The Levi-Flat structure dates back to the seminal work of L.Nirenberg generalizing the Newlander-Nirenberg theorem on complex structures, it has been one of the main topics in theory of involutive structures. We introduce a notion of convexity for Levi-flat structures motivated by Morse theory and Grauert type convexity from SCV. Applications to global/local solvability of the Treves complex will be presented. We will also talk about the extension problem for the canonical bundle of a Levi flat structure.
