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arXiv · 1810.05242

Small eigenvalues and thick-thin decomposition in negative curvature

Ursula Hamenstaedt
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Abstract

Let $M$ be a finite volume oriented Riemannian manifold of dimension $n\geq 3$ and curvature in $[-b^2,-1]$, with thick-thin decomposition $M=M(thick)\cup M(thin)$. Denote by $λ_k(M(thick))$ the k-th eigenvalue for the Laplacian on $M(thick)$, with Neumann boundary conditdions. We show that $λ_k(M(thick))/3\leq λ_k(M)$ for all k for which $λ_k(M)<(n-2)^2/12$. If $M$ is hyperbolic and of dimension 3 then $λ_k(M)< C \log(vol(M(thin))+2)λ_k(M(thick))$ for a fixed number $C>0$ provided that $λ_k(M(thick))<1/96$.

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Ursula Hamenstaedt. 2020-01-09. Small eigenvalues and thick-thin decomposition in negative curvature. https://arxiv.org/abs/1810.05242

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