Result 340, Differential geometry

A counterexample to the nearby Lagrangian conjecture

Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in T∗(S9×SN−1)T^*(S^9\times S^{N-1}) that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.

Disproof or counterexample

The bigger picture

Why it matters

The manuscript reports a counterexample to the unrestricted nearby Lagrangian conjecture: a geometric object with the same smooth shape as a standard reference object, but which cannot be moved onto it by Hamiltonian motion.

What changes?

For some sufficiently large even integer N, the claimed construction lies in the cotangent bundle of the product of a nine-dimensional sphere and an (N minus one)-dimensional sphere. This bundle records positions and covectors, the geometric counterparts of momenta. The constructed object is a closed exact smoothly embedded Lagrangian: a compact submanifold without boundary, with half the ambient dimension, on which the symplectic form vanishes and the canonical one-form is the differential of a function.

What does that help mathematicians do?

The reported example is diffeomorphic to the base, meaning they are equivalent as smooth manifolds, yet is not Hamiltonian isotopic to the zero section, the copy of the base with every momentum set to zero. Hamiltonian isotopy means deforming by motion generated by a Hamiltonian function. Thus, even matching the base's smooth shape does not ensure this symplectic equivalence. The obstruction is not simply a difference in smooth topology.

Are there practical applications?

Its immediate value is foundational: it draws a boundary around how broadly exact Lagrangians in cotangent bundles can be classified by comparison with the zero section. Researchers must account for distinctions invisible to smooth shape. The stated example does not settle versions restricted to other bases or dimensions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A counterexample to the nearby Lagrangian conjecture

September 23, 2026 28 pages

We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in T∗(S9×SN−1)T^*(S^9\times S^{N-1}) that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to the nearby Lagrangian conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.