, Houstonia species. A lovely native plant trying to fit in with suburbia, ;).

Can I transplant them into my garden. I will have to mow sooner or later.

Yes, you can. Or wait until they set seed and then gather that and spread it in a new place.

Mine seem to grow the thickest where they get above average moisture and part sun, if that helps. They grow in areas that get quite a bit of sun and less water but tend to grow rather thinly.

BRMG

If they are in a lawn and doing OK, then they should survive mowing just fine, since they must already experience mowing on a regular basis.

I am sure they will survive to a certain extent, but they seem to be dwindling in numbers. The very back of the lawn I can mow high, since it is by a line of pines and does not look too bad. There are many patches of them there. (I am guessing 50 or more.) Right now in the rest of the lawn, it looks like two patches. Or perhaps too patches just managed to blossom enough to be noticed.

Besides all that, I think they would look nice in the garden. I just thought I would not move them if it would just finish them off.