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.