## On the spectrum of a finite-volume negatively-curved manifold

Venue: | Amer. J. Math |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Lott_onthe,

author = {John Lott},

title = {On the spectrum of a finite-volume negatively-curved manifold},

journal = {Amer. J. Math},

year = {},

pages = {185--205}

}

### OpenURL

### Abstract

Abstract. We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently collapsed, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to −1 and with an infinite number of gaps in the spectrum of the function Laplacian. 1.

### Citations

30 |
Pure point spectrum and negative curvature for noncompact manifold
- Donnelly, Li
- 1979
(Show Context)
Citation Context ...ion Proposition 3. The operator DD ∗ + D ∗ D + B ∗ B + CC ∗ has vanishing essential spectrum. Proof. Without loss of generality, we consider the neighborhood UI of a single end. By standard arguments =-=[3]-=-, it suffices to show that as c → ∞, the infimum of |DJ| 2 + |D ∗ J| 2 + |BJ| 2 + |C ∗ J| 2 , (6.1) where J runs over smooth unit-length elements of H1 with compact support in [c, ∞) × NI, goes to inf... |

24 | Linear Operators, Interscience - Dunford, Schwartz - 1958 |

22 |
Geometry of horospheres
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(Show Context)
Citation Context ...nite-dimensional. This proves the theorem.ON THE SPECTRUM OF A FINITE-VOLUME NEGATIVELY-CURVED MANIFOLD 5 3. Geometry of Finite-Volume Negatively-Curved manifolds We review some results from [5] and =-=[6]-=-. Let (M, g) be a complete connected Riemannian manifold of finite volume whose sectional curvatures satisfy −b 2 ≤ K ≤ −a 2 , with 0 < a ≤ b. Then M is diffeomorphic to the interior of a smooth compa... |

21 |
L-subgroups in spaces of nonpositive curvature
- Eberlein
- 1986
(Show Context)
Citation Context ...order ordinary differential operator, the solution space of (2.21) is finite-dimensional. This proves the theorem. 3. Geometry of Finite-Volume Negatively-Curved manifolds We review some results from =-=[6]-=- and [7]. Let (M, g) be a complete connected Riemannian manifold of finite volume whose sectional curvatures satisfy −b 2 ≤ K ≤ −a 2 , with 0 < a ≤ b. Then M is diffeomorphic to the interior of a smoo... |

13 | Collapsing and differential form Laplacian: the case of a smooth limit space - Lott - 2002 |

11 |
Eigenvalues of the Laplacian on forms
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- 1982
(Show Context)
Citation Context .... From [1], there is an ǫ > 0 such that for all s ∈ [1, ∞), there is a metric h0(s) on N(s), coming from an F-invariant left-invariant inner product on N, with e −ǫ h0(s) ≤ h(s) ≤ e ǫ h0(s). (4.1) By =-=[2]-=-, there is an integer J > 0 such that the j-th eigenvalue λp,j of the p-form Laplacian satisfies e −Jǫ λp,j(h0(s)) ≤ λp,j(h(s)) ≤ e Jǫ λp,j(h0(s)). (4.2) Thus without loss of generality, we may assume... |

7 |
Nilpotent Structures and
- Cheeger, Fukaya, et al.
- 1992
(Show Context)
Citation Context ...ry conditions on ∂UI. From [14, Theorem 2], NI is diffeomorphic to an infranilmanifold. The proof of [14, Theorem 2] uses the collapsing results of Cheeger, Fukaya and Gromov, as given for example in =-=[1]-=-. In particular, it uses Fukaya’s fibration theorem, along with the fact that UI is Gromov-Hausdorff close to a ray which passes through it. Strictly speaking, as in the proof of [14, Theorem 2], one ... |

7 |
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(Show Context)
Citation Context ...all p ∈ Z ∩ [0, n], the essential spectrum of △M p equals the essential spectrum of (A A∗ + A∗ A)p. Theorem 2 was previously known in the case when M is a finite-volume rank-1 locally symmetric space =-=[11]-=-. As an example of Theorem 2, we consider the case of the Laplacian on functions. It is well-known that if M is a noncompact finite-volume [ ) hyperbolic manifold then[ the spectrum (n−1) 2 of its fun... |

6 | L 2 -Cohomology of Geometrically Infinite Hyperbolic 3-Manifolds - Lott |

2 |
The Zero-in-the-Spectrum Question”, L’Enseignement Mathématique 42
- Lott
- 1996
(Show Context)
Citation Context ... while this research was performed. 2. Proof of Theorem 1 The vector space H p (2) (M) is isomorphic to the p-dimensional (reduced) L2 -cohomology of M. For background on L 2 -cohomology, we refer to =-=[8]-=-, [9, Section 2] and references therein. Suppose that the sectional curvatures of M are all bounded in absolute value by b 2 . From [14, Theorem 1], if the number δ is sufficiently small and M satisfi... |

1 |
On the Differential Form Spectrum of Negatively Curved
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(Show Context)
Citation Context ... NSF Grant DMS-9704633. 12 JOHN LOTT Corollary 1 was previously known to be true when p /∈ { n−1 2 and when p = n , n 2 n+1 b , } and 2 a < b and satisfies a certain inequality for which we refer to =-=[4]-=-. n−1 2min(p,n−p) , 2 a The other results in this paper concern manifolds M as in Corollary 1. Recall that the essential spectrum of △M p consists of all numbers in the spectrum of △M p other than tho... |

1 |
On Complete Manifolds of Nonnegative kth-Ricci
- Shen
- 1993
(Show Context)
Citation Context ...a compact manifold-with-boundaryON THE SPECTRUM OF A FINITE-VOLUME NEGATIVELY-CURVED MANIFOLD 3 M. (In fact, for this conclusion it is enough to just have the lower bound on the sectional curvatures =-=[13]-=-.) In particular, if {NI} B I=1 are the connected components of ∂M then there of the connected components of is a compact set K ⊂ M such that the closures {UI} B I=1 M − K are homeomorphic to {[0, ∞) ... |

1 | On Complete Riemannian Manifolds with Collapsed Ends”, Pac - Shen - 1994 |