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

## TL;DR

This paper constructs the first known examples of geometric transition from hyperbolic to anti-de Sitter structures in four dimensions, using deformations of hyperbolic 4-polytopes and a novel collapsing process.

## Contribution

It introduces a new four-dimensional geometric transition framework, extending three-dimensional examples by employing deformations of hyperbolic and anti-de Sitter polytopes.

## Key findings

- Existence of a family of collapsing hyperbolic 4-polytopes.
- Existence of a similar family of collapsing anti-de Sitter polytopes.
- Construction of geometric transitions via gluing of polytopes.

## 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.