38 research outputs found
On the multiplicity of eigenvalues of conformally covariant operators
Let be a compact Riemannian manifold and an elliptic, formally
self-adjoint, conformally covariant operator of order acting on smooth
sections of a bundle over . We prove that if has no rigid eigenspaces
(see Definition 2.2), the set of functions for which
has only simple non-zero eigenvalues is a residual set in
. As a consequence we prove that if has no rigid
eigenspaces for a dense set of metrics, then all non-zero eigenvalues are
simple for a residual set of metrics in the -topology. We also prove that
the eigenvalues of depend continuously on in the -topology,
provided is strongly elliptic. As an application of our work, we show
that if acts on (e.g. GJMS operators), its non-zero
eigenvalues are generically simple.Comment: To appear in Annales de l'Institut Fourie
Improvements for eigenfunction averages: An application of geodesic beams
Let be a smooth, compact Riemannian manifold and
an -normalized sequence of Laplace eigenfunctions, . Given a smooth submanifold of
codimension , we find conditions on the pair , even when
, for which as . These
conditions require no global assumption on the manifold and instead relate
to the structure of the set of recurrent directions in the unit normal bundle
to . Our results extend all previously known conditions guaranteeing
improvements on averages, including those on sup-norms. For example, we show
that if is a surface with Anosov geodesic flow, then there are
logarithmically improved averages for any . We also find weaker
conditions than having no conjugate points which guarantee improvements for the norm of eigenfunctions. Our results
are obtained using geodesic beam techniques, which yield a mechanism for
obtaining general quantitative improvements for averages and sup-norms.Comment: 70 pages, 4 figures. The new version includes a major revision of
Appendix A, parts of which have been replaced by section
Eigenfunction concentration via geodesic beams
In this article we develop new techniques for studying concentration of
Laplace eigenfunctions as their frequency, , grows. The
method consists of controlling by decomposing
into a superposition of geodesic beams that run through the point . Each
beam is localized in phase-space on a tube centered around a geodesic whose
radius shrinks slightly slower than . We control
by the -mass of on each geodesic tube and
derive a purely dynamical statement through which can be
studied. In particular, we obtain estimates on by decomposing
the set of geodesic tubes into those that are non self-looping for time and
those that are. This approach allows for quantitative improvements, in terms of
, on the available bounds for norms, norms, pointwise Weyl
laws, and averages over submanifolds.Comment: 61 pages, 2 figures. Improved exposition and includes new explanatory
material in the introduction as well as an examples section (1.5) and a full
section on comparison with previous work (1.6). Appendices A.1 (Index of
notation) and B were also adde
Scalar curvature and -curvature of random metrics
We study Gauss curvature for random Riemannian metrics on a compact surface,
lying in a fixed conformal class; our questions are motivated by comparison
geometry. Next, analogous questions are considered for the scalar curvature in
dimension , and for the -curvature of random Riemannian metrics.Comment: The proof of Proposition 3.10 has been correcte
High Frequency Eigenfunction Immersions and Supremum Norms of Random Waves
A compact Riemannian manifold may be immersed into Euclidean space by using
high frequency Laplace eigenfunctions. We study the geometry of the manifold
viewed as a metric space endowed with the distance function from the ambient
Euclidean space. As an application we give a new proof of a result of
Burq-Lebeau and others on upper bounds for the sup-norms of random linear
combinations of high frequency eigenfunctions.Comment: This article supersedes arXiv:1310.1361, which has now been withdraw
Averages of eigenfunctions over hypersurfaces
Let be a compact, smooth, Riemannian manifold and an
-normalized sequence of Laplace eigenfunctions with defect measure .
Let be a smooth hypersurface. Our main result says that when is
concentrated conormally to , the eigenfunction restrictions
to and the restrictions of their normal derivatives to have integrals
converging to 0 as .Comment: 18 pages, 1 figur