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.