As an additional comment AA is a great hand to check on a board like JJ9 for the reason, that when its ahead, its often way ahead. So AA does not benefit as much from protection as lower pairs like A9 or TT. If for instance the opponent had a hand like Q5s, then he would beat A9 or TT on the turn. But when we have AA, we are happy for him to spike top pair, because then we can get some value on the turn and river. Whereas if we bet the flop, he is probably just going to fold, unless he has the right suit for the flushdraw.
I think, its very common for people to feel, they have to bet the flop, when there are draw available. But this is kind of a flawed way of thinking. Draws only make up a small percentage of our opponents range. And in top of that good draws are not folding on the flop, and we are not that fat ahead of them, unless we have 3 of a kind. So instead of blindly focusing on draws, we want to focus on our opponents entire range. And in this case that will contain a lot of hands, that missed completely and are drawing almost dead. Which makes checking a good play with AA or KK specifically.