A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

Content Provider	Semantic Scholar
Author	Muller, Jorn
Copyright Year	2010
Abstract	A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of Q−rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology H(X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge-Laplace-Operator on Ω(X). Introduction and Statement of Results The classical theorems of geometric topology, such as the Hodge theorem, the signature theorem and the index theorem, reveal a profound relationship between analysis of classical linear operators over a smooth compact manifold and the topology of the manifold. Since their proofs in the middle of the last century, there has been much interest in extending these theorems to more general settings, where the manifolds may either be noncompact or may have singularities. The analytic approaches have often involved creation of pseudodifferential operator calculi suited to certain geometric settings, such as the b-calculus of Melrose. Another approach has sprung from techniques in analytic number theory, and the analysis of the Laplacian over noncompact locally symmetric spaces, as for example in the work of G. Harder. Recall that the classical Hodge theorem states that the natural map from harmonic forms over a compact smooth manifold to de Rham cohomology classes is an isomorphism. In the situation when X is not compact or is not smooth, there is no such general statement. One possible extension is to consider square integrable harmonic forms H(2)(X) and identify them with a topologically defined space. For several geometric situations, including manifolds with cylindrical ends ([APS]), conical singularities ([Ch]) and locally symmetric spaces (e.g. [Z], [Sa-St]), such theorems of “Hodge type” have been found. Manifolds with fibered cusp metrics can be considered as a geometrical generalization of Q-rank one locally symmetric spaces at “infinity” as well as of manifolds with cusps or cylindrical ends. In [HHM] methods from the φ−calculus developed by Melrose [Me], Mazzeo, Vaillant [Vai] and others have been used to find an identification of H p (2)(X) with a subspace of the intersection cohomology. We want to take another approach and identify the de Rham cohomology of a manifold with fibered cusps with a space of harmonic forms. Generally these forms will not be square integrable. In the early paper [Har] of G. Harder such a theorem for locally symmetric spaces of Q-rank one was proved. In this situation the representatives of the de Rham cohomology classes are either L2−harmonic forms, or they are defined by special harmonic values of Eisenstein series. In [Mu4] W. Müller suggested to use analytical arguments to find a similar theorem in the context of manifolds with cusps, by replacing Eisenstein series with generalized eigenforms of the Laplacian. His method relies on an explicit parametrix construction for the resolvent which allows an investigation of the 1
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Alternate Webpage(s)	https://arxiv.org/pdf/1005.4606v1.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article