Reference : On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/29979
On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds
English
Bonsante, Francesco [Università degli Studi di Pavia]
Seppi, Andrea [Università degli Studi di Pavia]
Tamburelli, Andrea mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
2017
Geometric & Functional Analysis
Springer Science & Business Media B.V.
Yes (verified by ORBilu)
1016-443X
1420-8970
[en] We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in Teichmüller space of S provided by Mess’ parameterization - namely on two isotopy classes of hyperbolic metrics h and h' on S. The main result of the paper is that the volume coarsely behaves like the minima of the L1-energy of maps from (S, h) to (S, h'). The study of Lp-type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston’s Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behaviour is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil-Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from (S, h) to (S, h'), and to the minima of the holomorphic 1-energy.
http://hdl.handle.net/10993/29979
To appear in GAFA.

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