My name's Korra. [She glances at him from the corner of her eyes, checking to see if he recognizes her name. Her name is fairly well known, after all.]

I was in the South Pole, checking on the spirit portal there before... I don't know. Something happened and I got knocked out for a second. When I woke up, I was here.

[And now she would know for sure--even if there is someone else named Korra around, linking herself to the spirits like this definitely gives her away as the Avatar.]