Paper · Sep 24, 2020
On Four-dimensional Steady gradient Ricci solitons that dimension reduce
As on arXiv
Abstract
In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to -manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form or weakly dimension reduce to the -dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE -manifolds or dimension reduce to -dimensional manifolds. As an application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE Ricc-flat -manifolds.
- Submitted
- Sep 24, 2020
- Last revised
- Mar 18, 2022 (v2)
- Subjects
- math.DG
Comments: A volume comparison result is corrected. See the discussion in the appendix. The paper is accepted by Adv in Math
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Background reading
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In the guidebook
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.