Integrable Convex Hamiltonians on Exotic Tori

Definitions and mathematical statement

Let $\Sigma$ be an exotic smooth $n$-torus, meaning a smooth manifold homeomorphic to $\mathbb T^n=(S^1)^n$ but not diffeomorphic to it. The question concerns dimensions and smooth structures for which such $\Sigma$ exists. Equip $T^*\Sigma$ with its canonical symplectic structure. The source asks whether there exists a completely integrable convex Hamiltonian $H:T^*\Sigma\to{\mathbb{R}}$ for a given such $\Sigma$. Under the usual smooth Liouville interpretation, complete integrability means functions $F_1=H,F_2,\ldots,F_n$ with \[\{F_i,F_j\}=0\quad(1\leq i,j\leq n),\qquad dF_1\wedge\cdots\wedge dF_n\ne0\] on an appropriate open dense regular set. The extract does not specify the regular set, regularity, or whether the integrals are globally defined on the whole cotangent bundle or in a designated region. Convexity is normally fiberwise in the cotangent variable; strict fiber convexity would require the vertical Hessian of $H$ to be positive definite. Weak versus strict convexity, properness, and superlinear growth are not fixed by the supplied wording. No kinetic-energy, quadratic, reversible, or real-analytic restriction is inserted. The requested existence for an arbitrary given exotic torus is not reduced to existence on at least one example. The 2026-08-21 triage labels the question open and reports Butler's strong obstructions for finitely geometrically semisimple and real analytically integrable convex Hamiltonians, while explicitly saying these do not settle existence of a merely completely integrable convex Hamiltonian. The stronger obstruction hypotheses are not imposed on the remaining question.

Short English statement

Can an exotic smooth torus carry a convex completely integrable Hamiltonian on its cotangent bundle? The exact convexity and integrability conventions are missing, and the reported obstructions under stronger tameness or analyticity hypotheses do not answer the stated smooth existence question.

Sources

• Local UnsolvedMath registry, permanent ID 30001027, source problem catalogue entry, Integrable Convex Hamiltonians on Exotic Tori, proposed year 2008; no individual proposer supplied.
• Supplied citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406: https://doi.org/10.4171/owr/2008/42. Supplied literature label: original report.
• Supplied literature label: Butler, Maslov cocycle and complete integrability, https://arxiv.org/abs/0708.3157.
• Literature assessment date: 2026-08-21. The status and progress claims are supplied dated curation, not independently verified here.