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 proceedingConference proceeding contributionpeer-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 languageEnglish
    Title of host publicationDifferential geometry in the large
    EditorsOwen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner, Diarmuid Crowley
    Place of PublicationCambridge, UK ; New York, US ; Victoria, AU ; New Delhi, India
    PublisherCambridge University Press
    Chapter3
    Pages75-97
    Number of pages23
    ISBN (Electronic)9781108884136
    ISBN (Print)9781108812818
    DOIs
    Publication statusPublished - 2021
    EventAustralian–GermanWorkshop on Differential Geometry in the Large (2019) - Creswick, Australia
    Duration: 4 Feb 201915 Feb 2019

    Publication series

    NameLondon Mathematical Society Lecture Note Series
    PublisherCambridge University Press
    Volume463

    Conference

    ConferenceAustralian–GermanWorkshop on Differential Geometry in the Large (2019)
    CountryAustralia
    CityCreswick
    Period4/02/1915/02/19

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