Transition of convex core doubles from hyperbolic to Anti-de sitter geometry

TL;DR

This paper introduces a method to construct geometric transitions from hyperbolic to Anti-de Sitter structures on surfaces with cone singularities, using Half-pipe geometry and convex core deformations.

Contribution

It provides a new construction framework for geometric transitions between hyperbolic and Anti-de Sitter geometries via convex core deformation and Half-pipe geometry.

Findings

Established a deformation process for convex core structures.

Demonstrated the transition via collapsing bending laminations.

Extended the construction to surfaces with cone singularities.

Abstract

Let $\Sigma$ be a surface of negative Euler characteristic, homeomorphic to a closed surface, possibly with a finite number of points removed. In this paper, we present a construction method for a wide range of examples of geometric transition from hyperbolic to Anti-de Sitter structures via Half-pipe geometry on $\Sigma\times\mathbb{S}^1$, with cone singularities along a link. The main ingredient lies in studying the deformation of a convex core structure as the bending laminations of the upper and lower boundary components of the convex core uniformly collapse to zero.