Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

Abstract

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from appropriate dilations of a Type IIb singularity is always Hamilton's cigar soliton solution.

On the long-time behavior of type-III Ricci flow solutions

“…Remark 5. 5 We wish to identify equivalent groupoids. A more intrinsic approach is to consider instead the category BG of G-sheaves.…”

“…The result of [8] was used in [5] to study three-dimensional type-II Ricci flow solutions. Corollary 5.14 Let (M, p, g(·)) be a type-III Ricci flow solution.…”

A brief survey on the Ricci flow in homogeneous manifolds

“…The lines γ 1 , γ 2 , γ 3 and γ 4 are solutions for (9) [and for (8)] passing through p 1 , p 2 , p 3 and p 4 respectively. Proof This proof is a straightforward calculation.…”

“…Theorem 3 provide us 4 solutions for (9) [and for (8)]. Now we study the stability of these solutions using Lyapunov exponents.…”

“…Thus it is natural to study the Ricci flow from a qualitative point of view, using tools from the theory of dynamical systems. Some recent papers that use this approach include [4,9,11] and [7].…”