Lagrangian neighborhood theorem

Statement

Let $L$ be an embedded Lagrangian submanifold of a symplectic manifold $(M,\omega)$ . There are neighborhoods $U$ of the zero section in the cotangent bundle $T^*L$ and $V$ of $L$ in $M$ , together with a symplectomorphism

\Phi:(U,\omega_{\mathrm{can}})\longrightarrow(V,\omega)

whose restriction to the zero section is the given inclusion $L\hookrightarrow M$ . Thus the germ of the ambient symplectic structure along $L$ is modeled by the canonical cotangent symplectic form. The neighborhoods and the symplectomorphism are generally not unique, and the theorem makes no global claim about all of $T^*L$ or $M$ .

Relative formulation

A useful equivalent form compares two symplectic manifolds containing copies of the same Lagrangian $L$ . If a diffeomorphism between those copies is fixed, then it extends, after shrinking neighborhoods, to a symplectomorphism near $L$ . Taking one ambient manifold to be $T^*L$ and one copy to be its zero section gives the stated form. This is the Lagrangian case of Weinstein's neighborhood results.

Proof idea

The symplectic form identifies the normal bundle $TM|_L/TL$ with $T^*L$ : the class of $v$ is sent to the covector $w\mapsto\omega(v,w)$ . A tubular-neighborhood choice realizes this bundle identification by a diffeomorphism near the zero section. The pulled-back ambient form and the canonical cotangent form agree along $L$ . A relative Moser argument then corrects the diffeomorphism to a symplectomorphism while fixing $L$ .

The correction step explains both the local nature of the theorem and its nonuniqueness: auxiliary tubular data and the Moser primitive can vary.

Consequences and limitations

Nearby sections of $T^*L$ are Lagrangian exactly when their defining one-forms are closed. Consequently, sufficiently small Lagrangian deformations of $L$ , once placed in a Weinstein neighborhood and transverse to the cotangent fibers, are represented by closed one-forms. Exact one-forms describe the corresponding local Hamiltonian deformations.

References

Alan Weinstein, “Symplectic Manifolds and Their Lagrangian Submanifolds,” Advances in Mathematics 6 (1971), 329–346. DOI record. Relevant: Theorem 6.1 and the neighborhood-equivalence argument.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §3.4, the Weinstein neighborhood theorem and local Lagrangian graphs.