Volume and angle structures on 3-manifolds

TL;DR

This paper introduces a variational approach using volume functions on angle structures of triangulated 3-manifolds to identify constant curvature metrics, revealing geometric properties through volume maximization.

Contribution

It develops a new method linking angle structures and volume maximization to the existence of constant curvature metrics on 3-manifolds.

Findings

Maximum volume points imply the existence of constant curvature metrics.

The volume functional extends continuously to the moduli space closure.

Presence of certain surfaces indicates the absence of constant curvature metrics.

Abstract

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.