Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds

Details Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds

2009-03-06

arxiv.org/abs/0903.1138v3

Mathematics

Geometric Topology

27 pages, 5 figures. We have rewritten sections 1 and 5 due to recent work of Futer-Gueritaud, Segerman-Tillmann and Luo-Tillm

Scientific paper

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.

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