Augmented reality (AR) experiences place users inside the user interface (UI), where they can see and interact with three-dimensional virtual content. This paper explores UI security for AR platforms, for which we identify three UI security-related properties: Same Space (how does the platform handle virtual content placed at the same coordinates?), Invisibility (how does the platform handle invisible virtual content?), and Synthetic Input (how does the platform handle simulated user input?). We demonstrate the security implications of different instantiations of these properties through five proof-of-concept attacks between distrusting AR application components (i.e., a main app and an included library)\dash including a clickjacking attack and an object erasure attack. We then empirically investigate these UI security properties on five current AR platforms: ARCore (Google), ARKit (Apple), Hololens (Microsoft), Oculus (Meta), and WebXR (browser). We find that all platforms enable at least three of our proof-of-concept attacks to succeed. We discuss potential future defenses, including applying lessons from 2D UI security and identifying new directions for AR UI security.