Title: Approximating and Prescribing Orbifold Spectra
Abstract: The question of whether it is possible for a manifold
to be Laplace isospectral to an orbifold having nontrivial singular
set is a central question of spectral geometry.
In this talk, I will explain a method that, for any N>0, produces a
sequence of smooth manifolds and a smooth orbifold having nontrivial
singular set such that the first N eigenvalues of the Laplace spectra
of the manifolds come arbitrarily close to the first N eigenvalues of
the orbifold. Here, the result holds whether the Laplacian acts on
functions or on forms. I will also explain how for any N>0, it is
possible to prescribe the first N eigenvalues of an oriented orbifold
of dimension greater than or equal to 3. These results are achieved
by generalizing results of Colin de Verdière, Jammes, and Rauch and
Taylor to the orbifold setting, and by proving a Hodge decomposition
theorem for orbifolds having manifold boundary. This is joint work
with Carla Farsi and Christopher Seaton.