The flatness conjecture for completely integrable geodesic flows

Let $(M,g)$ be a compact Riemannian manifold whose geodesic flow is completely integrable. Let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$ , and suppose that all of its orthonormal bases of eigenfunctions have uniformly bounded sup norms. Flatness conjecture for completely integrable geodesic flows. Then $(M,g)$ is flat. Without quantum complete integrability, the source notes that it is not known whether eigenfunctions localize on level sets of the classical moment map, so the conjecture concerns the proposed weakening to classical integrability.

Sources & referencesView supporting material

Primary source

John Toth and Steve Zelditch, “Riemannian Manifolds With Uniformly Bounded Eigenfunctions”, arXiv:math-ph/0002038 (2001).

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Progress summary

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Open

The conjecture remains unproved: bounded eigenfunctions are known to force flatness under a stronger quantum-integrability assumption, but classical integrability alone is unresolved.

For compact $(M,g)$ with completely integrable geodesic flow, the conjecture says that uniformly bounded eigenfunction sup norms for a Schrödinger operator $\Delta_g+V$ force $(M,g)$ to be flat. The source explicitly identifies the missing step: under classical, rather than quantum, integrability, eigenfunction localization on classical moment-map levels is unknown.

Known results

Under quantum complete integrability and the relevant properness and moment-map hypotheses, uniform bounds imply flatness of the Jacobi metric $(E-V)g$ at one energy, and flatness of $(M,g)$ with $V$ constant across an energy interval.
In the quantum-integrable setting, unless $(M,g)$ is a flat torus, joint eigenfunctions have sequences with $\|\varphi_k\|_{L^\infty}\ge C(\epsilon)\lambda_k^{1/4-\epsilon}$ under Eliasson’s non-degeneracy condition; this does not address the conjecture’s classical-integrability hypothesis.

Current status (as of August 2026): The classical-integrability flatness conjecture remains open; only the stronger quantum-completely-integrable versions are established.

Sources

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