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Collapsing and small eigenvalues of differential forms. (Effondrements et petites valeurs propres des formes différentielles.) (French) Zbl 1106.58024
Proceedings of the seminar on spectral theory and geometry. 2004–2005. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 23, 115-124 (2005).
Let $$(M^ n,g)$$ be a compact connected oriented Riemannian manifold of dimension $$n$$ and let $$\Delta$$ be the Laplacian acting on the space of differential $$p$$-forms on $$M^ n$$. The spectrum of $$\Delta$$ forms a discrete set of nonnegative real numbers $0=\lambda_{p,0}(M,g)<\lambda_{p,1}(M,g)\leq \lambda_{p,2}(M,g)\leq\ldots,$ where we repeat the nonzero eigenvalues with multiplicity, respectively. If the manifold admits a sequence of metrics $$(g_ i)$$ with bounded diameter and curvature such that $$\lambda_{p,1}(M, g_ i)$$ tends to $$0$$ for some $$1\leq p\leq n$$, then its volume tends to $$0$$ and we say that $$M$$ collapses.
In the paper under review the author presents the recent results concerning the asymptotic behavior of $$\lambda_{p,1}$$ when the manifold $$M^ n$$ collapses.
For the entire collection see [Zbl 1089.35003].

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58C40 Spectral theory; eigenvalue problems on manifolds
##### Keywords:
Laplacian; small eigenvalue; collapse; limit cohomology; spectrum
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