LA Noire Cosmic Rays how do I catch the suspect without him dieing?

Question

he keeps jumping off the roof and dies!

Answer 1

Unfortunately, there's not any way to save him. It's coded into the game. You still get credit for completing the Street Crime, though. For more information on LA Noire Street Crimes, see this post.

Answer 2

Instead of chasing him into the parking lot, go left around the cafe and meet the suspect in the lot facing the other way, you'll get into a fist fight with him.