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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
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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Article copyright: © Copyright 1995 American Mathematical Society