Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman

By T. Colding, W. Minicozzi. 2003

117 citations recorded by the source corpus.

We show that the Ricci flow becomes extinct in finite time on any Riemannian 3-manifold without aspherical summands. In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question that Grisha Perelman asked the first author at a dinner in New York City on April 25th of 2003. His question was "what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular, does the flow become extinct in finite time?" He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min-max argument where the minimal of all maximal slices of sweep-outs is a minimal surface; see, for instance, (2). The idea is then to look at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see (9)) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier. Let M 3 be a smooth closed orientable 3-manifold and let g(t) be a one-parameter family of metrics on M evolving by the Ricci flow, so

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/46d450ba5702acc9ef452ea0daa815716fe62c71

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. A. Neves
2. S. Brendle
3. Reto Muller
4. J. Lott
5. F. C. Marques
6. Bruce Kleiner
7. G. Tian
8. W. Minicozzi

In Computational Theory and Mathematics

1. Reto Muller
2. T. Colding
3. W. Minicozzi
4. F. C. Marques
5. A. Neves
6. J. Lott
7. Jonathan Dinkelbach
8. B. Leeb

In Geometry and Topology

1. S. Brendle
2. A. Neves
3. J. Lott
4. F. C. Marques
5. Reto Muller
6. Bruce Kleiner
7. G. Tian
8. Robert Haslhofer

Citations in the corpus

• Recent progress on the Poincaré conjecture and the classification of 3-manifolds (2004, background): “(There is a closely related result in [ 3 ] that can be used instead of [17].) In particular, if Σ3 is a homotopy three-sphere, then all of its prime factors are also homotopy three-spheres and hence none of its prime factors is acyclic.”
• Ricci flow coupled with harmonic map flow (2009, methodology): “Most importanty, Perelman’s work [31, 32] led to a completion of Hamilton’s program [15] and a complete proof of Thurston’s geometrization conjecture [42] and (using a finite extinction result from Perelman [33] or Colding and Minicozzi [8, 9]) of the Poincaré conjecture [34].”
• Completion of the Proof of the Geometrization Conjecture (2008, background): “The main part of this paper is devoted to giving a proof of Theorem 7.4 stated in [19] on locally volume collapsed 3-manifolds with curvature bounded from below which is the last step in the proof of the Geometrization Conjecture.”
• Minimal submanifolds (2005, unknown): “We close this survey with indicating how minimal surfaces can be used to show that on a homotopy 3-sphere the Ricci flow become extinct in finite time (see [25], [80] for details).”
• Width and finite extinction time of Ricci flow (2007, background): “M / is nontrivial (see, eg, Colding and Minicozzi [8]).”
• Area‐minimizing projective planes in 3‐manifolds (2010, background): “[2], [10], [13].”
• Ricci Flow and the Poincaré Conjecture (2007, background): “A result of Perelman (for which a simpler and more elegant proof was provided by Colding and Minicozzi [2]) says that if the manifold M is simply-connected, then all the components of the Ricci flow with surgery become extinct in finite time.”
• Rigidity of min-max minimal spheres in three-manifolds (2011, methodology): “These estimates are based on the so called Hersch’s trick which has been used, among other places, in [3, 4, 8, 11, 29, 30, 42, 43].”
• Notes on Perelman's papers (2006, background): “The papers [24, 53], which are not covered in these notes, each provide a shortcut in the case of the Poincaré Conjecture.”
• Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds (2008, methodology): “Our argument is based on several deep recent results concerning the Ricci flow with cutoff on closed 3–manifolds, namely its long time existence for arbitrary initial metrics (Perelman [26], Kleiner and Lott [15], Morgan and Tian [20], Bamler [1]), its extinction in finite time on non-aspherical pri”
• Dimensional reduction and the long-time behavior of Ricci flow (2007, background): “If this piece has Thurston type S(3) or S(1)×S(2) then its Ricci flow has a finite extinction time [10, 11, 45].”