Langford–Laugesen Neumann eigenvalue conjecture

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain equipped with the conformal metric $g=\omega\lvert dz\rvert^2$ , where $\omega\in C^2(\Omega)\cap L^\infty(\Omega)$ is positive in the interior and may tend to zero at the boundary. Suppose that the Gaussian curvature satisfies $K_g\leq K$ for some $K>0$ , and that $KM<4\pi$ , where $M=\int_\Omega\omega\,dA$ . If $0=\mu_1(\Omega;\omega)<\mu_2(\Omega;\omega)\leq\mu_3(\Omega;\omega)\leq\cdots$ are the Neumann eigenvalues and $D_K(M)$ is the geodesic disk of area $M$ in the simply connected surface of constant curvature $K$ , then $\frac{1}{\mu_2(\Omega;\omega)}+\frac{1}{\mu_3(\Omega;\omega)}\geq\frac{2}{\mu_2(D_K(M))}$ . Equality holds if and only if the interiors of $(\Omega,g)$ and $D_K(M)$ are isometric.

References

Primary source

arXiv

Additional references

Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres — arXiv — Hongtao Hu, Meiqi Liu, Qiaoran Wu, Wenming Zou

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to prove the Langford–Laugesen conjecture, with the model disk giving the sharp bound, but independent verification is not recorded.

The conjecture concerns sharp upper bounds for Neumann eigenvalues of simply connected curved surfaces, comparing them with a constant-curvature disk or two equal-area disks. Langford and Laugesen formulated the relevant comparison in their 2023 work.

Known results

First-positive eigenvalue comparison for a simply connected surface: $\mu_2(\Omega,g)\leq\mu_2(\Omega_K^\star)$ , with equality only for the model disk (March 2026).

September 2026 claimed proof

A preprint claims the conjectured two-disk comparison, including $\frac{1}{\lambda_2(\Omega,g)}+\frac{1}{\lambda_3(\Omega,g)}>\frac{2}{\lambda_1(D_K(A/2))}$ and $\lambda_2(\Omega,g)<\lambda_1(D_K(A/2))$ under the stated hypotheses. It also describes sharpness through degeneration to two equal-area disks; the claim is presented as a proof but remains unverified here.

Current status (as of September 2026): The conjecture is claimed proved under the stated regularity, curvature, and area hypotheses, but independent verification of the preprint remains open.

Sources

No solutions have been posted yet.