149 research outputs found
Quasi-Fuchsian AdS representations are Anosov
In a recent paper, Q. M\'erigot proved that representations in SO(2,n) of
uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are
quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in
the boundary of anti-de Sitter space. In the present paper, we prove the
reverse implication. It also includes: -- A construction of Dirichlet domains
in the context of anti-de Sitter geometry, -- A proof that spatially compact
globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit
locally CAT(-1) Cauchy hypersurfaces
Pseudo-Anosov flows in toroidal manifolds
We first prove rigidity results for pseudo-Anosov flows in prototypes of
toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered
manifold is up to finite covers topologically equivalent to a geodesic flow and
we show that a pseudo-Anosov flow in a solv manifold is topologically
equivalent to a suspension Anosov flow. Then we study the interaction of a
general pseudo-Anosov flow with possible Seifert fibered pieces in the torus
decomposition: if the fiber is associated with a periodic orbit of the flow, we
show that there is a standard and very simple form for the flow in the piece
using Birkhoff annuli. This form is strongly connected with the topology of the
Seifert piece. We also construct a large new class of examples in many graph
manifolds, which is extremely general and flexible. We construct other new
classes of examples, some of which are generalized pseudo-Anosov flows which
have one prong singularities and which show that the above results in Seifert
fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows.
Finally we also analyse immersed and embedded incompressible tori in optimal
position with respect to a pseudo-Anosov flow.Comment: 44 pages, 4 figures. Version 2. New section 9: questions and
comments. Overall revision, some simplified proofs, more explanation
Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space
We study the existence of surfaces with constant or prescribed Gauss
curvature in certain Lorentzian spacetimes. We prove in particular that every
(non-elementary) 3-dimensional maximal globally hyperbolic spatially compact
spacetime with constant non-negative curvature is foliated by compact spacelike
surfaces with constant Gauss curvature. In the constant negative curvature
case, such a foliation exists outside the convex core. The existence of these
foliations, together with a theorem of C. Gerhardt, yield several corollaries.
For example, they allow to solve the Minkowski problem in the 3-dimensional
Minkowski space for datas that are invariant under the action of a co-compact
Fuchsian group
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