Scaling inequalities for spherical and hyperbolic eigenvalues

Jeffrey J. Langford, Richard S. Laugesen

Research output: Contribution to journalArticlepeer-review


Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the (length)-2 scaling relation in Euclidean space. For a cap of aperture Θ on the sphere S2, normalizing the k-th eigenvalue by the square of the Euclidean radius of the boundary circle yields that μk(Θ) sin2 ‚ is strictly decreasing, while normalizing by the stereographic radius squared gives that μk(Θ)4 tan2 Θ/2 is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that μ2(Θ)4 sin2 Θ/2 is strictly increasing. Monotonicities of this kind are somewhat surprising, since the Neumann eigenvalues themselves can vary non-monotonically. Cheng and Bandle-type inequalities are deduced by assuming either fixed radius or fixed area and comparing eigenvalues of disks having different curvatures.

Original languageEnglish (US)
Pages (from-to)263-296
Number of pages34
JournalJournal of Spectral Theory
Issue number1
StatePublished - 2023


  • Cheng’s inequality
  • Dirichlet
  • Laplace–Beltrami operator
  • Laplacian
  • Neumann
  • geodesic disk
  • spherical cap
  • vibrating membrane

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Geometry and Topology


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