The spectrum of the Laplacian on forms over flat manifolds
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Mathematische Zeitschrift
The spectrum of the Laplacian on forms over flat manifolds Nelia Charalambous1 · Zhiqin Lu2 Received: 26 October 2017 / Accepted: 10 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this article we prove that the spectrum of the Laplacian on k-forms over a non compact flat manifold is always a connected closed interval of the non negative real line. The proof is based on a detailed decomposition of the structure of flat manifolds. Keywords Essential spectrum · Hodge Laplacian · Flat manifolds Mathematics Subject Classification Primary 58J50; Secondary 53C35
1 Introduction We consider the spectrum of the Hodge Laplacian on differential forms of any order k over a noncompact complete flat manifold M. It is well known that the Laplacian is a densely defined, self-adjoint and nonnegative operator on the space of L 2 integrable k-forms. The spectrum of the Laplacian consists of all points λ ∈ R for which − λI fails to be invertible. The essential spectrum consists of the cluster points in the spectrum and of isolated eigenvalues of infinite multiplicity. We will be denoting the spectrum of the Laplacian on k-forms over M by σ (k, M) and its essential spectrum by σess (k, M). The complement of the essential spectrum in σ (k, M), which consists of isolated eigenvalues of finite multiplicity, is often referred to as the discrete isolated spectrum and is denoted by σpt (k, M). Since is nonnegative, its spectrum is contained in the nonnegative real line. The spectrum, the essential spectrum, and the discrete isolated spectrum are closed subsets of R. When the manifold is compact, the essential spectrum is an empty set and the spectrum consists only of discrete eigenvalues. In the case of a noncompact complete manifold on the other hand, a continuous part in the spectrum might appear. Unlike the discrete spectrum,
The first author was partially supported by a University of Cyprus Start-Up Grant. The second author is partially supported by the DMS-1510232.
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Nelia Charalambous [email protected]; [email protected] Zhiqin Lu [email protected]
1
Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus
2
Department of Mathematics, University of California, Irvine, CA 92697, USA
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N. Charalambous, Z. Lu
which in most cases cannot be explicitly computed, the essential spectrum can be located either by the classical Weyl criterion as in [5], or by a generalization of it as we have shown in previous work [1]. Both criteria require the construction of a large class of test differential forms that act as generalized eigenforms. Our main goal in this article is to compute the spectrum and essential spectrum of a general noncompact complete flat manifold M n . The main result of this paper is the following: Theorem 1.1 Let M be a flat noncompact complete Riemannian manifold. Then σ (k, M) = σess (k, M) = [α, ∞) for some nonnegative constant α. The constant α in the above theorem is the first eigenvalue on -forms ( ≤ k) f
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