Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions

Abstract

We prove that every closed Bohr-Sommerfeld Lagrangian submanifold $Q$ of a symplectic/K\ahler manifold $X$ can be realised as a Morse-Bott minimum for some 'convex' exhausting function defined in the complement of a symplectic/complex hyperplane section $Y$. In the K\ahler case, 'convex' means strictly plurisubharmonic while, in the symplectic case, it refers to the existence of a Liouville pseudogradient. In particular, $Q\subset X\setminus Y$ is a regular Lagrangian submanifold in the sense of Eliashberg-Ganatra-Lazarev.