Generating functions for 0-exact Lagrangians

Determine whether 0-exact Lagrangian submanifolds in cotangent bundles of closed manifolds admit generating functions in the classical sense; specifically, ascertain whether for every 0-exact Lagrangian embedding L → T* M there exists a smooth function F: M × R^k → R that is quadratic at infinity and whose associated immersed Lagrangian, obtained by the standard transversality construction, coincides with L.

Background

The paper investigates exact Lagrangians in locally conformally symplectic (lcs) cotangent bundles and highlights limits of rigidity compared with the symplectic case. In particular, it discusses generating functions, a central tool in symplectic topology for studying Lagrangian submanifolds of cotangent bundles.

After constructing examples showing that certain exact Lagrangians in lcs settings lack generating functions, the text notes that the analogous question in the purely symplectic (β = 0) case remains unresolved. This connects to broader themes around Abouzaid–Kragh’s theorem and constraints derived from generating-function techniques.