Geometric transition from hyperbolic to anti-de Sitter structures in dimension four

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.

50 pages, 27 figures (many of the figures use colours). To appear in Annali della Scuola Normale Superiore, Classe di Scienze

References in corpus (5)

• Families of Geometries, Real Algebras, and Transitions
• 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

Cited by in corpus (3)

• Character varieties of a transitioning Coxeter 4-orbifold
• A small closed convex projective 4-manifold via Dehn filling
• A small cusped hyperbolic 4-manifold