Triangulated 3-Manifolds: from Haken's normal surfaces to Thurston's algebraic equation

TL;DR

This paper explores solving Thurston's and Haken's equations on triangulated 3-manifolds using a variational principle based on volume, aiming to prove conjectures that could offer a new proof of the Poincaré conjecture.

Contribution

It introduces a finite-dimensional variational approach linking solutions of Thurston's and Haken's equations to critical points of volume, generalizing previous work by Casson and Rivin.

Findings

Proposes conjectures on solutions to Thurston's and Haken's equations.

Establishes a variational principle where volume acts as an action functional.

Suggests potential to prove the Poincaré conjecture without Ricci flow.

Abstract

We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurston's equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincar\'e conjecture without using the Ricci flow. We approach these conjectures by a finite dimensional variational principle so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary.