Maybe the Belgian-Australian songwriter/performer was nervous. He's an international star, but is finally starting to make a splash in the U.S. thanks to his hit song "Somebody That I Used to Know," and this appearance on SNL will surely boost his visibility. When he was interacting with his fellow singer Kimbra, he seemed at his best and most approachable, and he was fine in the skit with Andy Samberg and Taran Killam. During the quiet bars of his tunes, however, he closed his eyes, a habit that he shares with many other singers, but it makes it no less frustrating to watch.
Viewers want to connect with performers they see on stage. When singers have their eyes shut, it's a barrier for the audience. It's fine to occasionally fall into the moment, but doing it for a prolonged time seems self-indulgent. Hopefully, Gotye will be a little more confident in future live productions and open up a bit more.