potentially progressing forever. Perelman proved this cannot happen by using minimal surfaces on the manifold. A minimal surface is essentially a soap film. Hamilton had shown that the area of a minimal surface decreases as the manifold undergoes Ricci flow. Perelman verified what happened to the area of the minimal surface when the manifold was sliced. He proved that, eventually, the area is so small that any cut after the area is that small can only be chopping off three-dimensional spheres and not more complicated pieces. This is described as a battle with a Hydra by Sormani in Szpiro's book cited below. This last part of the proof appeared in Perelman's third and final paper on the subject. References Further reading External links "The Poincaré Conjecture" – BBC Radio 4 programme In Our Time'', 2 November 2006. Contributors June Barrow-Green, Lecturer in the History of Mathematics at the Open University, Ian Stewart, Professor of Mathematics at the University of Warwick, Marcus du Sautoy, Professor of Mathematics at the University of Oxford, and presenter Melvyn Bragg. Geometric topology 3-manifolds Theorems in topology Millennium Prize Problems Conjecture Conjectures that have been proved 1904 introductions