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

Bryan, Paul, Ivaki, Mohammad N. and Scheuer, Julian ORCID: https://orcid.org/0000-0003-2664-1896 2020. Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow. Dearricott, Owen, Tuschmann, Wilderich, Nikolayevsky, Yuri, Leistner, Thomas and Crowley, Diarmuid, eds. Differential Geometry in the Large, London Mathematical Society Lecture Notes Series, Cambridge University Press, pp. 75-97. (10.1017/9781108884136.005)

Full text not available from this repository.

Abstract

This short note is a mostly expository article 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 embedability 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.

Item Type:	Book Section
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Publisher:	Cambridge University Press
ISBN:	9781108884136
Funders:	Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
Date of First Compliant Deposit:	9 October 2020
Date of Acceptance:	17 January 2020
Last Modified:	19 Jun 2025 14:30
URI:	https://orca.cardiff.ac.uk/id/eprint/135490

Actions (repository staff only)

Edit Item