Ends of locally symmetric spaces with maximal bottom spectrum

Lizhen Ji; Peter Li; Jiaping Wang

doi:10.1515/CRELLE.2009.048

Ends of locally symmetric spaces with maximal bottom spectrum

Lizhen Ji, Peter Li, Jiaping Wang

Mathematics

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Abstract

Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ₁(Γ\X) satisfies the inequality λ₁(Γ\X) ≦λ₁(X). We show that if equality λ ₁(⌈\X) = λ₁(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to ℝ¹×N endowed with a multi-warped metric, where N is compact.

Original language	English (US)
Pages (from-to)	1-35
Number of pages	35
Journal	Journal fur die Reine und Angewandte Mathematik
Issue number	632
DOIs	https://doi.org/10.1515/CRELLE.2009.048
State	Published - Jul 2009

Bibliographical note

Funding Information:
The first author was partially supported by NSF grant DMS-0604878. The second author was partially supported by NSF grant DMS-0503735. The third author was partially supported by NSF grant DMS-0706706.

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abstract = "Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≦λ1(X). We show that if equality λ 1(⌈\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to ℝ1×N endowed with a multi-warped metric, where N is compact.",

author = "Lizhen Ji and Peter Li and Jiaping Wang",

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AU - Li, Peter

AU - Wang, Jiaping

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Y1 - 2009/7

N2 - Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≦λ1(X). We show that if equality λ 1(⌈\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to ℝ1×N endowed with a multi-warped metric, where N is compact.

AB - Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≦λ1(X). We show that if equality λ 1(⌈\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to ℝ1×N endowed with a multi-warped metric, where N is compact.

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Ends of locally symmetric spaces with maximal bottom spectrum

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