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Abstract We introduce a finite dimensional variational approach to find constant curvature metrics on triangulated closed 3-manifolds. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced. Each angle structure has a nature volume. The main result shows that for a 1-vertex triangulation of a closed 3-manifold if the volume function on the space of all angle structures has a local maximum point, then either the manifold admits a constant curvature Riemannian metric, or the manifold is reducible.