Resonances for manifolds hyperbolic near infinity: Optimal lower bounds on order of growth

D. Borthwick; T. J. Christiansen; P. D. Hislop; P. A. Perry

doi:10.1093/imrn/rnq249

Resonances for manifolds hyperbolic near infinity: Optimal lower bounds on order of growth

D. Borthwick, T. J. Christiansen, P. D. Hislop, P. A. Perry

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Suppose that (X, g) is a conformally compact (n+1)-dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to-1 outside of a compact set K⊂X. We prove that the counting function for the resolvent resonances has maximal order of growth (n+1) generically for such manifolds. This is achieved by constructing explicit examples of manifolds hyperbolic at infinity for which the resonance counting function obeys optimal lower bounds.

Original language	English
Pages (from-to)	4431-4470
Number of pages	40
Journal	International Mathematics Research Notices
Volume	2011
Issue number	19
DOIs	https://doi.org/10.1093/imrn/rnq249
State	Published - 2011

Bibliographical note

Funding Information:
The authors are grateful for support from the Mathematical Sciences Research Institute and the

Funding Information:
DB supported in part by NSF grant DMS-0901937; TC supported in part by NSF grants DMS-0500267 and DMS-1001156; PDH supported in part by NSF grant DMS-0803379; PP supported in part by NSF grant DMS-0710477.

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General Mathematics

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10.1093/imrn/rnq249

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title = "Resonances for manifolds hyperbolic near infinity: Optimal lower bounds on order of growth",

abstract = "Suppose that (X, g) is a conformally compact (n+1)-dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to-1 outside of a compact set K⊂X. We prove that the counting function for the resolvent resonances has maximal order of growth (n+1) generically for such manifolds. This is achieved by constructing explicit examples of manifolds hyperbolic at infinity for which the resonance counting function obeys optimal lower bounds.",

author = "D. Borthwick and Christiansen, {T. J.} and Hislop, {P. D.} and Perry, {P. A.}",

note = "Funding Information: The authors are grateful for support from the Mathematical Sciences Research Institute and the Funding Information: DB supported in part by NSF grant DMS-0901937; TC supported in part by NSF grants DMS-0500267 and DMS-1001156; PDH supported in part by NSF grant DMS-0803379; PP supported in part by NSF grant DMS-0710477.",

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doi = "10.1093/imrn/rnq249",

language = "English",

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TY - JOUR

T1 - Resonances for manifolds hyperbolic near infinity

T2 - Optimal lower bounds on order of growth

AU - Borthwick, D.

AU - Christiansen, T. J.

AU - Hislop, P. D.

AU - Perry, P. A.

N1 - Funding Information: The authors are grateful for support from the Mathematical Sciences Research Institute and the Funding Information: DB supported in part by NSF grant DMS-0901937; TC supported in part by NSF grants DMS-0500267 and DMS-1001156; PDH supported in part by NSF grant DMS-0803379; PP supported in part by NSF grant DMS-0710477.

PY - 2011

Y1 - 2011

N2 - Suppose that (X, g) is a conformally compact (n+1)-dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to-1 outside of a compact set K⊂X. We prove that the counting function for the resolvent resonances has maximal order of growth (n+1) generically for such manifolds. This is achieved by constructing explicit examples of manifolds hyperbolic at infinity for which the resonance counting function obeys optimal lower bounds.

AB - Suppose that (X, g) is a conformally compact (n+1)-dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to-1 outside of a compact set K⊂X. We prove that the counting function for the resolvent resonances has maximal order of growth (n+1) generically for such manifolds. This is achieved by constructing explicit examples of manifolds hyperbolic at infinity for which the resonance counting function obeys optimal lower bounds.

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Resonances for manifolds hyperbolic near infinity: Optimal lower bounds on order of growth

Abstract

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