  The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an
annulus than on any other surface of revolution in $\mathbb{R}^3$ 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 $\mathbb{R}^2 \times
\mathbb{S}^1$ with the same boundary.
