Gluing equations for real projective structures on 3-manifolds

TL;DR

This paper introduces a unified system of real equations for constructing and analyzing projective structures on 3-manifolds, extending classical and recent geometric frameworks.

Contribution

It formulates a comprehensive set of equations that generalize Thurston's gluing equations to various geometries and detects convex structures on 3-manifolds.

Findings

Unified framework for Thurston, Anti-de Sitter, and half-pipe geometries

Equations can identify properly convex structures

Explicit examples demonstrating construction of convex structures

Abstract

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more recent counterparts in Anti-de Sitter and half-pipe geometry. Moreover, these equations can be used to detect properly convex structures on $M$. The paper also includes a few explicit examples where the equations are used to construct properly convex structures.