The Dirichlet eigenvalues of the Laplace–Beltrami operator are larger on an annulus than on any other surface of revolution in with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace–Beltrami operator are larger on a half-helicoid than on any other screw surface in with the same boundary.
Cite this article
Sinan Ariturk, An annulus and a half-helicoid maximize Laplace eigenvalues. J. Spectr. Theory 8 (2018), no. 2, pp. 315–346DOI 10.4171/JST/198