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December 2007 Volume and Angle Structures on 3-Manifolds
Feng Luo
Asian J. Math. 11(4): 555-566 (December 2007).


We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the volume functional always exists in the compactification. The main result shows that for a 1-vertex triangulation of a closed 3-manifold if the volume function on the moduli space has a local maximum point, then either the manifold admits a constant curvature Riemannian metric or the manifold contains a non-separating 2-sphere or real projective plane.


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Feng Luo. "Volume and Angle Structures on 3-Manifolds." Asian J. Math. 11 (4) 555 - 566, December 2007.


Published: December 2007
First available in Project Euclid: 2 May 2008

zbMATH: 1170.57014
MathSciNet: MR2402938

Primary: 57M30

Keywords: 3-manifolds , angle structures , constant curvature metrics , normal surfaces , Schlaefli formula , triangulations , Volume

Rights: Copyright © 2007 International Press of Boston

Vol.11 • No. 4 • December 2007
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