Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow

Paul Bryan; Mohammad Ivaki; Julian Scheuer

doi:10.1017/9781108884136.005

Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow

Paul Bryan, Mohammad Ivaki, Julian Scheuer

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

Abstract

This short chapter is a mostly expository note examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the cross curvature flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston’s geometrisation. The relationship between integrability and embeddability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a question posed by Chow and Hamilton when they introduced the XCF.

Original language	English
Title of host publication	Differential geometry in the large
Editors	Owen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner, Diarmuid Crowley
Place of Publication	Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India
Publisher	Cambridge University Press (CUP)
Chapter	3
Pages	75-97
Number of pages	23
ISBN (Electronic)	9781108884136
ISBN (Print)	9781108812818
DOIs	https://doi.org/10.1017/9781108884136.005
Publication status	Published - 2021
Event	Australian–GermanWorkshop on Differential Geometry in the Large (2019) - Creswick, Australia Duration: 4 Feb 2019 → 15 Feb 2019

Publication series

Name	London Mathematical Society Lecture Note Series
Publisher	Cambridge University Press
Volume	463

Conference

Conference	Australian–GermanWorkshop on Differential Geometry in the Large (2019)
Country/Territory	Australia
City	Creswick
Period	4/02/19 → 15/02/19

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10.1017/9781108884136.005

Cite this

Bryan, P., Ivaki, M., & Scheuer, J. (2021). Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow. In O. Dearricott, W. Tuschmann, Y. Nikolayevsky, T. Leistner, & D. Crowley (Eds.), Differential geometry in the large (pp. 75-97). (London Mathematical Society Lecture Note Series; Vol. 463). Cambridge University Press (CUP). https://doi.org/10.1017/9781108884136.005

Bryan, Paul ; Ivaki, Mohammad ; Scheuer, Julian. / Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow. Differential geometry in the large. editor / Owen Dearricott ; Wilderich Tuschmann ; Yuri Nikolayevsky ; Thomas Leistner ; Diarmuid Crowley. Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India : Cambridge University Press (CUP), 2021. pp. 75-97 (London Mathematical Society Lecture Note Series).

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title = "Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow",

abstract = "This short chapter is a mostly expository note examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the cross curvature flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston{\textquoteright}s geometrisation. The relationship between integrability and embeddability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a question posed by Chow and Hamilton when they introduced the XCF.",

author = "Paul Bryan and Mohammad Ivaki and Julian Scheuer",

year = "2021",

doi = "10.1017/9781108884136.005",

language = "English",

isbn = "9781108812818",

series = "London Mathematical Society Lecture Note Series",

publisher = "Cambridge University Press (CUP)",

pages = "75--97",

editor = "Owen Dearricott and Wilderich Tuschmann and Yuri Nikolayevsky and Thomas Leistner and Diarmuid Crowley",

booktitle = "Differential geometry in the large",

address = "United Kingdom",

note = "Australian–GermanWorkshop on Differential Geometry in the Large (2019) ; Conference date: 04-02-2019 Through 15-02-2019",

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Bryan, P, Ivaki, M & Scheuer, J 2021, Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow. in O Dearricott, W Tuschmann, Y Nikolayevsky, T Leistner & D Crowley (eds), Differential geometry in the large. London Mathematical Society Lecture Note Series, vol. 463, Cambridge University Press (CUP), Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India, pp. 75-97, Australian–GermanWorkshop on Differential Geometry in the Large (2019), Creswick, Victoria, Australia, 4/02/19. https://doi.org/10.1017/9781108884136.005

Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow. / Bryan, Paul; Ivaki, Mohammad; Scheuer, Julian.
Differential geometry in the large. ed. / Owen Dearricott; Wilderich Tuschmann; Yuri Nikolayevsky; Thomas Leistner; Diarmuid Crowley. Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India: Cambridge University Press (CUP), 2021. p. 75-97 (London Mathematical Society Lecture Note Series; Vol. 463).

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

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AB - This short chapter is a mostly expository note examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the cross curvature flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston’s geometrisation. The relationship between integrability and embeddability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a question posed by Chow and Hamilton when they introduced the XCF.

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DO - 10.1017/9781108884136.005

M3 - Conference proceeding contribution

SN - 9781108812818

T3 - London Mathematical Society Lecture Note Series

SP - 75

EP - 97

BT - Differential geometry in the large

A2 - Dearricott, Owen

A2 - Tuschmann, Wilderich

A2 - Nikolayevsky, Yuri

A2 - Leistner, Thomas

A2 - Crowley, Diarmuid

PB - Cambridge University Press (CUP)

CY - Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India

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Bryan P, Ivaki M, Scheuer J. Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow. In Dearricott O, Tuschmann W, Nikolayevsky Y, Leistner T, Crowley D, editors, Differential geometry in the large. Cambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India: Cambridge University Press (CUP). 2021. p. 75-97. (London Mathematical Society Lecture Note Series). doi: 10.1017/9781108884136.005

Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow

Abstract

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Analysis of fully non-linear geometric problems and differential equations

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Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in minkowski space and the cross curvature flow

Abstract

Publication series

Conference

Access to Document

Other files and links

Fingerprint

Projects

Analysis of fully non-linear geometric problems and differential equations

Cite this