Hodge theory on hyperbolic manifolds. (English) Zbl 0712.58006

Hodge-de Rham theory is a bridge between topology of and analysis on closed Riemannian manifolds. Generally, it fails for nonclosed manifolds. The paper discusses what survives of Hodge theory for a not too difficult class of manifolds, namely geometrically finite complete manifolds M of constant negative curvature. Such an M is the quotient \({\mathbb{H}}^ n/\Gamma\) of the standard hyperbolic space \({\mathbb{H}}^ n\) by a discrete group \(\Gamma\) of isometries, and the fundamental domain has only finitely many sides. The space of \(L_ 2\)-harmonic k-forms is proved to be naturally isomorphic to a modified de Rham-cohomology space \(H^ k:\) only differential forms which vanish near some portion (depending on k!) of the boundary \(\partial M\) enter the definition of \(H^ k\). Here \(\partial M\) originates from fixed points of \(\Gamma\) or from compactification of M. More results concern the asymptotics of \(L_ 2\)- harmonic forms and the essential spectrum of the Laplace operator.
Let us complete the literature quotations by J. Eichhorn, Elliptic differential operators on noncompact manifolds. Teubner-Verlag, Leipzig (1988; Zbl 0683.58045)].
Reviewer: R.Schimming


58A14 Hodge theory in global analysis
58A12 de Rham theory in global analysis
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58A10 Differential forms in global analysis
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds


Zbl 0683.58045
Full Text: DOI


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