Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

By G. Perelman. 2003

950 citations recorded by the source corpus.

Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0dafb7b15ce112ff6e5c071e141674e09233ecb6

How researchers see this paper

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In Applied Mathematics

1. Reto Muller
2. J. Lott
3. Bruce Kleiner
4. W. Minicozzi
5. T. Colding
6. Ashot Minasyan
7. Henry Wilton
8. Nicola Gigli

In Geometry and Topology

1. J. Lott
2. Bruce Kleiner
3. Reto Muller
4. Henry Wilton
5. Ashot Minasyan
6. W. Minicozzi
7. Denis V. Osin
8. T. Colding

Citations in the corpus

• The Geometry and Topology of Coxeter Groups (2025, background): “In dimension 3 we have the famous Geometrization Conjecture of Thurston [276] (now a theorem of Perelman [237, 239, 238]).”
• Profinite properties of graph manifolds (2008, background): “Préaux [21] has shown that the conjugacy problem is solvable in the fundamental group of an orientable, geometrizable 3-manifold. assume that M is orientable.2 Under these hypotheses, M has a canonical JSJ decomposition, the pieces of which are either Seifert-fibred or, according to the Geometrizat”
• Understanding over-squashing and bottlenecks on graphs via curvature (2021, methodology): “One notion of curvature that has received attention for graph learning is Ricci curvature (Hamilton, 1988), also known in geometry for its use in Ricci flow and the subsequent proof of the Poincaré conjecture (Perelman, 2003).”
• Dimensional reduction and the long-time behavior of Ricci flow (2007, background): “In [45], Perelman showed that for large t, any connected component of Mt is aspherical or S(3).”
• Width and finite extinction time of Ricci flow (2007, methodology): “In fact, we get as a corollary finite extinction time for the Ricci flow on all 3manifolds without aspherical summands (see 1.5 of [Pe] or section 4 of [CM1] for why this easily follows): Corollary 1.11. ([CM1], [ Pe ]).”