AA $5NL good line?

Villain's 4th hand at the table, so no read. Obviously this is standard up to the turn, but is a check better on the turn here, and what do you think of the river play?

PokerStars No-Limit Hold'em, $0.05 BB (9 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

saw flop

SB ($1.95)
BB ($4.80)
UTG ($2.52)
UTG+1 ($10.35)
MP1 ($6.92)
Hero (MP2) ($6.37)
MP3 ($7.13)
Villain (CO) ($4.93)
Button ($13.85)

Preflop: Hero is MP2 with A, A
3 folds, Hero bets $0.25, 1 fold, Villain calls $0.25, 3 folds

Flop: ($0.57) J, 4, 9 (2 players)
Hero bets $0.40, Villain calls $0.40

Turn: ($1.37) 8 (2 players)
Hero bets $1, Villain calls $1

River: ($3.37) 3 (2 players)
Hero checks, Villain bets $3.28 (All-In), Hero ???

wut?
Why aren't you shoving the river?

betbetbetbetbet... No reason to slow down anywhere on that board imo.

As played I think his range is prolly 89,QT,88,99,JJ,QQ,KK plus one or two stupid random hands.

You need to think about what you're doing when villain shoves before you check! - If you're going to call thinking that you're probably behind, then you should always bet instead.

i'm shoving this river for him

Hmm, kinda makes sense I guess. Maybe I'm just second guessing myself a lot because I seem to have developed a talent for paying off sets with AA.

Have run really good today (+$40 in 2k hands, roughly) and doubled up with AA twice ignoring this hand, but before today I was breakeven with AA over 15k hands so figure I'm probably doing something wrong.