# zbMATH — the first resource for mathematics

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

##### MSC:
 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
Full Text:
##### References:
 [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 · doi:10.1007/BFb0083043 [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 · doi:10.1090/S0273-0979-1985-15405-9 [3] A. Borel and J. P. Serre, Corners and arithmetic groups , Comment. Math. Helv. 48 (1973), 436-491. · Zbl 0274.22011 · doi:10.1007/BF02566134 · eudml:139559 [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 · doi:10.2307/2042193 · links.jstor.org [8] H. Donnelly, The differential form spectrum of hyperbolic space , Manuscripta Math. 33 (1981), no. 3-4, 365-385. · Zbl 0464.58020 · doi:10.1007/BF01798234 · eudml:154755 [9] H. Donnelly, On the essential spectrum of a complete Riemannian manifold , Topology 20 (1981), no. 1, 1-14. · Zbl 0463.53027 · doi:10.1016/0040-9383(81)90012-4 [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 · doi:10.2307/2374434 · links.jstor.org [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 · doi:10.2307/1969703 · links.jstor.org [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 · doi:10.1073/pnas.37.1.48 · links.jstor.org [14] B. Maskit, On Poincaré’s theorem for fundamental polygons , Advances in Math. 7 (1971), 219-230. · Zbl 0223.30008 · doi:10.1016/S0001-8708(71)80003-8 [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 · doi:10.2307/2374820 [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 · doi:10.1007/BF01390727 · eudml:142974
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.