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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The spectrum of the Hodge Laplacian for a degenerating family of hyperbolic three manifolds


Authors: Jozef Dodziuk and Jeffrey McGowan
Journal: Trans. Amer. Math. Soc. 347 (1995), 1981-1995
MSC: Primary 58G25; Secondary 35P15
DOI: https://doi.org/10.1090/S0002-9947-1995-1308007-X
MathSciNet review: 1308007
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Abstract: We consider a sequence $ ({M_n})_{n = 1}^\infty $ of compact hyperbolic manifolds converging to a complete hyperbolic manifold $ {M_0}$ with cusps. The Laplace operator acting on the space of $ {L^2}$ differential forms on $ {M_0}$ has continuous spectrum filling the half-line $ [0,\infty )$. One expects therefore that the spectra of this operator on $ {M_n}$ accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accumulation.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1308007-X
Article copyright: © Copyright 1995 American Mathematical Society