Geometric transition from hyperbolic to anti-de Sitter structures in dimension four
arXiv:1908.05112 · doi:10.2422/2036-2145.202005_031
Abstract
We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboctahedron. We show the existence of a similar family of collapsing anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope.
arXiv comment50 pages, 27 figures (many of the figures use colours). To appear in Annali della Scuola Normale Superiore, Classe di Scienze
References in corpus (10)
- A Geometric transition from hyperbolic to anti de Sitter geometry
- Hyperbolic four-manifolds with one cusp
- Deforming Euclidean cone 3-manifolds
- From the hyperbolic 24-cell to the cuboctahedron
- Families of Geometries, Real Algebras, and Transitions
- Hyperbolic structures from Sol on pseudo-Anosov mapping tori
- Quasi-Fuchsian co-Minkowski manifolds
- A small closed convex projective 4-manifold via Dehn filling
- Character varieties of a transitioning Coxeter 4-orbifold
- Some open questions on anti-de Sitter geometry