Long-time behavior of 3-dimensional Ricci flow. D: Proof of the main results.

*(English)*Zbl 1388.53061In Part D of this series of papers, the author proves the main two results of this study on long-time behaviour of 3-dimensional Ricci flows with surgery. The proofs are based on results of the three previous papers by the author [Zbl 1388.53058; Zbl 1388.53059; Zbl 1388.53060]. The first result states: “If the surgeries are performed correctly then the flow becomes non-singular eventually and the curvature is bounded by \(C(1/t)\).” (Using this bound it is possible to rule out the existence of surgeries, since the surgeries can only arise at points where the curvature goes to infinity.) The second result provides a qualitative description of the geometry of a Ricci flow with surgery as \(t\) goes to infinity. In Part A, the author recalled Perelman’s result on the long-time behaviour of Ricci flows (with surgery). It states that for every sufficiently large time there is a decomposition of the manifold into the thick part and the thin part; the metric on the thick part converges to a hyperbolic metric, while the metric on the thin part collapses locally at scale with a lower sectional curvature bound. So, the desired curvature bound holds on thick part and it remains to study the thin part and to show that the curvature bound holds there as well. The author analyzes the collapsed part and the geometric/topological consequences.

Reviewer: Corina Mohorianu (Iaşi)

##### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

49Q05 | Minimal surfaces and optimization |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

57M15 | Relations of low-dimensional topology with graph theory |

57M20 | Two-dimensional complexes (manifolds) (MSC2010) |

##### Keywords:

Ricci flow; Ricci flow with surgery; finitely many surgeries; asymptotics of Ricci flow; collapsing theory of 3-manifolds; topology of 3-manifolds; geometrization conjecture
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