Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions

We prove more convexity properties for Lagrangian submanifolds in symplectic and Kähler manifolds. Namely, every closed Bohr–Sommerfeld Lagrangian submanifold $Q$ of a symplectic/Kähler 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 Kähler 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.

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Received 12 July 2018

Accepted 17 October 2018

Published 25 March 2020