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REALIZING METRICS OF CURVATURE $\mathbf {\leq -1}$ ON CLOSED SURFACES IN FUCHSIAN ANTI-DE SITTER MANIFOLDS

Part of: Global differential geometry

Published online by Cambridge University Press:  19 February 2021

HICHAM LABENI Open the ORCID record for HICHAM LABENI [Opens in a new window]
HICHAM LABENI*
Affiliation:
CY Cergy Paris Université, Laboratoire AGM, UMR 8088 du CNRS, F-95000Cergy, France
*
e-mail: hicham.labeni@cyu.fr
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Abstract

We prove that any metric with curvature less than or equal to $-1$ (in the sense of A. D. Alexandrov) on a closed surface of genus greater than $1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$ . The proof is by approximation, using a result about isometric immersion of smooth metrics by Labourie and Schlenker.

Keywords

anti-de SitterCAT(-1)Fuchsian surfaces

MSC classification

Primary: 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 312 - 334
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Graeme Wilkin

The author was supported by: École Doctorale Économie, Management, Mathématiques, Physique et Sciences Informatiques (EM2PSI)ED no 405 – CY Cergy Paris Université.

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