135 research outputs found
Tracking eigenvalues to the frontier of moduli space I: Convergence and spectral accumulation
We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian
of a family of Riemannian metrics that degenerates on a hypersurface. Our
results generalize earlier work concerning the degeneration of hyperbolic
surfaces
Generic spectral simplicity of polygons
We study the Laplace operator with Dirichlet or Neumann boundary condition on
polygons in the Euclidean plane. We prove that almost every simply connected
polygon with at least four vertices has simple spectrum. We also address the
more general case of geodesic polygons in a constant curvature space form.Comment: length reduced to 6 pages, 1 figur
Hyperbolic triangles without embedded eigenvalues
We consider the Neumann Laplacian acting on square-integrable functions on a
triangle in the hyperbolic plane that has one cusp. We show that the generic
such triangle has no eigenvalues embedded in its continuous spectrum. To prove
this result we study the behavior of the real-analytic eigenvalue branches of a
degenerating family of triangles. In particular, we use a careful analysis of
spectral projections near the crossings of these eigenvalue branches with the
eigenvalue branches of a model operator.Comment: 65 pages, 4 figures, to appear in Annals of Mathematics
http://annals.math.princeton.edu/articles/1159
The maximum number of systoles for genus two Riemann surfaces with abelian differentials
In this article, we provide bounds on systoles associated to a holomorphic
-form on a Riemann surface . In particular, we show that if
has genus two, then, up to homotopy, there are at most systolic loops on
and, moreover, that this bound is realized by a unique translation
surface up to homothety. For general genus and a holomorphic 1-form
with one zero, we provide the optimal upper bound, , on the
number of homotopy classes of systoles. If, in addition, is hyperelliptic,
then we prove that the optimal upper bound is .Comment: 41 page
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