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Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman. (English) Zbl 1083.53058
The paper affirmatively answers a question of Grisha Perelman (April 2003): Starting with an arbitrary metric on \(S^3\), does the Ricci flow become extinct in finite time? Part of Perelman’s interest in this question comes from the following: if one were interested in geometrization of a homotopy 3-sphere (or, more generally, a 3-manifold without aspherical submanifolds) and knew that the Ricci flow became extinct in finite time, then one would not need to analyze what happens to the flow as time goes to infinity (in particular, one would not need collapsing arguments). The actual answer is given as a corollary from an upper bound for the derivative of the width \(W(g(t))\) established by the authors for the Ricci flow \(g(t)\) on a closed orientable prime non-aspherical 3-manifold (Theorem 1.1), Corollary 1.2: Let \(M^3\) be a closed orientable 3-manifold whose prime decomposition has only non-aspherical factors and is equipped with a Riemannian metric \(g=g(0)\). Under the Ricci flow with surgery, \(g(t)\) must become extinct in finite time.
The authors note that Perelman has answered his original question in his posted preprint ”Finite extinction time for the solution to the Ricci flow on certain three-manifolds”, DG/0307245. Though they think that their slightly different approach may be of interest, in particular because it avoids using the curve shortening flow that Perelman simultaneously with the Ricci flow needed to invoke.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57M50 General geometric structures on low-dimensional manifolds
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