Neumann eigenvalue comparison for spherical domains of at most hemispherical volume

Determine whether every Lipschitz subdomain of the unit sphere S^n with volume at most one hemisphere has first positive Neumann eigenvalue no greater than that of a geodesic ball of the same volume.

Background

Langford–Laugesen’s higher-dimensional conjecture asserts that a geodesic ball of the same volume maximizes the first positive Neumann eigenvalue among Lipschitz subdomains of the unit sphere. The paper constructs smooth simply connected counterexamples in every dimension n ≥ 3, but these counterexamples have volume greater than half the volume of the sphere.

The authors explicitly note that their construction does not resolve the restricted comparison problem under the volume constraint of at most one hemisphere. The unresolved question is therefore whether geodesic balls remain optimal for the first positive Neumann eigenvalue in this smaller-volume regime.

References

This construction does not decide the weaker problem under the volume constraint of at most one hemisphere. Whether that weaker version holds remains open.

— Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres  (2609.35360 - Hu et al., 28 Sep 2026) in Section 1, discussion following Theorem 1.3, p. 4

However, it isn't clear to us whether the results of imply our Lemma \ref{lem: no sets of pos capacity far from BAHP}, due to the technical conditions on the domains of integration in Theorems B,C.

— Characterizing higher-dimensional Hölder domains via spherical geometric functions  (2610.13866 - Casey et al., 9 Oct 2026) in Section 1, subsection “Strategy of the proof,” discussion preceding Lemma 3.1