  Elliptic partial differential equations on surfaces play an essential role in
geometry, relativity theory, phase transitions, materials science, image
processing, and other applications. They are typically governed by the
Laplace-Beltrami operator. We present and analyze approximations by Surface
Finite Element Methods (SFEM) of the Laplace-Beltrami eigenvalue problem. As
for SFEM for source problems, spectral approximation is challenged by two
sources of errors: the geometric consistency error due to the approximation of
the surface and the Galerkin error corresponding to finite element resolution
of eigenfunctions. We show that these two error sources interact for
eigenfunction approximations as for the source problem. The situation is
different for eigenvalues, where a novel situation occurs for the geometric
consistency error: The degree of the geometric error depends on the choice of
interpolation points used to construct the approximate surface. Thus the
geometric consistency term can sometimes be made to converge faster than in the
eigenfunction case through a judicious choice of interpolation points.
