Approximating orbifold spectra using collapsing connected sums
Abstract
For a closed Riemannian orbifold $O$, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain $U$ in $O$ whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of $O$ can be perturbed to ensure that the first $N$ eigenvalues of $U$ and $O$ are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the non-collapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2016
- DOI:
- 10.48550/arXiv.1611.07676
- arXiv:
- arXiv:1611.07676
- Bibcode:
- 2016arXiv161107676F
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Spectral Theory;
- Primary 58J53;
- 57R18;
- Secondary 53C20;
- 58A14
- E-Print:
- v2: 31 pages. Substantial rewriting and generalization of version 1, including extension of results to differential forms and inclusion of the Hodge decomposition. v3: 32 pages. Edited to improve exposition, added Section 8.4