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


[1] M. T. Anderson, \(L^2\) Harmonic Forms on Complete Riemannian Manifolds , Geometry and Analysis on Manifolds (Katata/Kyoto, 1987) ed. T. Sunada, Lecture Notes in Math., vol. 1339, Springer-Verlag, Berlin, 1988, pp. 1-19. · Zbl 0652.53030
[2] Michael T. Anderson, \(L^ 2\) harmonic forms and a conjecture of Dodziuk-Singer , Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 163-165. · Zbl 0573.53025
[3] A. Borel and J. P. Serre, Corners and arithmetic groups , Comment. Math. Helv. 48 (1973), 436-491. · Zbl 0274.22011
[4] R. Bott and L. Tu, Differential Forms in Algebraic Topology , Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York, 1982. · Zbl 0496.55001
[5] G. de Rham, Differentiable Manifolds , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 266, Springer-Verlag, Berlin, 1984. · Zbl 0534.58003
[6] J. Dodziuk, \(L^2\) Harmonic forms on complete manifolds , Ann. of Math. Studies 102 (1982), 292-302. · Zbl 0484.53033
[7] J. Dodziuk, \(L^2\) harmonic forms on rotationally symmetric Riemannian manifolds , Proc. Amer. Math. Soc. 77 (1979), no. 3, 395-400. JSTOR: · Zbl 0423.58002
[8] H. Donnelly, The differential form spectrum of hyperbolic space , Manuscripta Math. 33 (1981), no. 3-4, 365-385. · Zbl 0464.58020
[9] H. Donnelly, On the essential spectrum of a complete Riemannian manifold , Topology 20 (1981), no. 1, 1-14. · Zbl 0463.53027
[10] H. Donnelly and F. Xavier, On the differential form spectrum of negatively curved Riemannian manifolds , Amer. J. Math. 106 (1984), no. 1, 169-185. JSTOR: · Zbl 0547.58034
[11] C. Epstein, The spectral theory of geometrically periodic hyperbolic \(3\)-manifolds , Mem. Amer. Math. Soc. 58 (1985), no. 335, ix+161. · Zbl 0584.58047
[12] M. Gaffney, A special Stokes’s Theorem for complete Riemannian manifolds , Ann. of Math. (2) 60 (1954), 140-145. JSTOR: · Zbl 0055.40301
[13] M. Gaffney, The harmonic operator for exterior differential forms , Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 48-50. JSTOR: · Zbl 0042.10205
[14] B. Maskit, On Poincaré’s theorem for fundamental polygons , Advances in Math. 7 (1971), 219-230. · Zbl 0223.30008
[15] R. Mazzeo, Hodge cohomology of negatively curved manifolds , Ph.D. thesis, MIT, 1986.
[16] R. Mazzeo, The Hodge cohomology of a conformally compact metric , J. Differential Geom. 28 (1988), no. 2, 309-339. · Zbl 0656.53042
[17] R. Mazzeo, Unique Continuation of Infinity and Embedded Eigenvalues for Asymptotically Hyperbolic Manifolds , · Zbl 0725.58044
[18] R. Melrose, Analysis on Manifolds with Corners , Lecture Notes, MIT, 1988.
[19] W. Thurston, The Geometry and Topology of \(3\)-Manifolds , Princeton University. · Zbl 0483.57007
[20] S. Zucker, \(L_2\) cohomology of warped products and arithmetic groups , Invent. Math. 70 (1982/83), no. 2, 169-218. · Zbl 0508.20020
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