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Compactness and finiteness theorems for isospectral manifolds. (English) Zbl 0737.53038

In this paper, we prove finiteness and compactness results for families of isospectral manifolds under some auxiliary curvature assumptions. Specifically, we prove the following finiteness result: Suppose that \(\{M_ i\}\) is a family of isospectral manifolds of dimension \(n\) and that either (i) all of the \(M_ i\)’s have negative sectional curvature or (ii) all of the \(M_ i\)’s have sectional curvatures bounded below. Then the \(M_ i\)’s contain at most finitely many homeomorphism types, and if \(n\neq 4\), at most finitely many diffeomorphism types. We also prove the following compactness result in \(n=3\): Suppose that \(\{M_ i\}\) is a sequence of isospectral 3-manifolds, and that either (i) The \(M_ i\)’s all have negative sectional curvatures or (ii) there is a uniform lower bound for the Ricci curvatures of the \(M_ i\). Then there are finitely many model manifolds \(M_ 1',\cdots,M_ k'\) and families of metrics \({\mathcal M}_ 1,\cdots,{\mathcal M}_ n\), with \({\mathcal M}_ i\) compact in \(C^ \infty\) sense on \(M_ i'\), such that each of the manifolds \(M_ i\) is diffeomorphic to one of the \(M_ j'\) and isometric to an element of \({\mathcal M}_ j\). Our proof uses spectral invariants together with geometric finiteness and compactness theorems.

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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