Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds
Abstract
We propose a finite dimensional variational principle on triangulated 3-manifolds 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. Combining the result in this paper and the work of Futer-Gu\'eritaud, Segerman-Tillmann and Luo-Tillmann, we obtain a new finite dimensional variational formulation of the Poncare-conjecture. This provides a step toward a new proof the Poincar\'e conjecture without using the Ricci flow.
Cite
@article{arxiv.0903.1138,
title = {Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds},
author = {Feng Luo},
journal= {arXiv preprint arXiv:0903.1138},
year = {2010}
}
Comments
27 pages, 5 figures. We have rewritten sections 1 and 5 due to recent work of Futer-Gueritaud, Segerman-Tillmann and Luo-Tillmann. The results obtained are stronger then those in the first version