Spherical lunes with angle pi/p have explicitly determined spectra and satisfy Polya's eigenvalue conjecture eventually when p>1.
Bang, Eigenvalues of the Laplacian on a geodesic ball in the n-sphere, Chinese J
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Spherical n-lunes: billiards and eigenvalues
Spherical lunes with angle pi/p have explicitly determined spectra and satisfy Polya's eigenvalue conjecture eventually when p>1.