For convex domains in the sphere and hyperbolic plane, the second Neumann eigenfunction has no interior critical points when μ2 D² ≤ j_{1,1}², and planar convex domains satisfy C(Ω) ≤ 2.4828.
Baginski, Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap, Quart
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Critical points of second Neumann eigenfunctions on some convex domains in two-dimensional space forms
For convex domains in the sphere and hyperbolic plane, the second Neumann eigenfunction has no interior critical points when μ2 D² ≤ j_{1,1}², and planar convex domains satisfy C(Ω) ≤ 2.4828.