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Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions
Journal of Symplectic Geometry  (IF0.707),  Pub Date : 2020-01-01, DOI: 10.4310/jsg.2020.v18.n1.a9
Alexandre Vérine

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.