[Paper Review] Small eigenvalues and thick-thin decomposition in negative curvature

Ursula Hamenstaedt|arXiv (Cornell University)|Oct 11, 2018

Geometric Analysis and Curvature Flows13 references4 citations

TL;DR

This paper establishes sharp spectral comparison estimates between the Laplacian eigenvalues of finite-volume, pinched negatively curved manifolds and those of their thick-thin decompositions. It proves that for $ k $ such that $ \lambda_k(M) < (n-2)^2/12 $, the $ k $-th eigenvalue of the full manifold $ M $ is at least one-third that of its thick part $ M_{\rm thick} $ with Neumann boundary conditions. For 3-dimensional hyperbolic manifolds, it further shows $ \lambda_k(M) \leq C \log({\rm vol}(M_{\rm thin})+2) \lambda_k(M_{\rm thick}) $ when $ \lambda_k(M_{\rm thick}) < 1/96 $, providing a quantitative link between small eigenvalues and the geometry of the thin part.

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 $λ_k(M)0$ provided that $λ_k(M(thick))&lt;1/96$.

Motivation & Objective

• To understand the small spectrum of the Laplacian on finite-volume, pinched negatively curved manifolds of dimension $ n \geq 3 $, particularly in relation to their thick-thin decomposition.
• To bridge the gap in understanding the small eigenvalues of higher-dimensional negatively curved manifolds, which are less understood than in the 2-dimensional hyperbolic case.
• To establish quantitative comparisons between the eigenvalues of the full manifold $ M $ and its thick part $ M_{\rm thick} $, especially for small eigenvalues.
• To extend known spectral results from hyperbolic surfaces to higher-dimensional manifolds with pinched negative curvature.
• To analyze how the geometry of the thin part, particularly its volume, influences the spectrum of the full manifold.

Proposed method

• Use the thick-thin decomposition of $ M $, where $ M_{\rm thick} $ consists of points with injectivity radius at least $ \varepsilon $, and $ M_{\rm thin} $ is the union of Margulis tubes and cusps.
• Apply Neumann boundary conditions on $ M_{\rm thick} $, ensuring the eigenvalue problem is well-posed and compatible with the manifold's geometry.
• Construct a function $ F $ on $ M $ by extending an eigenfunction $ f $ from $ M_{\rm thick} $ to the thin part using a radial extension with controlled $ L^2 $-norm and energy.
• Use bilipschitz control of radial projections between boundary hypersurfaces to bound the Rayleigh quotient of the extended function $ F $.
• Leverage the fact that $ M_{\rm thick} $ is uniformly quasi-isometric to a finite graph of bounded valence, allowing spectral comparison via graph Laplacian eigenvalues.
• Employ the Schwarz inequality and energy estimates to compare the $ L^2 $-norm and Dirichlet energy of the original and extended functions, ensuring the Rayleigh quotient of $ F $ is bounded in terms of $ \lambda_k(M_{\rm thick}) $.

Experimental results

Research questions

• RQ1 How do the small eigenvalues of the Laplacian on a finite-volume, pinched negatively curved manifold relate to those on its thick part with Neumann boundary conditions?
• RQ2 Can a universal lower bound be established for $ \lambda_k(M) $ in terms of $ \lambda_k(M_{\rm thick}) $ when $ \lambda_k(M) $ is small?
• RQ3 What is the dependence of $ \lambda_k(M) $ on the volume of the thin part $ M_{\rm thin} $ in 3-dimensional hyperbolic manifolds?
• RQ4 Does the spectral gap of $ M_{\rm thick} $ control the small spectrum of the full manifold $ M $, and if so, with what uniform constants?
• RQ5 How does the geometry of the thin part—specifically its volume and structure—constrain the small eigenvalues of the full manifold?

Key findings

• For any finite-volume, oriented Riemannian $ n $-manifold with $ n \geq 3 $ and curvature in $ [-b^2, -1] $, it holds that $ \lambda_k(M) \geq \frac{1}{3} \lambda_k(M_{\rm thick}) $ whenever $ \lambda_k(M) < \frac{(n-2)^2}{12} $.
• In the case of 3-dimensional hyperbolic manifolds, $ \lambda_k(M) \leq C \log({\rm vol}(M_{\rm thin}) + 2) \cdot \lambda_k(M_{\rm thick}) $ for a fixed $ C > 0 $, provided $ \lambda_k(M_{\rm thick}) < \frac{1}{96} $.
• The constant $ \frac{(n-2)^2}{12} $ is sharp in the sense that it cannot be improved without additional assumptions, as it arises from curvature and dimension constraints.
• The thick part $ M_{\rm thick} $ is uniformly quasi-isometric to a finite graph of bounded valence, enabling spectral comparison via graph Laplacian eigenvalues.
• The construction of the extended function $ F $ ensures that its $ L^2 $-norm and Dirichlet energy are controlled by those of the original eigenfunction on $ M_{\rm thick} $, with uniform constants depending only on $ n $ and $ b $.
• The proof relies on bilipschitz control of radial projections and energy estimates via the Schwarz inequality, ensuring the Rayleigh quotient of $ F $ is bounded in terms of $ \lambda_k(M_{\rm thick}) $.