Blowdowns along Ricci flow in dimension four
Abstract
We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C14-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C14-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂℙ2#ℓℂℙ̅2, through finitely many blowdown singularities. (2) On the standard 2ℂℙ2#9ℂℙ̅2, there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.
Related
- Classification of cohomogeneity one steady gradient Ricci solitons in dimension four and five
- Conjugate radius and parabolicity along the Ricci flow
- Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds
- O(3)-symmetry of four-dimensional steady Ricci solitons
- Codimension-four regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
BibTeX
@misc{kim2026blowdowns,
title = {{Blowdowns along Ricci flow in dimension four}},
author = {Dain Kim and Tristan Ozuch},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00139},
eprint = {2611.00139},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00139},
}
RIS
TY - UNPB
TI - Blowdowns along Ricci flow in dimension four
AU - Dain Kim
AU - Tristan Ozuch
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00139
UR - https://arxiv.org/abs/2611.00139
AB - We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C^14-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C^14-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂℙ^2#ℓℂℙ̅^2, through finitely many blowdown singularities. (2) On the standard 2ℂℙ^2#9ℂℙ̅^2, there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.
ER -