Bohr-Sommerfeld Lagrangian submanifolds as minima of convex functions

TL;DR

The paper demonstrates that every closed Bohr-Sommerfeld Lagrangian submanifold in a symplectic or Kähler manifold can be realized as a Morse-Bott minimum of a specially constructed convex function outside a hyperplane section.

Contribution

It establishes a method to realize Bohr-Sommerfeld Lagrangians as minima of convex functions in both symplectic and Kähler settings, linking geometric and analytic properties.

Findings

Every Bohr-Sommerfeld Lagrangian can be realized as a Morse-Bott minimum.

Construction of convex functions with prescribed Lagrangian minima.

Extension of the concept of convexity in symplectic and Kähler geometry.

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.