Orbifold Spectral Theory

Hörmander's theorem on the asymptotics of the spectral function of an elliptic operator is extended to the setting of compact Riemannian orbifolds. In contrast to the manifold case, the asymptotics depend on the isotropy type of the point at which the spectral function is computed. It is shown that "on average" the eigenfunctions of the operator are larger at singular points than at manifold points, by a factor of the order of the isotropy type. A sketch of a more direct approach to the wave trace formula on orbifolds is also given, obtaining results already shown separately by M. Sandoval and Y. Kordyukov in the setting of Riemmannian foliations.

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“…As a corollary we obtain the standard Weyl asymptotics first obtained for orbifolds by Farsi in [8] :…”

Section: Clean Intersection For T =mentioning

confidence: 71%

The spectral function of a Riemannian orbifold

Stanhope

Uribe

2010

Ann Glob Anal Geom

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“…The following regularity result is a central one in the subject -we shall derive it from results of Harvey and Polking [9] but there are many other proofs see, for example, the discussion in [1,2] for p = 0. It identifies the eigenforms on M with the smooth equivariant eigenforms on the desingularization.…”

Section: -Manifolds and V -Submersionsmentioning

confidence: 97%

“…Again, this definition coincides with the usual definition in the smooth setting. We summarize the discussion given above in the following Lemma and refer to [2,11,18] for additional related materials.…”

Section: -Manifolds and V -Submersionsmentioning

confidence: 99%

Eigenforms of the Laplacian for Riemannian V-submersions

Gilkey

Kim

Park

2005

Tohoku Math. J. (2)

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“…A Riemannian orbisurface is an orbisurface which is endowed with a Riemannian metric. The Gauss-Bonnet theorem gives the usual relationship between topology and geometry for these objects: Since the volume of a Riemannian orbisurface is spectrally determined via Weyl's asymptotic formula (see [5]), we see that for a Riemannian orbisurface with given curvature, the spectrum determines the orbifold Euler characteristic. However, since the orbifold Euler characteristic involves both the genus of the underlying surface and the orders of the cone points in the orbisurface, it is not immediately clear that the spectrum determines the genus.…”

Section: Consequencesmentioning

confidence: 99%