Ricci flow with surgery on three-manifolds

Ricci flow with surgery on three-manifolds

February 1, 2008 | Grisha Perelman
This paper, a continuation of [I], verifies most of the assertions made in [I, §13]. It focuses on the Ricci flow with surgery on 3-manifolds, addressing issues such as the lower bound for sectional curvature and the smoothness of the solution. The author introduces two scale bounds: a cutoff radius \( h \) for surgeries and a larger radius \( r \) for standard geometry. The paper also reviews results on ancient solutions with bounded entropy and classifies their asymptotic behavior. It constructs a standard solution that converges to a round infinite cylinder and discusses the structure of solutions at the first singular time. The paper then defines the Ricci flow with cutoff and justifies the canonical neighborhood assumption, showing that solutions satisfying certain conditions are defined for all time. Finally, it summarizes the existence of decreasing functions \( r(t) \) and \( \bar{\delta}(t) \) such that solutions to the Ricci flow with cutoff satisfy the canonical neighborhood assumption and the pinching estimate. The long-time behavior of solutions is also discussed, including the extinction of solutions with positive scalar curvature and the need for additional work to understand solutions with negative scalar curvature.This paper, a continuation of [I], verifies most of the assertions made in [I, §13]. It focuses on the Ricci flow with surgery on 3-manifolds, addressing issues such as the lower bound for sectional curvature and the smoothness of the solution. The author introduces two scale bounds: a cutoff radius \( h \) for surgeries and a larger radius \( r \) for standard geometry. The paper also reviews results on ancient solutions with bounded entropy and classifies their asymptotic behavior. It constructs a standard solution that converges to a round infinite cylinder and discusses the structure of solutions at the first singular time. The paper then defines the Ricci flow with cutoff and justifies the canonical neighborhood assumption, showing that solutions satisfying certain conditions are defined for all time. Finally, it summarizes the existence of decreasing functions \( r(t) \) and \( \bar{\delta}(t) \) such that solutions to the Ricci flow with cutoff satisfy the canonical neighborhood assumption and the pinching estimate. The long-time behavior of solutions is also discussed, including the extinction of solutions with positive scalar curvature and the need for additional work to understand solutions with negative scalar curvature.
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Understanding Ricci flow with surgery on three-manifolds