A Geometric transition from hyperbolic to anti de Sitter geometry

Abstract

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the (2,m,m) triangle orbifold for m at least 5.

47 pages, 20 figures

References in corpus (3)

• Locally homogeneous geometric manifolds
• Collisions of particles in locally AdS spacetimes
• Some open questions on anti-de Sitter geometry

Cited by in corpus (19)

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• Generalized Cusps in Real Projective Manifolds: Classification
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• Families of Geometries, Real Algebras, and Transitions
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• Divisible convex sets with properly embedded cones
• Affine deformations of quasi-divisible convex cones
• Solving Thurston Equation in a Commutative Ring
• Character varieties of a transitioning Coxeter 4-orbifold
• A small closed convex projective 4-manifold via Dehn filling
• On branched coverings of singular $(G,X)$-manifolds
• On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces
• Group actions and scattering problems in Teichmüller theory
• Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds
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• The $\mathrm{PSL}(2,\mathbb{R}{)}^2$-configuration space of four points in the torus $S^1\times S^1$
• The Heisenberg Plane
• Higher signature Delaunay decompositions