Width and finite extinction time of Ricci flow

By T. Colding, W. Minicozzi. 2007

93 citations recorded by the source corpus.

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given here are quite useful and seem to be of interest to a wide audience.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e982fd37e42242896015d33737e8cc9c8a44acd7

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. Yu Li
5. J. Lott
6. Bruce Kleiner
7. H. Bray
8. M. Eichmair

In Geometry and Topology

1. S. Brendle
2. A. Neves
3. Reto Muller
4. Bruce Kleiner
5. J. Lott
6. Yu Li
7. Yuxiang Li
8. Younde Wang

In Mathematical Physics

1. Reto Muller
2. Yu Li
3. J. Lott
4. Bruce Kleiner
5. A. Neves
6. Yuxiang Li
7. S. Brendle
8. Reto Müller

Citations in the corpus

• A weak energy identity and the length of necks for a sequence of Sacks–Uhlenbeck α-harmonic maps☆ (2010, background): “Lamm gave a simple proof of this energy identity [11] (also see [4] and [14]).”
• Notes on Perelman's papers (2006, background): “In conjunction with one of the finite extinction time results [24, 25, 53], this completes the proof of the Poincaré Conjecture.”
• Ricci flow on asymptotically Euclidean manifolds (2016, methodology): “Robert Haslhofer obtained the same result, see details in [40, Corollary 6], by using the min-max argument of Colding-Minicozzi [15].”
• On the Multiplicity One Conjecture in min-max theory (2019, unknown): “We refer to [39, 6, 10, 17, 9, 23, 7, 47, 50, 32] for other variants of this theory.”
• Area‐minimizing projective planes in 3‐manifolds (2010, result): “M; g/ is also related to the notion of width studied by Colding and Minicozzi [5, 6].”
• 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].”
• 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].”