Volume and angle structures on 3-manifolds

Feng Luo

Geometric Analysis and Geometric Topology mathscidoc:1912.43799

Asian Journal of Mathematics, 11, (4), 555-566, 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
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  title={Volume and angle structures on 3-manifolds},
  author={Feng Luo},
  booktitle={Asian Journal of Mathematics},
Feng Luo. Volume and angle structures on 3-manifolds. 2007. Vol. 11. In Asian Journal of Mathematics. pp.555-566. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205858049329363.
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