  Eigenvalue problems are fundamental to mathematics and science. We present a
simple algorithm for determining eigenvalues and eigenfunctions of the
Laplace--Beltrami operator on rather general curved surfaces. Our algorithm,
which is based on the Closest Point Method, relies on an embedding of the
surface in a higher-dimensional space, where standard Cartesian finite
difference and interpolation schemes can be easily applied. We show that there
is a one-to-one correspondence between a problem defined in the embedding space
and the original surface problem. For open surfaces, we present a simple way to
impose Dirichlet and Neumann boundary conditions while maintaining second-order
accuracy. Convergence studies and a series of examples demonstrate the
effectiveness and generality of our approach.
