 If this theorem can be extended to large regions (namely, to the whole hemisphere), then the filling area conjecture is true.It has been conjectured that all simple Riemannian manifolds (those that are convex at their boundary, and where every two points are joined by a unique geodesic) are volume minimizers.The proof that each almost flat manifold is a volume minimizer involves embedding in formula_64, and then showing that any isometric replacement of can also be mapped into the same space formula_65, and projected onto , without increasing its volume.