Math Conjectures

Nearby Lagrangian conjecture

OPENMajorConjectureProposed c. 1980 · Full conjecture

Canonical statement

Let QQ be a closed connected smooth manifold and π:T∗Q→Q\pi:T^*Q\to Q its cotangent projection. Define the canonical Liouville 11-form by λ(q,p)(ξ)=p(dπ(q,p)ξ)\lambda_{(q,p)}(\xi)=p(d\pi_{(q,p)}\xi). If L⊂T∗QL\subset T^{*}Q is a closed connected Lagrangian submanifold and λ∣L=df\lambda|_L=df for some smooth f:L→Rf:L\to\mathbb R, then a compactly supported Hamiltonian isotopy of T∗QT^{*}Q carries LL to the zero section.
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Let \(Q\) be a closed connected smooth manifold and \(\pi:T^*Q\to Q\) its cotangent projection. Define the canonical Liouville \(1\)-form by \(\lambda_{(q,p)}(\xi)=p(d\pi_{(q,p)}\xi)\). If \(L\subset T^{*}Q\) is a closed connected Lagrangian submanifold and \(\lambda|_L=df\) for some smooth \(f:L\to\mathbb R\), then a compactly supported Hamiltonian isotopy of \(T^{*}Q\) carries \(L\) to the zero section.
GEO-TOP-014:[Arnold1986LagrangianSingularities][Abouzaid2012NearbyLagrangians][Kragh2013ParametrizedSpectra][Porcelli2025LagrangianMonodromy]
Geometry and topologyDynamical systems

Notes

Let QQ be a closed connected manifold and T∗QT^*Q its cotangent bundle with the canonical Liouville form λ=p dq\lambda=p\,dq. A closed Lagrangian L⊂T∗QL\subset T^*Q is exact if λ∣L\lambda|_L is the differential of a function. The nearby Lagrangian conjecture asserts that every closed exact Lagrangian is carried to the zero section by a compactly supported Hamiltonian isotopy — that is, up to Hamiltonian equivalence the zero section is the only closed exact Lagrangian in a cotangent bundle. The problem grew out of Arnold's nearby-Lagrangian program of the late 1970s and early 1980s [Arnold1986LagrangianSingularities], which is why its date is given only approximately as c. 1980.

The topological side is now well understood. Work of Abouzaid on nearby Lagrangians with vanishing Maslov class [Abouzaid2012NearbyLagrangians] and Kragh's parametrized-spectrum methods [Kragh2013ParametrizedSpectra] led to the conclusion that the projection L→QL\to Q is a homotopy equivalence, indeed a simple homotopy equivalence, and more recent work probes finer C0C^0 and monodromy phenomena [Porcelli2025LagrangianMonodromy].

These conclusions fall short of the conjecture itself: homotopy-theoretic control does not by itself produce a Hamiltonian isotopy, and the full statement remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.